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Document related concepts
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Transcript 
01_Math_intro/e_01_1_011.html
Distance in polar coordinates
Find the distance between two points
and
(polar coordinates).
01_Math_intro/e_01_1_012.html
Distance in spherical coordinates
Find the distance between
and
(spherical coordinates).
01_Math_intro/e_01_1_013.html
Elliptical coordinates
We define elliptical coordinates as follows:
as functions of
.
and
as in polar coordinates. Find
01_Math_intro/e_01_1_014.html
Distance in elliptical coordinates
In elliptical coordinates (
points
and
and
as in polar coordinates) find distance between two
.
01_Math_intro/e_01_1_015.html
Coordinates and vectors
Two different coordinate systems are established on a straight line,
and , which are related as follows:
, where and are constants. The distance element in terms of
is given by
.
Find the expression for the distance element in terms of .
01_Math_intro/e_01_1_016.html
Distance element
Two different coordinate systems are established on a plane,
where
and
. Find the distance element in terms of coordinates
, which are related as follows:
, if
are Cartesian.
01_Math_intro/e_01_1_017.html
Distance element
Same as above but the relation reads
.
01_Math_intro/e_01_1_018.html
Distance element
New coordinates
are introduced which are related to the ordinary polar coordinates
,
. Find the distance element.
as follows:
01_Math_intro/e_01_1_019.html
Coordinates and vectors
Starting with cartesian coordinates we define new ones as follows:
,
, where
are some constant parameters. What conditions on these parameters should be satisfied in order that the
new coordinates also be cartesian and the measure of the distance remain the same ?
01_Math_intro/e_01_1_020.html
Volume of tetrahedron
Four vertices of the tetrahedron are given by four radius-vectors
,
. Find the volume.
01_Math_intro/e_01_1_021.html
Coordinates and vectors
Let
be the vector from the center of the Earth to New-York and
is the vector from the center of the Earth
to Jerusalem. Find the angle between the two vectors (find the lacking data in a geographical atlas).
01_Math_intro/e_01_1_022.html
Elliptical coordinates
Find the distance element
in the elliptical coordinates
,
unit vectors tangential and normal to
curves and derive their relation to
. Define
and
.
01_Math_intro/e_01_1_023.html
Legitimate coordinates?
Are the coordinates
,
legitimate ? If yes, find the distance element.
01_Math_intro/e_01_1_024.html
Length of parabola
Calculate the length of the parabola
,
, from
to
.
01_Math_intro/e_01_1_025.html
Circumference
Ellipse is given by the equation
. Calculate the circumference.
01_Math_intro/e_01_1_026.html
Equation of plane
A plane passes through the point
and its normal is
. Write down the equation of the plane.
01_Math_intro/e_01_1_027.html
Spherical coordinates
Find the relation between the unit vectors of 3D Cartesian coordinates and unit vectors of spherical coordinates.
01_Math_intro/e_01_1_028.html
Cylindrical and spherical coordinates
Decompose unit vectors of cylindrical coordinates using unit vectors of spherical coordinates.
01_Math_intro/e_01_1_029.html
Coordinates and vectors
Let
(cylindrical coordinates). Derive
,
, and
.
01_Math_intro/e_01_1_030.html
The shortest distance between cities
Two cities on Earth surface (assume it is a sphere with the radius ) have the coordinates
and
, where
is the latitude, and
is the longitude. Find the shortest distance between the cities. Hint:
Arc of the circle passing through the center of the Earth.
01_Math_intro/e_01_1_031.html
Coordinates and vectors
Four vertices of the tetrahedron are given by four radius-vectors
,
. Find the volume.
01_Math_intro/e_01_1_032.html
Distance element
in the elliptical coordinates
Find the distance element
in the elliptical coordinates
,
unit vectors tangential and normal to
curves and derive their relation to
. Define
and
.
01_Math_intro/e_01_1_033.html
Cross product
Simplify
.
01_Math_intro/e_01_1_034.html
Coordinates and vectors
Given two non-parallel unit vectors
and , find another unit vector
plane and
divides the angle between the first two in the way
01_Math_intro/e_01_1_035.html
Projection
What is the projection of the vector
01_Math_intro/e_01_1_036.html
Projection
onto the unit vector
?
such that all three are in the same
.
What is the projection of the vector
onto the vector
?
01_Math_intro/e_01_1_037.html
Coordinates and vectors
Given two vectors
and
.
and , represent vector
as a sum of two vectors,
and
,
, such that
01_Math_intro/e_01_1_038.html
Coordinates and vectors
On the Earth a man is at the point
east longitude and
latitude moves in the north-east direction.
Express the unit vector along the velocity in spherical and cartesian coordinates.
01_Math_intro/e_01_1_039.html
Coordinates and vectors
Show that a straight line is given by the relation
.
, where
and
are constant vectors and
01_Math_intro/e_01_1_040.html
Intersection line of two planes
Find the intersection line of the two planes
and
.
01_Math_intro/e_01_1_041.html
Equation for a circle
Write in the vector form the equation for a circle with the radius
center at .
, normal direction
(unit vector !), and
01_Math_intro/e_01_1_042.html
Cylindrical surface
Write in the vector form the equation for a cylindrical surface with the radius
crossing the point .
and the axis parallel to
and
01_Math_intro/e_01_1_043.html
Coordinates and vectors
Prove that four different points
sphere.
,
, are always on a sphere and find the center and the radius of the
01_Math_intro/e_01_1_044.html
Calculate a)
, b)
01_Math_intro/e_01_1_045.html
, c)
, d)
.
Prove a)
, b)
, c)
.
01_Math_intro/e_01_1_046.html
Express
with the use of
and
.
01_Math_intro/e_01_1_047.html
Angular momentum
Angular momentum is defined as
particle.
. Express in terms of angular velocity for a circularly moving
01_Math_intro/e_01_1_048.html
Coordinates and vectors
Given two nonparallel vectors
and
build three mutually perpendicular unit vectors.
01_Math_intro/e_01_1_049.html
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซ ื—ืฉื‘ ืืชโ€ฌ.
โ€ซื•โ€ฌ
โ€ซื ืชื•ื ื™ื ื”ื•ืงื˜ื•ืจื™ืโ€ฌ
.โ€ซืโ€ฌ
.โ€ซื‘โ€ฌ
.โ€ซื’โ€ฌ
01_Math_intro/e_01_1_050.html
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซ ื—ืจืื”โ€ฌ.
,
,
,
โ€ซื ืชืŸื ื™ื ื”ื•ืงื˜ื•ืจื™ืโ€ฌ
:โ€ซื›ื™โ€ฌ
.โ€ซืโ€ฌ
.โ€ซื‘โ€ฌ
.โ€ซื’โ€ฌ
01_Math_intro/e_01_1_051.html
โ€ซืชืจื’ื™ืœโ€ฌ
?
01_Math_intro/e_01_1_052.html
โ€ซ ืžื”ื™โ€ฌ.
โ€ซ ื”ื™ืโ€ฌr2 โ€ซ ื•โ€ฌr1 โ€ซ ื”ื–ื•ื™ืช ื‘ื™ืŸโ€ฌ.
โ€ซื•โ€ฌ
โ€ซื ืชื•ื ื™ื ื”ื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
.
โ€ซื”ืžืงื™ื™ืžื™ืโ€ฌ
โ€ซื•โ€ฌ
โ€ซื ืชื•ื ื™ื ื”ื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื”ืจืื” ื›ื™โ€ฌ
. โ€ซืโ€ฌ
.โ€ซื‘โ€ฌ
01_Math_intro/e_01_1_053.html
Vector algebra
Given two vectors,
and
, with norms, v and w respectively. The angle between them is
a) Find the norm of the vector - , i.e.
b) Find the angle
between the vectors
c) Calculate (a) and (b) for:
v=12 w=9
and
d) Draw the vectors
,
and
.
and
for
.
.
01_Math_intro/e_01_1_054.html
Relations between two coordinate systems
Given two coordinate systems
and
Show that the norm of the vector
stays unchanged when moving to the
and the relations between them
, where =(Ax ,Ay) =(Bx , By) in the
coordinate system.
01_Math_intro/e_01_1_055.html
Linear dependence of vectors
Given two vectors
and
Find
such that
.
Find
such that
.
01_Math_intro/e_01_1_056.html
Partial derivatives and differentials
For the function:
Compute:
,
and the full differential 01_Math_intro/e_01_1_057.html
,
.
:
coordinate system
Vector algebra
Given two vectors,
and
1.Find
.
2. Find the angle
with norms,
between
,
and the angle
between them,
and .
01_Math_intro/e_01_1_058.html
Coordinates
The angle between two sets of Cartesian coordinates,
1. Find the relations between the unit vectors of
2. For an arbitrary vector
with components
in the
set.
01_Math_intro/e_01_1_059.html
Coordinates
A vector
has three angles with the axes,
Prove that :
.
01_Math_intro/e_01_1_060.html
Derivatives
Given the function
compute:
,
,
,
and show that
01_Math_intro/e_01_1_061.html
Differential equations
Solve the system of differential equations
where
.
01_Math_intro/e_01_1_062.html
Differential equations
.
and
, is .
in terms of the unit vectors of
.
in the
set, find its components,
,
Solve the system of differential equations
where
.
01_Math_intro/e_01_1_063.html
Differential equations
Solve the system of differential equations
where
.
01_Math_intro/e_01_1_064.html
Differential equations
Solve the system of differential equations
where
.
01_Math_intro/e_01_1_065.html
Differential equations
Solve the system of differential equations
where
.
01_Math_intro/e_01_1_066.html
Fourier series
For the sawtooth wave f(t)
1. Expand f(t) into the Fourier series both with real and complex coefficients (i.e. in the "sin/cos" and "exp"
forms)
2. Make sure that both series are equal.
3. Plot the function f(t) and its approximation using the first N members of the infinite Fouries series you
found. Check (qualitatively) how good the approximation is for different N's. (use any mathematical
program for this purpose, e.g. Matlab).
01_Math_intro/e_01_1_067.html
The power of curvilinear coordinates
a disk of radius R has a density of
.
1) in Cartesian coordinates, try to find the disk's mass,
2) now do it in polar coordinates.
.
01_Math_intro/e_01_2_001.html
โ€ซื˜ื•ืจื™ ื˜ื™ื™ืœื•ืจโ€ฌ
:
โ€ซ ืกื‘ื™ื‘โ€ฌIII โ€ซื™ ืืช ื”ืคื•ื ืงืฆื™ื•ืช ื”ื‘ืื•ืช ืœื˜ื•ืจ ื˜ื™ื™ืœื•ืจ ืขื“ ืœืกื“ืจโ€ฌ/โ€ซืคืชื—โ€ฌ
.1
.2
.3
01_Math_intro/e_01_2_002.html
โ€ซื™ืขืงื•ื‘ื™ืื ื™ืโ€ฌ
:โ€ซื™ ืืช ื”ื™ืขืงื•ื‘ื™ืืŸ ื‘ืžืขื‘ืจ ืžืงื•ืื•ืจื“ื™ื ื˜ื•ืช ืงืจื˜ื–ื™ื•ืช ืœืงื•ืื•ืจื“ื™ื ื˜ื•ืช ื”ื‘ืื•ืชโ€ฌ/โ€ซืžืฆืโ€ฌ
.โ€ซ ืคื•ืœืจื™ื•ืชโ€ฌ.1
.โ€ซ ื›ื“ื•ืจื™ื•ืชโ€ฌ.2
01_Math_intro/e_01_2_003.html
โ€ซืื™ื ื˜ื’ืจืœื™ืโ€ฌ
:โ€ซื™ ืืช ื”ืื™ื ื˜ื’ืจืœื™ื ื”ื‘ืื™ืโ€ฌ/โ€ซืคืชื•ืจโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_2_004.htmlโ€ฌโ€ฌ
โ€ซืžืฉื•ื•ืื” ื“ื™ืคืจื ืฆื™ืืœื™ืชโ€ฌ
โ€ซืคืชื•ืจโ€ช/โ€ฌื™ ืืช ื”ืžืฉื•ื•ืื” ื”ื“ืคืจื ืฆื™ืืœื™ืช ื”ื‘ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช R,L,Cโ€ฌื”ื ืงื‘ื•ืขื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_2_005.htmlโ€ฌโ€ฌ
โ€ซื–ื”ื•ื™ื•ืช ื•ืงื˜ื•ืจื™ื•ืชโ€ฌ
โ€ซื”ื•ื›ื—โ€ช/โ€ฌื™ ืืช ื”ื–ื”ื•ื™ื•ืช ื”ื‘ืื•ืชโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_2_006.htmlโ€ฌโ€ฌ
โ€ซืžืฉืคื˜ ืกื™ื ื•ืกื™ื ื•ืงื•ืกื™ื ื•ืกื™ืโ€ฌ
โ€ซื”ื•ื›ื—โ€ช/โ€ฌื™ ื‘ืืžืฆืขื•ืช ื—ืฉื‘ื•ืŸ ื•ืงื˜ื•ืจื™โ€ช:โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžืฉืคื˜ ื”ืงื•ืกื™ื ื•ืกื™ื )ื”ืžื™ืฉื•ืจื™(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืžืฉืคื˜ ื”ืกื™ื ื•ืกื™ื )ื”ืžื™ืฉื•ืจื™(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_2_007.htmlโ€ฌโ€ฌ
โ€ซืžืฉืคื˜ ืกื™ื ื•ืกื™ื ื•ืงื•ืกื™ื ื•ืกื™ืโ€ฌ
โ€ซื‘ืžื›ืคืœื” ื”ื•ืงื˜ื•ืจื™ืช โ€ช:โ€ฌโ€ฌ
โ€ซืžื” ื”ื•ื โ€ช) Bโ€ฌืžืกืคืจื™ืช( ื›ืืฉืจโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™โ€ช:โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_2_008.htmlโ€ฌโ€ฌ
โ€ซืžืฉืคื˜ ื”ืงื•ืกื™ื ื•ืกื™ืโ€ฌ
โ€ซื”ื•ื›ื™ื—ื• ืืช ืžืฉืคื˜ ื”ืงื•ืกื™ื ื•ืกื™ื ืข"ื™ ื—ืฉื‘ื•ืŸ ื•ื•ืงื˜ื•ืจื™โ€ช:โ€ฌโ€ฌ
01_Math_intro/e_01_2_009.html
Vectors
A vector that points from the origin to a moving object is given by:
r=(4t,sin(wt),Exp(-at))
find the velocity and acceleration of the object.
01_Math_intro/e_01_2_010.html
Vectors
Given the vectors: A=(1,2,3) and B=(4,3,1).
a. calculate the length of both vectors.
b. find a unit vector C, that points the same direction as A.
c. calculate the length of C to verify that it is unity.
d. find the angle between the two vectors A and B.
e. find the vector D that points from B to A.
f. find the vector E that points from A to B.
01_Math_intro/e_01_2_103.html
โ€ซืื™ื ื˜ื’ืจืœื™ืโ€ฌ
โ€ซื ื’ื“ื™ืจโ€ฌ
โ€ซืœื”ื•ื›ื™ื—โ€ฌ
โ€ซื›ืืฉืจโ€ฌ
01_Math_intro/e_01_3_121.html
โ€ซืžืฉื•ื•ืื•ืช ื“ื™ืคืจื ืฆื™ืืœื™ื•ืชโ€ฌ
.
โ€ซื‘ืืžืฆืขื•ืช ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ
.โ€ซื”ื•ื ืคืชืจื•ืŸโ€ฌ
.
โ€ซื•ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซื™ ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ/โ€ซ ื‘ื˜ืโ€ฌ.โ€ซืโ€ฌ
โ€ซื™ ื›ื™โ€ฌ/โ€ซ ื”ืจืื”โ€ฌ.โ€ซื‘โ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื•ืืชโ€ฌ
โ€ซื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_3_122.htmlโ€ฌโ€ฌ
โ€ซืžืฉื•ื•ืื•ืช ื“ื™ืคืจื ืฆื™ืืœื™ื•ืชโ€ฌ
โ€ซื›ืืฉืจ ื•โ€ช -โ€ฌืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ ื‘ืืžืฆืขื•ืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืคืชืจื•ืŸ ื›ืœืœื™ ื ื™ืชืŸ ืœืจืฉื•ื ื‘ืฆื•ืจื”โ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื“โ€ช .โ€ฌื ืชื•ืŸ ื›ื™โ€ฌ
โ€ซื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซื”โ€ช .โ€ฌืžื”ื• ื”ืคืชืจื•ืŸ ื›ืืฉืจโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื•ืืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืฉืœื™ืœื™? ื”ืกื‘ืจโ€ช/โ€ฌื™!โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_3_123.htmlโ€ฌโ€ฌ
โ€ซืžืฉื•ื•ืื•ืช ื“ื™ืคืจื ืฆื™ืืœื™ื•ืชโ€ฌ
โ€ซื›ืืฉืจ ื•โ€ช -โ€ฌืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ ื‘ืืžืฆืขื•ืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืคืชืจื•ืŸ ื›ืœืœื™ ื ื™ืชืŸ ืœืจืฉื•ื ื‘ืฆื•ืจื”โ€ฌ
โ€ซื•โ€ช. -โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ื•ืืชโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌื ืชื•ืŸ ื›ื™โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_011.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซืฉื’ื•ื“ืœื ื–ื”ื” ื•ืฉื•ื•ื” ืœโ€ช 12.7 -โ€ฌื™ื—ื™ื“ื•ืชโ€ช .โ€ฌื›ื™ื•ื•ื ื ืžืชื•ืืจ ื‘ืฆื™ื•ืจ )ืžื›ืœืฉืืจ ื”ื‘ื—ื™ื ื•ืช ื”ืฆื™ื•ืจ ื”ื•ื ืกื›ืžื˜ื™(โ€ช .โ€ฌื ืกืžืŸ ืืชโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื•ื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซืกื›ื•ืžื ื”ื•ื•ืงื˜ื•ืจื™ ื‘โ€ช . -โ€ฌืžื”ื โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืจื›ื™ื‘ โ€ช xโ€ฌื•ืจื›ื™ื‘ โ€ช yโ€ฌืฉืœ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื”ื’ื•ื“ืœ ืฉืœ โ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌื”ื–ื•ื•ื™ืช ืฉื™ื•ืฆืจ ืขื ืฆื™ืจ โ€ช? xโ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_012.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื•ื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซืโ€ช.โ€ฌืžืฆืโ€ช/โ€ฌื™ ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžืฆืโ€ช/โ€ฌื™ ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซื”ืžืงื™ื™ื โ€ช:โ€ฌโ€ฌ
โ€ซื”ืžืงื™ื™ื โ€ช:โ€ฌโ€ฌ
โ€ซื•ึฐโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_013.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ื”ื•ื•ืงื˜ื•ืจื™ื ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืžื›ืคืœื” ื”ืกืงืœืจื™ืชโ€ฌ
โ€ซื‘โ€ช.โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืžื›ืคืœื” ื”ื•ื•ืงื˜ื•ืจื™ืช โ€ช:โ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื’ื•ื“ืœ ืฉืœ ื”ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซโ€ช ,โ€ฌืžื”ื™ ื”ื–ื•ื•ื™ืช ื‘ื™ืŸ ืฉื ื™ ื”ื•ื•ืงื˜ื•ืจื™ื ?โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_014.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉืœื•ืฉื” ื•ื•ืงื˜ื•ืจื™ื โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื—ืฉื‘ื• ืืช ื”ืื•ืจืš ืฉืœ ื›ืœ ื•ื•ืงื˜ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ื–ื•ื•ื™ืชื‘ื™ืŸ ื”ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซืœื‘ื™ืŸ ื›ืœ ืื—ื“ ืžื”ื•ื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซื’โ€ช.โ€ฌื”ืื ื”ื•ื•ืงื˜ื•ืจื™ื ื™ื•ืฆืจื™ื ืžืฉื•ืœืฉ ?โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_015.htmlโ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžื›ื•ื ื™ืช ื ื•ืกืขืช ื‘ืžื”ื™ืจื•ืช โ€ช 50โ€ฌืงืž"ืฉโ€ช .โ€ฌื”ื™ื ื™ื•ืฆืืช ืžื ืงื•ื“ื” ื ื•ืกืขืช โ€ช 30โ€ฌื“ืงื•ืช ืžืขืจื‘ื” ื•ืœืื—ืจ ืžื›ืŸ ื ื•ืกืขืช โ€ช 70โ€ฌื“ืงื•ืช ื‘ื›ื™ื•ื•ืŸ ื“ืจื•ืโ€ช-โ€ฌืžืขืจื‘โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื™ื”ื™ื” ื”ืžืจื—ืง ื•ื”ื›ื™ื•ื•ืŸ ืฉืœ ื”ืžื›ื•ื ื™ืช ืžื ืงื•ื“ืช ื”ืžื•ืฆื ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื™ื”ื™ื”ื”ืžืจื—ืง ื•ื”ื›ื™ื•ื•ืŸ ืฉืœ ื”ืžื›ื•ื ื™ืช ืžื ืงื•ื“ืช ื”ืžื•ืฆื ื‘ืžื™ื“ื” ื•ืžื”ื™ืจื•ืชื” ื”ื™ื™ืชื” โ€ช 75โ€ฌืงืž"ืฉ ?โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_016.htmlโ€ฌโ€ฌ
โ€ซ ืกืคื™ื ื” ืžืคืœื™ื’ื” โ€ช 50โ€ฌืง"ืž ื‘ื–ื•ื•ื™ืช โ€ช 70โ€ฌืœืฆื™ืจ ื”โ€ฌโ€ซื•ืœืื—ืจ ืžื›ืŸ โ€ช 130โ€ฌืง"ืž ื‘ื–ื•ื•ื™ืช โ€ช 40โ€ฌืœืฆื™ืจโ€ฌ
โ€ซืžืฆืื• ืืช ื”ื›ื™ื•ื•ืŸ ื•ื”ืžืจื—ืง ืฉืœ ื”ืกืคื™ื ื” ืžื ืงื•ื“ืช ื”ืžื•ืฆืโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_017.htmlโ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื•ืงื˜ื•ืจ ื‘ื”ืฆื’ื” ืงืจื˜ื–ื™ืชโ€ฌ
โ€ซื”ืฆื’โ€ช/โ€ฌื™ ืืช ื”ื•ืงื˜ื•ืจ ื‘ื”ืฆื’ื” ืคื•ืœืจื™ืชโ€ฌ
โ€ซื ืชื•ืŸ ื•ืงื˜ื•ืจ ืฉืื•ืจื›ื• โ€ช 10โ€ฌืžื˜ืจื™ื ื•ื ืžืฆื ื‘ืžื™ืฉื•ืจ ื‘ื–ื•ื•ื™ืช โ€ช 53โ€ฌืœืฆื™ืจโ€ช2.โ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื”ื”ืฆื’ื” ื”ืงืจื˜ื–ื™ืช ืฉืœ ื”ื•ืงื˜ื•ืจ ื‘ืžืจื—ื‘โ€ฌ
โ€ซโ€ช-โ€ฌื—ืฉื‘\ื™ ื‘ืขื–ืจืช ืžื›ืคืœื” ืกืงืœืจื™ืช ืืช ื”ื–ื•ื•ื™ืช ื‘ื™ืŸ ื”ื•ืงื˜ื•ืจื™ืโ€ช3.โ€ฌโ€ฌ
โ€ซืžืฆื\ื™ ืืช ื”ื ื’ื–ืจืช ืฉืœ ื”ื•ืงื˜ื•ืจโ€ช4.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_018.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืžื•ืงื ื‘ืžืขืจื›ืช ื”ืฆื™ืจื™ื ืžืชื—ื™ืœ ืœื ื•ืข ืœื ืงื•ื“ื”โ€ฌ
โ€ซืžืฉื ืžืžืฉื™ืš ืœื ื•ืข ืฆืคื•ื ื” ื‘ืžื”ื™ืจื•ืช โ€ช 30โ€ฌืงืž"ืฉ ืœืžืฉืš โ€ช 10โ€ฌื“ืงื•ืช ื›ืฉื”ื•ื ืžื’ื™ืข ืœืฉื ืžื™ื“ ื—ื•ื–ืจ ืœืจืืฉื™ืชโ€ฌ
โ€ซืžืฆืื• ืืช ื•ืงื˜ื•ืจื™ ื”ื”ืขืชืง ื”ื—ืœืงื™ื™ื ื•ืชืืจื• ืื•ืชื ื‘ืื•ืคืŸ ืงืจื˜ื–ื™ ื•ืคื•ืœืจื™โ€ฌ
โ€ซืžืฆืื• ืืช ื”ื•ืงื˜ื•ืจ ื”ืฉืงื•ืœ )ื”ื”ืขืชืง ื”ื›ื•ืœืœโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_022.htmlโ€ฌโ€ฌ
โ€ซื—ืงื™ืจืช ืคื•ื ืงืฆื™ื”โ€ฌ
โ€ซื”ืคื•ื ืงืฆื™ื” )โ€ช h(tโ€ฌืžืชืืจืช ืืช ื”ื’ื•ื‘ื” ืฉืœ ื›ื“ื•ืจ ืงื˜ืŸ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ โ€ช .tโ€ฌื”ื›ื“ื•ืจ ื ื–ืจืง ื›ืœืคื™ ืžืขืœื” ื‘ืจื’ืข โ€ช t0=0โ€ฌืžื’ื•ื‘ื” ืžืกื•ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืื™ื–ื” ื’ื•ื‘ื” ื ื–ืจืง ื”ื›ื“ื•ืจ? )ืจืžื–โ€ช :โ€ฌื–ื”ื• ื”ืขืจืš ืฉืœ ื”ืคื•ื ืงืฆื™ื” )โ€ช h(tโ€ฌื‘ืจื’ืข ื”ื–ืจื™ืงื”โ€ช( .โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ื’ื•ื‘ื” ื”ืžืงืกื™ืžืœื™ ืฉืืœื™ื• ื™ื’ื™ืข ื•ื‘ืื™ื–ื” ืจื’ืข ื™ื’ื™ืข ืœื’ื•ื‘ื” ื–ื”? )ืจืžื–โ€ช :โ€ฌื–ื”ื• ืขืจืš ื”ืžืงืกื™ืžื•ื ืฉืœ ื”ืคื•ื ืงืฆื™ื” )โ€ช( . h(tโ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌื”ื›ื“ื•ืจ ื™ื’ื™ืข ืœืงืจืงืข ื‘ืจื’ืข ืžืกื•ื™ืโ€ช .โ€ฌืžืฆืื• ืืช ื”ื–ืžืŸ ืฉื‘ื• ื–ื” ื™ืงืจื”โ€ช) .โ€ฌืจืžื–โ€ช :โ€ฌื”ื’ื•ื‘ื” ื ืžื“ื“ ืœืคื™ ื”ืžืจื—ืง ืžื”ืจืฆืคื”โ€ช ,โ€ฌื›ืœื•ืžืจโ€ช ,โ€ฌื”ื’ื•ื‘ื” ืฉืœ ื”ืจืฆืคื”โ€ฌ
โ€ซื”ื•ื โ€ช(. 0โ€ฌโ€ฌ
โ€ซโ€ชh(t) = h0+v0 t-g t2 /2โ€ฌโ€ฌ
โ€ซโ€ชh0,g,v0>0โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_023.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื โ€ช 2โ€ฌื•ืงื˜ื•ืจื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืขื‘ื•ืจ ืื™ืœื• ืขืจื›ื™ โ€ช aโ€ฌื”ื•ืงื˜ื•ืจื™ื ื ื™ืฆื‘ื™ื ื–ื” ืœื–ื”โ€ช,โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืขื‘ื•ืจ ืื™ืœื• ืขืจื›ื™ โ€ช aโ€ฌื”ื•ืงื˜ื•ืจื™ื ืžืงื‘ื™ืœื™ื ื–ื” ืœื–ื”โ€ช,โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_5_024.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื•ื•ืงื˜ื•ืจื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžืฆืโ€ช/โ€ฌื™ ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžืฆืโ€ช/โ€ฌื™ ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซื”ืžืงื™ื™ื โ€ช:โ€ฌโ€ฌ
โ€ซื”ืžืงื™ื™ื โ€ช:โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_011.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซืžื›ื•ื ื™ืช ื ื•ืกืขืช ืœื›ื™ื•ื•ืŸ ื“ืจื•ื ืžื–ืจื—ื‘ืžื”ื™ืจื•ืช โ€ช 72โ€ฌืงืž"ืฉโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื ื”ื™ื˜ืœื™ ื”ืžื”ื™ืจื•ืช ืœื›ื™ื•ื•ืŸ ืžื–ืจื— ื•ืœื›ื™ื•ื•ืŸ ื“ืจื•ื?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื‘ืžื”ืœืš ื”ื ืกื™ืขื” ืžื—ื•ื’ ื”ืžื”ื™ืจื•ืช ืžืชืงืœืงืœ )ื›ื ืจืื” ื‘ื’ืœืœ ืจื™ื‘ื•ื™ ื”ื‘ื•ืจื•ืช ื‘ื›ื‘ื™ืฉื™ ื‘"ืฉ( ื•ืžืจืื”ืืช ื”ืžื”ื™ืจื•ืช ื‘โ€ช) m/sec -โ€ฌืžื˜ืจื™ื ืœืฉื ื™ื™ื”(โ€ช.โ€ฌโ€ฌ
โ€ซืื™ื–ื• ืžื”ื™ืจื•ืช ื™ืจืื” ื”ืžื—ื•ื’ ื›ืขืช ื‘ื”ื ื—ื”ืฉืžื”ื™ืจื•ืช ื”ื ืกื™ืขื” ืœื ื”ืฉืชื ืชื”?โ€ฌ
โ€ซื’โ€ช.โ€ฌื™ื“ื•ืข ื›ื™ ื‘ื›ื‘ื™ืฉื™ ื”ื ื’ื‘ื” ืžื”ื™ืจื•ืช ื”ืžื•ืชืจืช ื”ื™ื โ€ช 90โ€ฌืงืž"ืฉโ€ช .โ€ฌืžื—ื•ื’ ื”ืžื”ื™ืจื•ืช ื‘ืžื›ื•ื ื™ืช ืขื•ืžื“ ืขืœ โ€ช,[m/sec] 30โ€ฌื”ืื ืื ื—ื ื• ืขื‘ืจื™ื™ื ื™ ืชื ื•ืขื”?โ€ฌ
โ€ซื•ืื ื›ืŸโ€ช ,โ€ฌืžื” ื’ื•ื‘ื” ื”ื“ื•"ื— ืฉื ืงื‘ืœ?โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_012.htmlโ€ฌโ€ฌ
โ€ซืคื•ื ืงืฆื™ื•ืชโ€ฌ
โ€ซื”ืคื•ื ืงืฆื™ื” ื”ืžืชืืจืช ืืช ืื—ื•ื– ื”ืืœื›ื•ื”ื•ืœ ื‘ื“ื )ืฉืœ ื—ื•ื’ื’ื‘ืžืกื™ื‘ืช ื‘ืจ ืžืฆื•ื•ื”( ื›ืชืœื•ืช ื‘ื–ืžืŸโ€ช ,โ€ฌื‘ื˜ื•ื•ื— โ€ช 5-0โ€ฌืฉืขื•ืช ื”ื™ืโ€ฌ
โ€ซ)ืจืื”โ€ช/โ€ฌื™ ืฆื™ื•ืจ(โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื›ืขื‘ื•ืจื›ืžื” ื–ืžืŸ ืžืชื—ื™ืœืช ื”ืžืกื™ื‘ื” ืื—ื•ื– ื”ืืœื›ื•ื”ื•ืœ ื”ื•ื ืžืงืกื™ืžืœื™?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ืื—ื•ื– ื”ืืœื›ื•ื”ื•ืœ ื”ืžืžื•ืฆืข ืฉืœ ื”ื—ื•ื’ื’?โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื• ื’ื™ืœื•ืฉืœ ื—ืชืŸ ื”ื‘ืจโ€ช-โ€ฌืžืฆื•ื•ื”?โ€ฌ
โ€ซื’โ€ช.โ€ฌื‘ื“ื•ืงโ€ช/โ€ฌื‘ื“ืงื™ ืชืฉื•ื‘ืชืš ืข" ื™ ื—ื™ืฉื•ื‘ ื”ื–ื•ื•ื™ื•ืช ื‘ืฆื•ืจื” ื’ื™ืื•ืžื˜ืจื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช.โ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื”ื–ื•ื•ื™ืช ืฉืœ ื‘ื™ื—ืก ืœืฆื™ืจ โ€ช xโ€ฌื›ื“ื™ ืœืงื™ื™ืโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_013.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื”โ€ฌ
โ€ซืžื˜ื•ืก ื˜ืก ื‘ืžื”ื™ืจื•ืช โ€ช 500โ€ฌืงืž"ืฉ ื‘ืžืฉืš โ€ช 30โ€ฌื“ืงื•ืช ืžืขืจื‘ื” ื•ืœืื—ืจ ืžื›ืŸ โ€ช 70โ€ฌื“ืงื•ืช ื‘ื›ื™ื•ื•ืŸ ื“ืจื•ืโ€ช-โ€ฌืžืขืจื‘โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื™ื”ื™ื” ื”ืžืจื—ืง ืฉืœื”ืžื˜ื•ืก ืžื ืงื•ื“ืช ื”ืžื•ืฆื ืœืื—ืจ โ€ช 100โ€ฌื“ืงื•ืช?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ื”ื›ื™ื•ื•ืŸ ืžื ืงื•ื“ืช ื”ืžื•ืฆื ืœืžื™ืงื•ื ื”ื—ื“ืฉ ืฉืœ ื”ืžื˜ื•ืก?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ื™ื• ื”ืžืจื—ืง ื•ื”ื›ื™ื•ื•ืŸ ืฉืœ ื”ืžื˜ื•ืก ืžื ืงื•ื“ืช ื”ืžื•ืฆื ื‘ืžื™ื“ื” ื•ืžื”ื™ืจื•ืชื• ื”ื™ื™ืชื” โ€ช 750โ€ฌืงืž"ืฉ ?โ€ฌ
โ€ซื“โ€ช.โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ืชืฉื•ื‘ื•ืช ืœืฉืœื•ืฉืช ื”ืกืขื™ืคื™ื ืโ€ช,โ€ฌื‘โ€ช,โ€ฌื’ ื‘ื™ื—ื™ื“ื•ืช ืฉืœ ืžื˜ืจื™ืืกื ื˜ื™ืžื˜ืจื™ื ื•ืžื™ืœื™ืžื˜ืจื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_014.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื•ื•ืงื˜ื•ืจื™ืโ€ช,โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืœืžื” ืฉื•ื•ื” ื”ื•ื•ืงื˜ื•ืจ ื”ืฉืงื•ืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื–ื•ื•ื™ื•ืช ื‘ื™ืŸ ืœโ€ช , -โ€ฌื‘ื™ืŸ ืœโ€ช-โ€ฌโ€ฌ
โ€ซื•ื‘ื™ืŸ ืœโ€ช -โ€ฌื‘ืขื–ืจืช ื”ืžื›ืคื™ืœื” ื”ืกืงืœืจื™ืช ื‘ื™ื ื™ื”ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_021.htmlโ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจื™ืโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื”ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซืโ€ช .โ€ฌื›ืชื•ื‘โ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ืœื•ื•ืงื˜ื•ืจ ื”ื™ื—ื™ื“ื” โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื›ืชื•ื‘โ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ืœื•ื•ืงื˜ื•ืจ ืฉื›ื™ื•ื•ื ื• ื–ื”ื” ืœื›ื™ื•ื•ืŸ ื”ื•ื•ืงื˜ื•ืจโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ื–ื•ื•ื™ืช ืฉืœ ื”ื•ื•ืงื˜ื•ืจ ื‘ื™ื—ืก ืœืฆื™ืจ โ€ช? xโ€ฌโ€ฌ
โ€ซืื‘ืœ ืื•ืจื›ื• โ€ช 2โ€ฌื™ื—ื™ื“ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_022.htmlโ€ฌโ€ฌ
โ€ซืžืฉื•ื•ืื•ืช ื“ื™ืคืจื ืฆื™ืืœื™ื•ืชโ€ฌ
โ€ซื›ืืฉืจ ื•โ€ช -โ€ฌืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ฌ
โ€ซื”ื•ื ื‘ื™ื—ื™ื“ื•ืช ืฉืœ ืื•ืจืš ื• ื‘ื™ื—ื™ื“ื•ืช ืฉืœ ืžืกื”โ€ฌ
โ€ซืื ื ืชื•ืŸ ืฉโ€ฌ
โ€ซโ€ช.โ€ฌื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซื•โ€ฌ
โ€ซโ€ช.โ€ฌืโ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌื‘โ€ช .โ€ฌืžืฆื ืคืชืจื•ืŸ ื›ืœืœื™ ืœืžืฉื•ื•ืื”โ€ฌ
โ€ซืžืฆื ืืช ืงื‘ื•ืขื™ ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซื’โ€ช .โ€ฌื ืชื•ืŸ ื›ื™โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_023.htmlโ€ฌโ€ฌ
โ€ซืžืขื‘ืจ ื™ื—ื™ื“ื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌื“ื ื” ื ื•ืกืขืช ื›ืœ ื™ื•ื ืœืขื‘ื•ื“ื” ื•ื—ื–ืจื” ืžืจื—ืง ืฉืœ โ€ช 52โ€ฌืง"ืžโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ืžื—ื™ืจ ื”ื“ืœืง ื”ื•ื โ€ช 6.1โ€ฌืฉ"ื— ืœืœื™ื˜ืจโ€ช ,โ€ฌืชืฆืจื•ื›ืช ื”ื“ืœืง ืฉืœ ื”ืจื›ื‘ ืฉืœื” ื”ื•ื โ€ช 13โ€ฌืง"ืž ืœืœื™ื˜ืจโ€ฌ
โ€ซโ€ช.โ€ฌื•ืขืœื•ืช ื”ืชื™ืงื•ื ื™ื ื”ืžืžื•ืฆืขืช ืœืจื›ื‘ ื”ื•ื โ€ช 1200โ€ฌืฉ"ื— ืœืขืฉืจืช ืืœืคื™ื ืง"ืžโ€ฌ
โ€ซโ€ช.โ€ฌื›ืžื• ื›ืŸ ื™ื“ื•ืข ืฉื‘ื—ื•ื“ืฉ ื™ืฉื ื ื‘ืžืžื•ืฆืข ื›โ€ช 22โ€ฌื™ืžื™ ืขื‘ื•ื“ื”โ€ฌ
โ€ซืžืฆืื• ืืช ื”ืขืœื•ืช ื”ื—ื•ื“ืฉื™ืช ืฉืœ ื”ื ืกื™ืขื” ืœืขื‘ื•ื“ื”โ€ฌ
โ€ซโ€ช01_Math_intro/e_01_8_024.htmlโ€ฌโ€ฌ
โ€ซืžืฉื•ื•ืื•ืช ื“ื™ืคืจื ืฆื™ืืœื™ื•ืชโ€ฌ
โ€ซื›ืืฉืจ ื•โ€ช -โ€ฌืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ ื‘ืืžืฆืขื•ืช ื”ื™ื—ื™ื“ื•ืช ืฉืœ โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืคืชืจื•ืŸ ื›ืœืœื™ ื ื™ืชืŸ ืœืจืฉื•ื ื‘ืฆื•ืจื”โ€ฌ
โ€ซื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื“โ€ช .โ€ฌื ืชื•ืŸ ื›ื™โ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื•ืืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช04_Dimensions/e_04_2_002.htmlโ€ฌโ€ฌ
โ€ซื ื™ืชื•ื— ืžื™ืžื“ื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ื”ืงื‘ื•ืขื™ื ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืช ื”ืื•ืจโ€ช:โ€ฌโ€ฌ
โ€ซืงื‘ื•ืข ืคืœื ืงโ€ช:โ€ฌโ€ฌ
โ€ซืงื‘ื•ืข ื’ืจื‘ื™ื˜ืฆื™ื”โ€ช:โ€ฌโ€ฌ
โ€ซื›ืœื•ืžืจโ€ช:โ€ฌโ€ฌ
โ€ซืขืœ ื™ื“ื™ ืฉื™ืžื•ืฉ ื‘ื’ื“ืœื™ื ืืœื• ืžืฆืื•โ€ช:โ€ฌโ€ฌ
โ€ซื( ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœ ืื•ืจืšโ€ช.โ€ฌโ€ฌ
โ€ซื’( ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœ ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื“( ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœ ืชื“ื™ืจื•ืชโ€ฌ
โ€ซื”( ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœ ืฆืคื™ืคื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช04_Dimensions/e_04_2_004.htmlโ€ฌโ€ฌ
โ€ซื ื™ืชื•ื— ืžื™ืžื“ื™ืโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืฉื•ื•ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช a,b,Dโ€ฌื”ื ืคืจืžื˜ืจื™ืโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ื”ื™ื—ื™ื“ื•ืช ืฉืœ โ€ช uโ€ฌื”ื ืฉืœ ืฆืคื™ืคื•ืชโ€ช:โ€ฌโ€ฌ
โ€ซื( ืžืฆืื• ืืช ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช,a,b,D,au,โ€ฌโ€ฌ
โ€ซื‘( ื”ืคื›ื• ืืช ื”ืžืฉื•ื•ืื” ื”ื ืชื•ื ื” ืœืžืฉื•ื•ืื” ื”ื‘ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจโ€ฌ
โ€ซื”ื™ื ื ื—ืกืจื™ ืžื™ืžื“โ€ช .โ€ฌืžื”ืโ€ฌ
โ€ซ?โ€ฌ
โ€ซโ€ช04_Dimensions/e_04_2_005.htmlโ€ฌโ€ฌ
โ€ซืžืขื‘ืจ ื™ื—ื™ื“ื•ืชโ€ฌ
โ€ซโ€ช .1โ€ฌื›ืžื” โ€ช dyneโ€ฌื™ืฉ ื‘โ€ช Neuton-โ€ฌืื—ื“?โ€ฌ
โ€ซโ€ช .2โ€ฌื›ืžื” ืœื™ื˜ืจ ื™ืฉ ื‘ืžื˜ืจ ืžืขื•ืงื‘ )ืงื•ื‘( ืื—ื“?โ€ฌ
โ€ซโ€ช cran .3โ€ฌื”ื™ื ื™ื—ื™ื“ืช ื ืคื— ื‘ืจื™ื˜ื™ืช ืขื‘ื•ืจ ื“ื™ื™ื’ ืฉืœ ื“ื’ื™ ื”ืจื™ื ื’โ€ช ,โ€ฌื›ืืฉืจ โ€ช - 1cran=170.474Lโ€ฌื‘ืขืจืš โ€ช 750โ€ฌื“ื’ื™ืโ€ช .โ€ฌืžืฉืœื•ื— ืฉืœ โ€ชcran 1255โ€ฌโ€ฌ
โ€ซืฉืœ ื“ื’ื™ื ืžื’ื™ืข ืœืขืจื‘ ื”ืกืขื•ื“ื™ืชโ€ช .โ€ฌื‘ืžื›ืก ื”ืกืขื•ื“ื™ ืฆืจื™ืš ืœื”ืฆื”ื™ืจ ืœืคื™ ืžื™ื“ืช ื”ื ืคื— ื”ืžืงื•ืžื™ืช ืฉื”ื™ื โ€ช covidoโ€ฌืžืขื•ืงื‘โ€ช ,โ€ฌื›ืืฉืจ โ€ชcovido 1โ€ฌโ€ฌ
โ€ซ= โ€ช .cm 48.26โ€ฌื›ืžื” ื”ื“ื’ื™ื ื”ื•ืฆื”ืจื• ื‘ืžื›ืก? ืžื” ื”ื•ื ื ืคื— ื”ื“ื’ื™ื ื‘ื™ื—ื™ื“ื•ืช โ€ช?SIโ€ฌโ€ฌ
04_Dimensions/e_04_4_001.html
โ€ซื”ืžืจืช ื–ื•ื™ืช ืžืจื“ื™ืืŸโ€ฌ
.
โ€ซ ืจื“ื™ืืŸ ื”ื•ื ื”ืจื“ื™ื•ืกโ€ฌ1 โ€ซืฉื ืฉืขื ืช ืขืœ ื–ื•ื™ืช ืฉืœโ€ฌ
โ€ซ ืจื“ื™ืืŸ ืžื•ื’ื“ืจืช ื›ืš ืฉืื•ืจืš ื”ืงืฉืชโ€ฌ1 โ€ซื–ื•ื™ืช ืฉืœโ€ฌ
.
โ€ซื ืชื•ืŸ ืฉืื•ืจืš ืงืฉืช ืฉื ืฉืขื ืช ืขืœ ื–ื•ื™ืช ืฉื•ื•ื” ืœื–ื•ื™ืช ื‘ืจื“ื™ืื ื™ื ื›ืคื•ืœ ื”ืจื“ื™ื•ืกโ€ฌ
.
โ€ซืฉื•ื•ื”โ€ฌ
โ€ซื•ื ืชื•ืŸ ืฉื”ื™ืงืฃ ืžืขื’ืœ ื‘ืขืœ ืจื“ื™ื•ืกโ€ฌ
.โ€ซ ืจื“ื™ืืŸโ€ฌ1 โ€ซื—ืฉื‘ื• ื›ืžื” ืžืขืœื•ืช ืฉื•ื•ื”โ€ฌ
04_Dimensions/e_04_8_006.html
โ€ซืžืขื‘ืจ ื™ื—ื™ื“ื•ืชโ€ฌ
? โ€ซ ื‘ื™ื—ื™ื“ื•ืช ืฉืœ ืžื˜ืจ ืœืฉื ื™ื”โ€ฌ,โ€ซ ืžื™ื™ืœื™ื )ื™ื‘ืฉืชื™ื™ื( ืœืฉืขื”โ€ฌ55 โ€ซ ( ืžื”ืโ€ฌ1
.โ€ซ ืžื˜ืจโ€ฌ1609 -โ€ซื ืชื•ืŸ ื›ื™ ืžื™ื™ืœ ื™ื‘ืฉืชื™ ื”ื•ื ื›โ€ฌ
? โ€ซ ื’ืœื•ื ื™ื ื‘ืกืž"ืงโ€ฌ16 โ€ซ( ื›ืžื” ื”ืโ€ฌ2
.โ€ซ ืก"ืžโ€ฌ2.54 -โ€ซ ื•ื›ื™ ืื™ื ืฅ' ื”ื•ื ื›โ€ฌ,โ€ซ ืื™ื ืฅ' ืžืขื•ืงื‘โ€ฌ231 โ€ซื ืชื•ืŸ ื›ื™ ื’ืœื•ืŸ ืื—ื“ ื”ื•ืโ€ฌ
07_Kinematics/e_07_1_011.html
Velocity, acceleration, trajectory
A particle moves in the
plain according to the law:
acceleration, c) distance from the coordinate origin.
,
. Find: a) velocity, b)
07_Kinematics/e_07_1_012.html
Velocity, acceleration, trajectory
A particle moves along the ellipse
acceleration. Find the radius of curvature.
so that
. Find the velocity and
07_Kinematics/e_07_1_013.html
Velocity, acceleration, trajectory
Trajectory is given by
,
, and
,
and
. Find
and .
07_Kinematics/e_07_1_014.html
Velocity, acceleration, trajectory
Trajectory is given by
and
,
and
between the velocity and acceleration as a function of time ?
. Find
and . What is the angle
07_Kinematics/e_07_1_015.html
Velocity, acceleration, trajectory
Two spacecraft are orbiting Earth. The orbit radii and angular velocities are the same but one (A) orbit is
always above the equator, while the other (B)passes above the poles. When B is above the equator the
spacecraft A is on the opposite side of the diameter. Find the vectors connecting A and B as a function of time.
07_Kinematics/e_07_1_016.html
Velocity, acceleration, trajectory
Express the velocity vector (in general) in terms of spherical coordinates and unit vectors
,
,
.
07_Kinematics/e_07_1_017.html
Velocity, acceleration, trajectory
A particle moves according to the law:
acceleration.
,
,
. Find the velocity and
07_Kinematics/e_07_1_018.html
Velocity, acceleration, trajectory
Derive the expression for the trajectory length for the motion with constant acceleration.
07_Kinematics/e_07_1_019.html
Velocity, acceleration, trajectory
A particle moves along the trajectory
Here , , and are constant parameters. Find
in cylindrical coordinates so that
as a function of . Same as a function of
.
.
07_Kinematics/e_07_1_020.html
Velocity, acceleration, trajectory
Given
as a function of time.
,
. Find the angle between the vectors of velocity and acceleration
07_Kinematics/e_07_1_021.html
Velocity, acceleration, trajectory
Given
,
of velocity and acceleration as a function of time.
. Find the angle between the vectors
07_Kinematics/e_07_1_022.html
Velocity, acceleration, trajectory
Given
,
. Find the tangential and normal acceleration as functions of time.
07_Kinematics/e_07_1_023.html
Velocity, acceleration, trajectory
Given
,
parameters for the absence of self-intersection.
. Find
and
. What are the conditions on the
07_Kinematics/e_07_1_024.html
Velocity, acceleration, trajectory
Given
,
origin to the particle as a function of time if
07_Kinematics/e_07_1_025.html
Velocity, acceleration, trajectory
,
. Find the distance from the coordinate
.
In a universe all bodies move away from the coordinate origin with the velocities
What would see an observer at an arbitrary position
?
, where
.
07_Kinematics/e_07_1_026.html
Velocity, acceleration, trajectory
A rabbit starts to run at
from the point
in the positive direction of axis
(magnitude !). A fox starts to run from
at the same moment and its velocity
the rabbit. How much time does it take to catch the rabbit.
with the velocity
always points towards
07_Kinematics/e_07_1_027.html
Velocity, acceleration, trajectory
Given
,
. Find
.
07_Kinematics/e_07_1_028.html
Velocity, acceleration, trajectory
Given
length (integral).
,
,
. Write down the expression for the path
07_Kinematics/e_07_1_029.html
Velocity, acceleration, trajectory
Given
acceleration.
,
,
, find the tangential and normal components of the
07_Kinematics/e_07_1_030.html
Velocity, acceleration, trajectory
A body starts from the equator of the sphere (``Earth'') with the radius
and moves all the time in the northeast direction so that the velocity magnitude remains constant. Where does it stop and how much time does it
take ?
07_Kinematics/e_07_1_031.html
Velocity, acceleration, trajectory
A cannonball is fired in an angle
over an inclined plane of angle
to hit the plane horizontally, the angles must fulfill the relation
. Show that in order for the cannonball
.
figure 1 - A cannonball is fired over an inclined plane.
07_Kinematics/e_07_1_032.html
Velocity, acceleration, trajectory
Find the minimal velocity that is needed in order to throw a body over a building with height h and width L.
07_Kinematics/e_07_1_033.html
Velocity Acceleration and Trajectory
Particle's motion is given by
and
,
Find
and
.
07_Kinematics/e_07_1_034.html
Velocity Acceleration and Trajectory
A ball is thrown in an angle
ground).
Find:
1.
.
2. The vector
and initial velocity
, and hits a building in a distance
at a hight
(above
(final velocity).
07_Kinematics/e_07_1_035.html
Velocity Acceleration and Trajectory
Particle moves according to
,
,
Find the normal acceleration and the curvature radius
.
07_Kinematics/e_07_1_036.html
Velocity Acceleration and Trajectory
Particle's trajectory is given by
,
where
and
are constants.
Find the magnitude of the normal and tangential acceleration.
07_Kinematics/e_07_1_037.html
โ€ซืชื ื•ืขื”โ€ฌ
โ€ซ ื›ืชืœื•ืช ื‘ื–ืžืŸ( ื•ืชืืจ ื‘ื’ืจืฃ ืืช ื”ืžื”ื™ืจื•ืช ื•ืืช ื”ืชืื•ืฆื” ื›ืคื•ื ืงืฆื™ื”โ€ฌx) โ€ซ ืžืฆื ืืช ื”ืคื•ื ืงืฆื™ืช ื”ืžื™ืงื•ืโ€ฌ.x โ€ซื ืชื•ืŸ ื’ืจืฃ ื”ืžืชืืจ ืชื ื•ืขื” ืฉืœ ื’ื•ืฃ ืขืœ ืฆื™ืจโ€ฌ
.โ€ซืฉืœ ื”ื–ืžืŸโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_038.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื”โ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืœืื•ืจืš ืฆื™ืจ โ€ช zโ€ฌืคื•ื ืงืฆื™ืช ื”ืžื™ืงื•ื ืฉืœื• ื ืชื•ื ื” ืข"ื™โ€ช:โ€ฌโ€ฌ
โ€ซืžื”ื• ืžืจื—ืงื• ืฉืœ ื”ื—ืœืงื™ืงื™ ืžื”ืจืืฉื™ืช ื›ืืฉืจ ื”ื•ื ื ืขืฆืจ? ื‘ืื™ื–ื” ื–ืžืŸ ื–ื” ืงื•ืจื”?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_039.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื”โ€ฌ
โ€ซื’ื•ืฃ ื™ื•ืฆื ืžื ืงื•ื“ื” ื›ืœืฉื”ื™ื ื•ืžื”ื™ืจื•ืชื• ืžืชื•ืืจืช ื‘ื’ืจืฃโ€ฌ
โ€ซืโ€ช .โ€ฌืชืืจ ืืช ื”ืชื ื•ืขื” ื‘โ€ช 15-โ€ฌืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืชโ€ฌ
โ€ซื‘โ€ช .โ€ฌืจืฉื•ื ืžืฉื•ื•ืืช ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ื“ืจืš ืฉืขื•ื‘ืจ ื”ื’ื•ืฃ ื‘ โ€ช 6โ€ฌืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืชโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื™ ืžืฉื•ื•ืืช ื”ืžืจื—ืงโ€ฌ
โ€ซื”โ€ช .โ€ฌืžืชื™ ื”ื—ืœืงื™ืง ื™ื—ื–ื•ืจ ืœืจืืฉื™ืชโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_040.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื’ื•ืฃ ื ื–ืจืง ื›ืœืคื™ ืžืขืœื” ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 30โ€ฌืžื˜ืจ ืœืฉื ื™ื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ื™ื›ืŸ ื™ืžืฆื ื”ื’ื•ืฃ ืœืื—ืจ โ€ช 2โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื• ืœืื—ืจ โ€ช 3โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌื›ืžื” ื–ืžืŸ ืชื™ืžืฉืš ืขืœื™ื™ืชื•โ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื• ื”ื’ื•ื‘ื” ื”ืžืงืกื™ืžืœื™ ื‘ืžืกืœื•ืœื•โ€ช.โ€ฌโ€ฌ
โ€ซื”โ€ช .โ€ฌื‘ืื™ื–ื” ืžื”ื™ืจื•ืช ื™ื’ื™ืข ืœื ืงื•ื“ืช ื”ื–ืจื™ืงื”โ€ช.โ€ฌโ€ฌ
โ€ซื•โ€ช .โ€ฌืœืื—ืจ ื›ืžื” ื–ืžืŸ ื”ื’ื•ืฃ ื™ื”ื™ื” โ€ช 10โ€ฌืžื˜ืจ ืžืชื—ืช ืœื ืงื•ื“ืช ื”ื–ืจื™ืงื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_041.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื’ื•ืฃ ื ื–ืจืง ื”ืžื”ื™ืจื•ืช ืฉืœ โ€ช 30โ€ฌืžื˜ืจ ืœืฉื ื™ื” ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 57โ€ฌืžืขืœื•ืช ืžืขืœ ื”ืื•ืคืงโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ื™ื›ืŸ ื™ืžืฆื ื”ื’ื•ืฃ ืœืื—ืจ โ€ช 5โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืœืื—ืจ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ืœืฉื™ื ื”ื’ื•ื‘ื” ? ืžื”ื• ืฉื™ื ื”ื’ื•ื‘ื”โ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืœืื—ืจ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ื—ื–ืจื” ืœื’ื•ื‘ื” ื”ื”ืชื—ืœืชื™โ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื• ื˜ื•ื•ื— ื”ื–ืจื™ืงื” ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_042.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžืžื˜ื•ืก ื”ื˜ืก ืื•ืคืงื™ืช ื‘ืžื”ื™ืจื•ืช โ€ช 360โ€ฌืง"ืž ืœืฉืขื” ื•ื‘ื’ื•ื‘ื” โ€ช 980โ€ฌืžื˜ืจ ืžื•ื˜ืœืช ืคืฆืฆื” ืœืขื‘ืจ ืจื›ื‘ ื”ื ืข ืœื›ื™ื•ื•ืŸ ื”ืžื˜ื•ืก ื‘ืžื”ื™ืจื•ืช โ€ช 90โ€ฌืง"ืž ืœืฉืขื”โ€ช.โ€ฌโ€ฌ
โ€ซืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ื”ืžืจื—ืง ื”ืื•ืคืงื™ ื‘ื ื™ื”ื ื‘ืจื’ืข ื”ื˜ืœืช ื”ืคืฆืฆื” ืขืœ ืžื ืช ืœื”ืฉื™ื’ ืคื’ื™ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_043.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืคื’ื– ื ื•ืจื” ื‘ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื‘ืžื”ื™ืจื•ืช ืœื•ืข ืฉืœ โ€ช 100โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื” )ืจืื” ืฆื™ื•ืจ(โ€ช .โ€ฌืื™ืคื” ื™ืคื•ืœ ื”ืคื’ื– ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_310.htmlโ€ฌโ€ฌ
โ€ซโ€ชVelocity, acceleration, trajectoryโ€ฌโ€ฌ
โ€ซโ€ช. Show that in order for the cannonballโ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ชA cannonball is fired in an angleโ€ฌโ€ฌ
โ€ซโ€ชover an inclined plane of angleโ€ฌโ€ฌ
โ€ซโ€ชto hit the plane horizontally, the angles must fulfill the relationโ€ฌโ€ฌ
โ€ซโ€ชfigure 1 - A cannonball is fired over an inclined plane.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_311.htmlโ€ฌโ€ฌ
โ€ซืชืื•ืฆื” ืชืœื•ื™ืช ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช .โ€ฌืžื“ื™ื“ื•ืช ื”ืจืื• ืฉืžื”ื™ืจื•ืช ื”ืกื™ืจื” ืื—ืจ ื›ืš ืคืจื•ืคื•ืจืฆื™ื•ื ื™ืช ืœโ€ช-โ€ฌโ€ฌ
โ€ซืžื•ืจื™ื“ื” ืžืคืจืฉ ื‘ืจื’ืขโ€ฌ
โ€ซืกื™ืจื” ื”ื ืขื” ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื( ืžืฆืื• ืืช ืชืœื•ืช ื”ืชืื•ืฆื” ื‘ืžื”ื™ืจื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืžืฆืื• ื‘ื™ื˜ื•ื™ ืžื“ื•ื™ืง ืœืžื”ื™ืจื•ืช ืœืื—ืจ ื”ื•ืจื“ืช ื”ืžืคืจืฉโ€ช.โ€ฌโ€ฌ
โ€ซื’( ืžืฆืื• ืืช ื”ื“ืจืš ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_1_312.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžืขื’ืœโ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ื‘ืžืขื’ืœ ื‘ืขืœ ืจื“ื™ื•ืกโ€ฌ
โ€ซื( ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื–ื•ื•ื™ืช ื”ืกื™ื‘ื•ื‘โ€ช.โ€ฌโ€ฌ
โ€ซื‘ืชืื•ืฆื” ืžืฉื™ืงื™ืช ืงื‘ื•ืขื”โ€ฌ
โ€ซื•ื‘ืœื™ ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืชโ€ช .โ€ฌืžืฆืื• ืืช ื’ื•ื“ืœ ื”ืชืื•ืฆื” ื”ื ื•ืจืžืœื™ืชโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_001.htmlโ€ฌโ€ฌ
โ€ซืžื ื™ืขืช ื”ืชืจืกืงื•ืชโ€ฌ
โ€ซื˜ื™ื™ืก ืงืจื‘ ื”ื˜ืก ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 1300โ€ฌืง"ืž ืœืฉืขื” ืžื‘ืฆืข ืชืžืจื•ื ื™ ื”ืชื—ืžืงื•ืช ืžืžื›"ื ื‘ื˜ื™ืกื” ื ืžื•ื›ื” ื‘ืจื•ื ืฉืœ โ€ช 35mโ€ฌืžืขืœ ืœืงืจืงืขโ€ช .โ€ฌื‘ืฉืœื‘ ืžืกื•ื™ืโ€ฌ
โ€ซื ืชืงืœ ื”ืžื˜ื•ืก ื‘ืงืจืงืข ื”ืขื•ืœื” ื‘ืฉื™ืคื•ืข ืงืœ ืฉืœ โ€ช) 4.3ºโ€ฌืฉื™ืคื•ืข ืฉืงืฉื” ืžืื•ื“ ืœื–ื”ื•ืช ื‘ืขื™ืŸ(โ€ช.โ€ฌื›ืžื” ื–ืžืŸ ื™ืฉ ืœื˜ื™ื™ืก ืœื‘ืฆืข ืชื™ืงื•ืŸ ืขืœ ืžื ืช ืœืžื ื•ืขโ€ฌ
โ€ซื”ืชืจืกืงื•ืช?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_002.htmlโ€ฌโ€ฌ
โ€ซื–ืจื™ืงืช ื’ื•ืฃโ€ฌ
โ€ซื‘ืœื•ืง ื ื–ืจืง ืžื‘ื ื™ื™ืŸ ื‘ื’ื•ื‘ื” โ€ช ,hโ€ฌื‘ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช โ€ช ,V0โ€ฌื•ื‘ื–ื•ื•ื™ืช โ€ช ฮธโ€ฌื™ื—ืกื™ืช ืœืื•ืคืงโ€ช.โ€ฌโ€ฌ
โ€ซื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ื”ืžื”ื™ืจื•ืช ื”ืกื•ืคื™ืช ืฉืœ ื”ื‘ืœื•ืง ืœื ืชืœื•ื™ื” ื‘ื–ื•ื•ื™ืช ื”ื–ืจื™ืงื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_003.htmlโ€ฌโ€ฌ
โ€ซื–ืจื™ืžื”โ€ฌ
โ€ซื ื”ืจ ื–ื•ืจื ืฆืคื•ื ื” ื‘ืžื”ื™ืจื•ืช โ€ช .Vrโ€ฌื‘ื’ื“ื” ื”ืžืขืจื‘ื™ืช ื ืžืฆื ืื“ื ื”ืžืฉื™ื˜ ืกื™ืจื” ืœืจื•ื—ื‘ ื”ื ื”ืจโ€ช ,โ€ฌื‘ืžื”ื™ืจื•ืช โ€ช Vbโ€ฌื™ื—ืกื™ืช ืœื ื”ืจโ€ช .โ€ฌื”ืื“ื ืžืขื•ื ื™ื™ืŸ ืœื”ื’ื™ืขโ€ฌ
โ€ซืืœ ื”ื’ื“ื” ื”ื ื’ื“ื™ืช ื‘ื“ื™ื•ืง ืžื–ืจื—ื™ืช ืœื ืงื•ื“ืช ืžื•ืฆืื•โ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืจื•ื—ื‘ ื”ื ื”ืจ โ€ช.dโ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌื‘ืื™ื–ื” ื›ื™ื•ื•ืŸ ื”ื•ื ื™ื”ื™ื” ื—ื™ื™ื‘ ืœื”ืฉื™ื˜ ืืช ื”ืกื™ืจื”?โ€ฌ
โ€ซโ€ช .2โ€ฌืžื” ืžื”ื™ืจื•ืช ื”ืกื™ืจื” ื™ื—ืกื™ืช ืœืื“ืžื”?โ€ฌ
โ€ซโ€ช .3โ€ฌื›ืžื” ื–ืžืŸ ืชืืจืš ื“ืจื›ื•?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_004.htmlโ€ฌโ€ฌ
โ€ซืืกื˜ืจื•ืื™ื“โ€ฌ
โ€ซื‘ืงื™ืฅ โ€ช 1989โ€ฌื—ืฆื” ืืกื˜ืจื•ืื™ื“ ) ื”ื•ื ื ืงืจื ื‘ื“ื™ืขื‘ื“ โ€ช (1989FCโ€ฌืืช ืžืกืœื•ืœ ื›ื“ื•"ื” ื‘ืžืจื—ืง ืฉืœ โ€ช km 650,000โ€ฌืžืื—ื•ืจื™ื•โ€ช ,โ€ฌื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืจื“ื™ื•ืก ืžืกืœื•ืœ ื›ื“ื•"ื” ื”ื™ื ื• โ€ช 150โ€ฌืžื™ืœื™ื•ืŸ ืง"ืžโ€ช ,โ€ฌื•ืžืฉืš ื”ื”ืงืคื” ื ืžืฉืš โ€ช 365.25โ€ฌื™ืžื™ืโ€ช .โ€ฌื‘ืขื–ืจืช ื ืชื•ื ื™ื ืืœื” ื—ืฉื‘โ€ช/โ€ฌื™ ื‘ื›ืžื” ื–ืžืŸ ืคืกืคืกโ€ฌ
โ€ซโ€ช 1989FCโ€ฌืืช ื›ื“ื•"ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืžื”ื™ืจื•ืช ื”ืืกื˜ืจื•ืื™ื“ ื”ื•ืขืจื›ื” ื›โ€ช -74014โ€ฌืงืž"ืฉโ€ช .โ€ฌื‘ื”ื ื—ื” ืฉืžืงื•ืจ ื”ืืกื˜ืจื•ืื™ื“ ื”ื•ื ื‘ื—ื’ื•ืจืช ื”ืืกื˜ืจื•ืื™ื“ื™ืโ€ช ,โ€ฌื”ื ืžืฆืืช ื‘ืžืจื—ืง ืฉืœ โ€ช300โ€ฌโ€ฌ
โ€ซืžื™ืœื™ื•ืŸ ืง"ืž ืžื›ื“ื•"ื”โ€ช ,โ€ฌื”ืขืจืšโ€ช/โ€ฌื™ ื‘ื›ืžื” )ื‘ืื—ื•ื–ื™ื( ืฆืจื™ืš ื”ื™ื” ืœืฉื ื•ืช ืืช ื”ืžื”ื™ืจื•ืช ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ืืกื˜ืจื•ืื™ื“ ื›ืš ืฉื”ื™ื™ืชื” ืžืชืจื—ืฉืชโ€ฌ
โ€ซืคื’ื™ืขื”?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_005.htmlโ€ฌโ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืขืœ ืžืขื’ืœโ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืขืœ ืžืขื’ืœ ื‘ืจื“ื™ื•ืก โ€ช .3mโ€ฌื‘ื–ืžืŸ โ€ช t=0โ€ฌื”ื—ืœืงื™ืง ื—ื•ืœืฃ ื“ืจืš ื ืงื•ื“ื” )โ€ช (5,4โ€ฌื‘ื™ื—ืก ืœืจืืฉื™ืช ื”ืฆื™ืจื™ื โ€ช .Oโ€ฌื ืชื•ืŸ ื›ื™ ืžืจื›ื– ื”ืžืขื’ืœ ื ืžืฆื ื‘โ€ช)-โ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™โ€ช:โ€ฌโ€ฌ
โ€ซโ€ช ,(5,7โ€ฌื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ื”ื™ืโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซโ€ช.5โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืžื™ืงื•ื ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืžื”ื™ืจื•ืช ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืชืื•ืฆื” ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ื‘ืื™ื ื˜ืจื•ื•ืœ ื–ืžืŸ โ€ช5โ€ฌืชื—ื•ื ื”ื–ื•ื™ื•ืช ื‘ื• ื ืข ื•ืงื˜ื•ืจ ื”ืžืงื•ืโ€ช.โ€ฌโ€ฌ
โ€ซืชื—ื•ื ื”ื’ื“ืœื™ื ื‘ื• ื ืžืฆื ื•ืงื˜ื•ืจ ื”ืžืงื•ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_006.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžื™ืžื“ โ€ช1โ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืชื• ืฉืœ ื—ืœืงื™ืง ื ืชื•ื ื” ืข"ื™โ€ช:โ€ฌโ€ฌ
โ€ซืงื‘ื•ืขื™ืโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ื‘ื–ืžืŸ โ€ช t=0โ€ฌืžื™ืงื•ื ื”ื—ืœืงื™ืง ื”ื™ื”โ€ฌ
โ€ซื›ืืฉืจโ€ฌ
โ€ซื—ืฉื‘ื• ืืช ืžื™ืงื•ืžื• ื•ืชืื•ืฆืชื• ืฉืœ ื”ื—ืœืงื™ืง ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_007.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžื™ืžื“ โ€ช1โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืœืื—ืจ โ€ช 6โ€ฌืฉื ื™ื•ืช ื ื’ืžืจ ื”ื“ืœืง ื‘ื˜ื™ืœ ื•ื”ื•ื ืžื‘ืฆืข ืชื ื•ืขื” ื‘ืœื™ืกื˜ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื˜ื™ืœ ื ื•ืจื” ื‘ื›ื™ื•ื•ืŸ ืื ื›ื™ ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ืฉืœโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ื’ื•ื‘ื” ื”ืžื™ืจื‘ื™ ืืœื™ื• ื™ื’ื™ืข ื”ื˜ื™ืœ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื›ืžื” ื–ืžืŸ ื™ื—ืœื•ืฃ ืžืจื’ืข ื”ื”ืžืจืื” ื•ืขื“ ื”ื ื—ื™ืชื”?โ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื™ืคื’ืข ื”ื˜ื™ืœ ื‘ืงืจืงืข?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_008.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžื™ืžื“ โ€ช1โ€ฌโ€ฌ
โ€ซื›ื“ื•ืจ ื ื–ืจืง ืื ื›ื™ืช ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื”ื›ื“ื•ืจื™ื ืžื’ื™ืขื™ื ื‘ื• ื–ืžื ื™ืช ืœืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซื—ืฉื‘ื• ืืช ื’ื•ื‘ื” ื”ื‘ื ื™ื™ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื›ืœืคื™ื” ืžืขืœื” ืžื’ื’ ืฉืœ ื‘ื ื™ืŸ ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซโ€ช .โ€ฌืœืื—ืจ โ€ช 2โ€ฌืฉื ื™ื•ืช ืžืคื™ืœื™ื ืžื”ื’ื’ ื›ื“ื•ืจ ื ื•ืกืฃ ื›ืš ืฉืฉื ื™โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_009.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžื”ื™ืจื•ืชื• ืฉืœ ืจืฅ ื‘ืื™ืžื•ืŸ ื ืชื•ื ื” ื‘ื’ืจืฃ ื”ื‘ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ืื•ืชื• ืขื•ื‘ืจ ื”ืจืฅ ื‘ืื™ืžื•ื ื• ?โ€ฌ
โ€ซ)ืžื™ืงื•ื ื”ืจืฅ ื›ืชืœื•ืช ื‘ื–ืžืŸ(โ€ช ,โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ื’ืจืฃ ืฉืœโ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ืฉืœ ื”ื’ื•ืฃ ื‘โ€ช 10-โ€ฌื”ืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืช ?โ€ฌ
โ€ซื›ื ื’ื“ โ€ช.tโ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_011.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซโ€ช ,โ€ฌื”ืกื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจื• ื‘ืžื”ื™ืจื•ืช ื–ื•ื™ืชื™ืช ืงื‘ื•ืขื” โ€ช.โ€ฌโ€ฌ
โ€ซืžืกื” ืžื•ื ื—ืช ื‘ืชื•ืš ื—ืจื•ื˜ ื‘ืขืœ ื–ื•ื™ืช ืจืืฉโ€ฌ
โ€ซืžืงื•ื“ืงื•ื“ ื”ื—ืจื•ื˜โ€ช,โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืื ื”ืžืกื” ื ืžืฆืืช ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ืžืกื” ื•ื”ื—ืจื•ื˜ ื”ื•ืโ€ฌ
โ€ซืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ื”ืžืงืกื™ืžืœื™ืช ื•ื”ืžื™ื ื™ืžืœื™ืช ืขื‘ื•ืจืŸ ื”ืžืกื” ืœื ืชื—ืœื™ืง ื‘ืžืขืœื” ืื• ื‘ืžื•ืจื“ ื”ื—ืจื•ื˜?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_012.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซืื˜ื•ื ื”ืžื™ืžืŸ ืžื•ืจื›ื‘ ืžืคืจื•ื˜ื•ืŸ ืกื˜ื˜ื™ ื•ืืœืงื˜ืจื•ืŸ ืฉืžืงื™ืฃ ืื•ืชื• ื‘ืžื”ื™ืจื•ืช ื’ื‘ื•ื”ื”โ€ช .โ€ฌืžื•ื“ืœ ื‘ื•ืจ ืžืชืืจ ืืช ืชื ื•ืขืช ื”ืืœืงื˜ืจื•ืŸ ืกื‘ื™ื‘ ื”ืคืจื•ื˜ื•ืŸ ื‘ืื•ืคืŸโ€ฌ
โ€ซืื ืœื•ื’ื™ ืœืชื ื•ืขืช ื”ื™ืจื— ืกื‘ื™ื‘ ื›ื“ื•ื”"ื โ€ช -โ€ฌื›ืืฉืจ ืืช ื›ื— ื”ื›ื‘ื™ื“ื” ืžื—ืœื™ืฃ ื”ื›ื— ื”ืืœืงื˜ืจื•ืกื˜ื˜ื™โ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจโ€ช:โ€ฌโ€ฌ
โ€ซืœืคื™ ืžื•ื“ืœ ื”ืื˜ื•ื ืฉืœ ื‘ื•ืจ ื”ืžื”ื™ืจื•ืช ื•ืจื“ื™ื•ืก ื”ืกื™ื‘ื•ื‘ ื‘ืจืžืช ื”ื™ืกื•ื“ ืงืฉื•ืจื™ื ืข"ื™ ื”ืžืฉื•ื•ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจโ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘ื• ืืช ืจื“ื™ื•ืก ื”ืกื™ื‘ื•ื‘ ืฉืœ ื”ืืœืงื˜ืจื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื—ืฉื‘ื• ืžื” ื”ื™ื” ืจื“ื™ื•ืก ื”ืกื™ื‘ื•ื‘ ืื ื‘ืžืงื•ื ื”ื›ื— ื”ืืœืงื˜ืจื•ืกื˜ื˜ื™ ื”ื™ื” ื›ื— ื”ื›ื‘ื™ื“ื” ืื—ืจืื™ ืขืœ ืชื ื•ืขืช ื”ืืœืงื˜ืจื•ืŸโ€ฌ
โ€ซโ€ช.โ€ฌื”ืฉื•ื• ืืช ื”ืจื“ื™ื•ืก ืœื’ื•ื“ืœื” ืฉืœ ืžืขืจื›ืช ื”ืฉืžืฉโ€ช.โ€ฌโ€ฌ
โ€ซื”ืฉืชืžืฉื• ื‘ืงื‘ื•ืขื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_2_201.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซืžื”ื™ืจื•ืช ืฉืœ ื’ื•ืฃ ื ืชื•ื ื” ืข"ื™โ€ฌ
โ€ซ?โ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื•ืงื˜ื•ืจ ื”ืžืงื•ื ืฉืœ ื”ื’ื•ืฃ ืื ื‘ื–ืžืŸ โ€ช t=0โ€ฌื”ื•ื ื”ื™ื” ื‘ืจืืฉื™ืช?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_131.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ฌ
โ€ซื‘ืืชืจ ืกืงื™ ืจื›ื‘ืœ ืžืขืœื” ืืช ื”ื’ื•ืœืฉื™ื ืœืคืกื’ืช ื”ื”ืจโ€ช .โ€ฌื”ืจื›ื‘ืœ ืขืฉื•ื™ ืงืจื•ื ื™ื•ืช ื–ื”ื•ืช ื‘ืขืœื•ืช ืื•ืจืšโ€ฌ
โ€ซืงื‘ื•ืขื” โ€ช.โ€ฌโ€ฌ
โ€ซื‘ืงื˜ืข ืžืกื•ื™ื™ื ื‘ืจื›ื‘ืœ ืฉืื•ืจื›ื• โ€ช ,โ€ฌืขื•ื‘ืจื•ืช ื”ืงืจื•ื ื™ื•ืช ืœื›ื‘ืœ ืื™ื˜ื™ ื™ื•ืชืจ ืฉืžื”ื™ืจื•ืชื•โ€ฌ
โ€ซื”ืชืœื•ื™ื•ืช ื‘ืืžืฆืขื™ืชืŸ ืขืœ ื›ื‘ืœ ื”ื ืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซ* ืฉื™ืžื• ืœื‘ ื›ื™ ื’ื•ื“ืœ ื”ืงืจื•ื ื™ืช ืกื•ืคื™ ืืš ื—ื™ื‘ื•ืจ ื”ืงืจื•ื ื™ืช ืœื›ื‘ืœ ื ืงื•ื“ืชื™โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืื ืžืจื—ืง ื‘ื™ืŸ ืฉืชื™ ืงืจื•ื ื™ื•ืช ื ืฉืืจ ืงื‘ื•ืข ืœืคื ื™ ื•ืื—ืจื™ ืžืขื‘ืจ ืฉืชื™ ื”ืงืจื•ื ื™ื•ืช ืืช ืงื˜ืข ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ื‘ืขโ€ช/โ€ฌื™ ื‘ืืžืฆืขื•ืช ื ืชื•ื ื™ ื”ืฉืืœื” ืืช ื”ืžืจื—ืง ื”ืžื™ื ื™ืžืœื™ ื‘ื™ืŸ ืฉืชื™ ืงืจื•ื ื™ื•ืช ืขื‘ื•ืจื• ืœื ืชืชืจื—ืฉ ื”ืชื ื’ืฉื•ืชโ€ช .โ€ฌื”ื ื—โ€ช/โ€ฌื™ ื›ื™โ€ฌ
โ€ซืžื“ื•ืข ืฆืจื™ืš ืœื”ื ื™ื— ืืช ื–ื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘ื’ื‘ื•ืœโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืงื•ืจื” ืœืžืจื—ืงโ€ฌ
โ€ซื•ื”ืกื‘ืจโ€ช/โ€ฌื™โ€ฌ
โ€ซ? ื”ืกื‘ืจโ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_132.htmlโ€ฌโ€ฌ
โ€ซื–ืจื™ืงื” ืื ื›ื™ืชโ€ฌ
โ€ซืื‘ืŸ ื ื–ืจืงืช ืื ื›ื™ืช ื›ืœืคื™ ืžืขืœื”โ€ช .โ€ฌื‘ื“ืจื›ื” ืžืขืœื” ื”ื™ื ืขื•ื‘ืจืช ืืช ื”ื ืงื•ื“ื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ื‘ื™ืŸ ื”ื ืงื•ื“ื” ื‘ื” ื’ื•ื‘ื” ื”ืื‘ืŸ ืžืงืกื™ืžืœื™ ืœื‘ื™ืŸ ื”ื ืงื•ื“ื” ?โ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืช โ€ช ,โ€ฌื•ืืช ื”ื ืงื•ื“ื”โ€ฌ
โ€ซื”ื ืžืฆืืชโ€ฌ
โ€ซืžื˜ืจื™ื ื’ื‘ื•ื” ื™ื•ืชืจโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_133.htmlโ€ฌโ€ฌ
โ€ซื–ืจื™ืงื” ืื ื›ื™ืชโ€ฌ
โ€ซืื‘ืŸ ื ื–ืจืงืช ืžื”ืงืจืงืข ืื ื›ื™ืช ื›ืœืคื™ ืžืขืœื” ื‘ืžื”ื™ืจื•ืช โ€ช .โ€ฌืื“ื ืขื•ืœื” ื‘ื›ื“ื•ืจ ืคื•ืจื— ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ฌ
โ€ซื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ืžืจื’ืข ื”ื–ืจื™ืงื” ืžืฉื ื” ื”ืื‘ืŸ ืืช ื›ื™ื•ื•ืŸ ืชื ื•ืขืชื”โ€ฌ
โ€ซืโ€ช .โ€ฌืžื ืงื•ื“ืช ืจืื•ืชื• ืฉืœ ืฆื•ืคื” ืขืœ ื”ืงืจืงืข?โ€ฌ
โ€ซโ€ช .โ€ฌื ืชื•ืŸโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื ืงื•ื“ืช ืจืื•ืชื• ืฉืœ ื”ืื“ื ื‘ื›ื“ื•ืจ ื”ืคื•ืจื—?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_134.htmlโ€ฌโ€ฌ
โ€ซื ืคื™ืœื” ื—ื•ืคืฉื™ืชโ€ฌ
โ€ซื›ื“ื•ืจ ืžื•ืฉืœืš ืžื‘ื ื™ืŸ ื‘ื’ื•ื‘ื” โ€ช hโ€ฌื”ื–ืžืŸ ืฉืœื•ืงื— ืœื›ื“ื•ืจ ืœืขื‘ื•ืจ ืืช ื”ืžื—ืฆื™ืช ื”ืฉื ื™ื” ืฉืœ ื”ื‘ื ื™ืŸ ื”ื•ื โ€ช 3โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ื’ื•ื‘ื” ื”ื‘ื ื™ืŸ?โ€ฌ
โ€ซื›ืœืคื™ ืžื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_135.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ื‘ืžื™ืžื“ ืื—ื“โ€ฌ
โ€ซื™ื•ืฆื ืืœืขื“ ืžื‘ืืจ ืฉื‘ืข ื•ื ื•ืกืข ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” โ€ช .โ€ฌื‘ื–ืžืŸ ื›ืœืฉื”ื•ื ื ื•ืกืข ื™ื•ืื‘ ืœืื•ืชื• ื›ื™ื•ื•ืŸ ื•ื ืข ื‘ืชืื•ืฆื” ืงื‘ื•ืขื”โ€ฌ
โ€ซืžื‘ืืจ ืฉื‘ืข ื•ื ืข ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซื™ื•ืื‘ ื ืžืฆื ื‘ืžืจื—ืง ืฉืœโ€ฌ
โ€ซโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ื‘ื–ืžืŸโ€ฌ
โ€ซื‘ื–ืžืŸโ€ฌ
โ€ซืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ืžืจื—ืงื• ืฉืœ ืืœืขื“ ืžื‘ืืจ ืฉื‘ืข ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ? ื‘ื˜ืโ€ช/โ€ฌื™ ืชืฉื•ื‘ืชืš ื‘ืืžืฆืขื•ืช ื ืชื•ื ื™ ื”ืฉืืœื”โ€ช.โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ื™ืชื” ืžื”ื™ืจื•ืชื• ืฉืœ ื™ื•ืื‘ ื‘ื–ืžืŸโ€ฌ
โ€ซ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ื™ื” ืžืจื—ืงื• ืฉืœ ื™ื•ืื‘ ืžื‘ืืจ ืฉื‘ืข ื‘ื–ืžืŸโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื• ืžืจื—ืงื• ืฉืœ ื™ื•ืื‘ ืžื‘ืืจ ืฉื‘ืข ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ? ื‘ื˜ืโ€ช/โ€ฌื™ ืชืฉื•ื‘ืชืš ื‘ืืžืฆืขื•ืช ื ืชื•ื ื™ ื”ืฉืืœื”โ€ช.โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซื”โ€ช .โ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ื’ื•ื“ืœื” ืฉืœ ื”ืžื”ื™ืจื•ืช ืขืœ ืžื ืช ืฉื™ื•ืื‘ ื™ื—ืœื•ืฃ ืขืœ ืคื ื™ ืืœืขื“ ื‘ื–ืžืŸโ€ฌ
โ€ซื•โ€ช .โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ืื™ื›ื•ืชื™ืช ื‘ืื•ืชื• ื”ื’ืจืฃ ืืช ืžืจื—ืงื ืฉืœ ืืœืขื“ ื•ืฉืœ ื™ื•ืื‘ ืžื‘ืืจ ืฉื‘ืข ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_136.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืฉื ื™ ืžื™ืžื“ื™ืโ€ฌ
โ€ซื ืชื•ื ื” ืกืคื™ืžืช ืžืœื—ืžื” ื”ื ืขื” ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 30โ€ฌืงืฉืจื™ื ืœื›ื™ื•ื•ืŸ ืžื–ืจื—โ€ช .โ€ฌืฆื•ืœืœืช ื”ืžืžื•ืงืžืช โ€ช 4โ€ฌืง"ืž ืžืžื ื” ื™ื•ืจื” ืœืขื‘ืจ ื”ืกืคื™ื ื” ื˜ื•ืจืคื“ื• ื”ื ืขโ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 65โ€ฌืงืฉืจื™ืโ€ช .โ€ฌื‘ื ื•ืกืฃโ€ช ,โ€ฌื ืชื•ืŸ ื›ื™ ื”ื–ื™ื”ื•ื™ ืฉืœ ื”ืฆื•ืœืœืช ื”ื™ื” ื‘ื›ื™ื•ื•ืŸ โ€ช 20) 020โ€ฌืžืขืœื•ืช ืœื›ื™ื•ื•ืŸ ืฆืคื•ืŸ ืžื–ืจื—(โ€ฌ
โ€ซื( ืžื” ื–ืžืŸ ื”ืจื™ืฆื” ืฉืœ ื”ื˜ื•ืจืคื“ื• )ื›ืžื” ื–ืžืŸ ื™ืขื‘ื•ืจ ืขื“ ืœืคื’ื™ืขื”(?โ€ฌ
โ€ซื‘( ืžื” ื›ื™ื•ื•ืŸ ื™ืจื™ ื”ื˜ื•ืจืคื“ื•?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_3_137.htmlโ€ฌโ€ฌ
โ€ซืคื™ืกื™ืงื” ื‘ื’ืฉืโ€ฌ
โ€ซื‘ืขืช ื™ืจื™ื“ืช ื’ืฉืโ€ช ,โ€ฌื”ืื ื ืจื˜ื‘ื™ื ื™ื•ืชืจ ื›ืืฉืจ ืจืฆื™ื ืื• ื›ืืฉืจ ื”ื•ืœื›ื™ื?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_001.htmlโ€ฌโ€ฌ
โ€ซื˜ื•ื•ื— ืงืจื‘ื™โ€ฌ
โ€ซื˜ื•ื•ื— ืงืจื‘ื™ ืžื•ื’ื“ืจ ื›ื˜ื•ื•ื— ื‘ื• ืคื’ื– ื”ื ื•ืจื” ื‘ื–ื•ื•ื™ืช ื”ื’ื‘ื”ื” ืžืกื•ื™ืžืช ื™ืคื’ืข ื‘ื›ืœ ืžื˜ืจื” ื”ื ืžืฆืืช ื‘ืชื•ืš ื”ื˜ื•ื•ื—โ€ช ,โ€ฌื›ืœื•ืžืจ ืœื ื™ืงืจื” ืžืฆื‘ ื‘ื• ื”ืคื’ื– ื™ื—ืœื•ืฃโ€ฌ
โ€ซืžืขืœ ืœืžื˜ืจื”โ€ช.โ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื”ื˜ื•ื•ื— ื”ืงืจื‘ื™ ืขื‘ื•ืจ ื”ื ืชื•ื ื™ื ื”ื‘ืื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื™ื ื ื•ืกืคื™ืโ€ช :โ€ฌืžื”ื™ืจื•ืช ืœื•ืขโ€ช 7200 :โ€ฌืงืž"ืฉโ€ฌ
โ€ซื’ื•ื‘ื” ื”ืžื˜ืจื”โ€ช 3.6 :โ€ฌืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซื’ื•ื‘ื” ื ืงื•ื“ืช ื”ื™ืฆื™ืื” ืฉืœ ื”ืคื’ื–โ€ช 1.8 :โ€ฌืžื˜ืจโ€ฌ
โ€ซื”ืคื’ื™ืขื” ื‘ืกื•ืฃ ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื‘ืžืจื›ื– ื”ืžื˜ืจื”โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_002.htmlโ€ฌโ€ฌ
โ€ซื‘ืœื™ืกื˜ื™ืงื” ืขื ื’ืจืจโ€ฌ
โ€ซื ืชื•ืŸ ืชื•ืชื— ื”ื ืžืฆื ื‘ืงืฆื” ืฆื•ืง ื‘ื’ื•ื‘ื” โ€ช m 200โ€ฌื™ื•ืจื” ืื•ืคืงื™ืช ืคื’ื– ื‘ืžืฉืงืœ ืฉืœ โ€ช ,kg 1โ€ฌื‘ืžื”ื™ืจื•ืช ืœื•ืข ืฉืœ โ€ช.m/s 635โ€ฌโ€ฌ
โ€ซื( ืžื” ื”ื˜ื•ื•ื— ืฉืœ ื”ืชื•ืชื— ื‘ื”ื ื—ื” ืฉืื™ืŸ ื’ืจืจ?โ€ฌ
โ€ซื‘( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื˜ื•ื•ื— ื‘ื”ื ื—ื” ืฉื”ืคื’ื– ื ื•ืจื” ื“ืจืš ื”ืื•ื•ื™ืจ )ื”ื ื—ื” ืกื‘ื™ืจื”โ€ช (...โ€ฌื”ืžืคืขื™ืœ ื›ื•ื— ื’ืจืจ ืœืคื™ ื”ื ื•ืกื—ื”โ€ฌ
โ€ซโ€ช ,โ€ฌื›ืืฉืจโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ื”ืฉื’ื™ืื” ื‘ื—ื™ืฉื•ื‘ ื”ื˜ื•ื•ื—โ€ฌ
โ€ซื‘ื”ื ื—ืช ืกืขื™ืฃ ื'? )ื™ืฉ ืœืคืชื•ืจ ื ื•ืžืจื™ืช(โ€ฌ
โ€ซื ื•ืกื—ืช ืขื–ืจโ€ช:โ€ฌโ€ฌ
โ€ซื’( ื‘ืื•ืคืŸ ืื™ื›ื•ืชื™โ€ช ,โ€ฌื›ืืฉืจ ืงื™ื™ื ื’ืจืจโ€ช ,โ€ฌื‘ืื™ื–ื” ืžืงืจื” ื”ื˜ื•ื•ื— ื™ื”ื™ื” ื’ื‘ื•ื”ื” ื™ื•ืชืจโ€ช ,โ€ฌื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 35°โ€ฌืื• โ€ช?55°โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_003.htmlโ€ฌโ€ฌ
โ€ซืžืจื—ืง ื‘ืœื™ืžื”โ€ฌ
โ€ซื ื”ื’ ืžืชืงืจื‘ ืœืจืžื–ื•ืจ ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื( ื ืชื•ืŸ ื›ื™ ื–ืžืŸ ื”ืชื’ื•ื‘ื” ื”ืื ื•ืฉื™ ื”ื•ืโ€ฌ
โ€ซโ€ช ,โ€ฌื›ืืฉืจ "ืœืคืชืข" )ื›ืš ืœืคื—ื•ืช ื”ื•ื ื˜ื•ืขืŸโ€ช (...โ€ฌื”ืชื—ืœืฃ ื”ืื•ืจ ืœื›ืชื•ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช ,โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ื’ืœื’ืœื™ื ืœื›ื‘ื™ืฉ ื”ื•ืโ€ฌ
โ€ซโ€ช .โ€ฌืžื” ื”ื•ื ื”ืžืจื—ืงโ€ฌ
โ€ซืžื”ืฆื•ืžืชโ€ช ,โ€ฌืฉื‘ื• ืฆืจื™ืš ืœื”ื™ืžืฆื ื”ื ื”ื’ ื›ืš ืฉื”ื•ื ื™ืกืคื™ืง ืœืขืฆื•ืจ ืœืคื ื™ ื”ืจืžื–ื•ืจ?โ€ฌ
โ€ซื”ืžื™ื ื™ืžืืœื™โ€ฌ
โ€ซื‘( ื‘ื”ื ื—ื” ืฉื”ืื•ืจ ื”ื›ืชื•ื ื“ื•ืœืง ื‘ืžืฉืš โ€ช 1.5โ€ฌืœืคื ื™ ืฉื”ื•ื ืžืชื—ืœืฃ ืœืื“ื•ืโ€ช ,โ€ฌื•ืฉืจื•ื—ื‘ ื”ืฆื•ืžืช ื”ื™ื โ€ช 25โ€ฌืžื˜ืจื™ืโ€ช ,โ€ฌืžื” ื”ื•ื ื”ืžืจื—ืง ื”ืžืงืกื™ืžืืœื™โ€ฌ
โ€ซืฉื‘ื• ืขืœ ื”ื ื”ื’ ืœื”ื™ืžืฆื ื›ื“ื™ ืฉื™ืกืคื™ืง ืœื—ืฆื•ืช ืืช ื”ืฆื•ืžืช ืœืคื ื™ ืฉื”ืื•ืจ ืžืชื—ืœืฃ ืœืื“ื•ื?โ€ฌ
โ€ซื’( ืคื™ ื›ืžื” ืชืฉืชื ื” ื”ืชืฉื•ื‘ื” ืœืกืขื™ืฃ ื' ื›ืืฉืจ ื™ื•ืจื“ ื’ืฉื ื•ื”ื›ื‘ื™ืฉ ืจื˜ื•ื‘?โ€ฌ
โ€ซ(โ€ฌ
โ€ซ)ืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ื’ืœื’ืœื™ื ืœืืกืคืœื˜ ื”ืจื˜ื•ื‘ ื”ื•ืโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_004.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ืœื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌื ืชื•ื ื” ื”ืžืขืจื›ืช ื”ืžืชื•ืืจืช ื‘ืื™ื•ืจ ื”ื‘ืโ€ช:โ€ฌโ€ฌ
โ€ซื( ืžืฆืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื‘( ืื ื ืชื•ืŸ ื›ื™โ€ฌ
โ€ซื›ืš ืฉื”ืžืขืจื›ืช ืชื™ืฉืืจ ื‘ืžื ื•ื—ื”โ€ฌ
โ€ซโ€ช ,โ€ฌืžื” โ€ช,โ€ฌืชื”ื™ื™ื ื” ืชืื•ืฆื•ืช ื”ื’ื•ืคื™ืโ€ฌ
โ€ซื•ื›ืžื•"ื› ื ืชื•ืŸ ื›ื™ ื”ื’ืœื’ืœื•ืช ื•ื”ื—ื•ื˜ื™ื ื—ืกืจื™ ืžืกื” ื•ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_005.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžื™ืžื“ ืื—ื“โ€ฌ
โ€ซืื•ืคื ื•ืข ื•ืžื›ื•ื ื™ืช ืžืชื—ื™ืœื™ื ืืช ืชื ื•ืขืชื ื‘ โ€ชt=0โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืžื™ืงื•ื ื”ืื•ืคื ื•ืข ืข"ื™ ื”ืžืฉื•ื•ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืžื™ืงื•ื ื”ืžื›ื•ื ื™ืช ืข"ื™ ื”ืžืฉื•ื•ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื(ืžื”ื ื”ืžื™ืงื•ืžื™ื ื•ื”ืžื”ื™ืจื•ื™ื•ืช ื”ื”ืชื—ืœืชื™ื•ืช ืฉืœ ื”ืื•ืคื ื•ืข ื•ื”ืžื›ื•ื ื™ืช?โ€ฌ
โ€ซื‘(ื‘ืื™ื–ื” ื–ืžืŸ ืžืฉื ื” ื”ื•ืคื ื•ืข ืืช ื›ื™ื•ื•ืŸ ื”ื”ืชืงื“ืžื•ืช ?โ€ฌ
โ€ซื’(ืžื”ื•ื ื”ืžืจื—ืง ื‘ื™ืŸ ื”ืžื›ื•ื ื™ืช ืœืื•ืคื ื•ืข ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซื“(ืžืชื™ ื”ื ื ืคื’ืฉื™ื?โ€ฌ
โ€ซื•ืคืชืจื• ื‘ืขื–ืจืชื ืืช ืกืขื™ืคื™ื ื’ ื• ื“โ€ฌ
โ€ซื”(ืฆื™ืจื• ื’ืจืคื™ืช ืืชโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_006.htmlโ€ฌโ€ฌ
โ€ซื ืคื™ืœื” ื—ื•ืคืฉื™ืชโ€ฌ
โ€ซื›ื“ื•ืจ ืžื•ืฉืœืš ืžื‘ื ื™ืŸ ื‘ื’ื•ื‘ื” โ€ช hโ€ฌื”ื–ืžืŸ ืฉืœื•ืงื— ืœื›ื“ื•ืจ ืœืขื‘ื•ืจ ืืช ื”ืžื—ืฆื™ืช ื”ืฉื ื™ื” ืฉืœ ื”ื‘ื ื™ืŸ ื”ื•ื โ€ช 3โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ื’ื•ื‘ื” ื”ื‘ื ื™ืŸ?โ€ฌ
โ€ซื›ืœืคื™ ืžื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_007.htmlโ€ฌโ€ฌ
โ€ซืชืื•ืฆื” ืžืฉืชื ื”โ€ฌ
โ€ซื—ืœืงื™ืง ืžืื™ืฅ ืžืžื ื•ื—ื” ื‘ืชืื•ืฆื”โ€ฌ
โ€ซืžื”ืจืืฉื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืฉืœืื—ืจ โ€ช 3โ€ฌืฉื ื™ื•ืช ื”ื—ืœืงื™ืง ื ืžืฆื ื‘ืžืจื—ืงโ€ฌ
โ€ซื—ืฉื‘ื• ืืช ืžื™ืงื•ื ื”ื—ืœืงื™ืง ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_008.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื‘ื–ื•ื•ื™ืช ืžืขืœ ืฆื™ืจ โ€ช , xโ€ฌืฆื•ืคื” ืžื•ื“ื“ ืืช ื”ืžืจื—ืง ืฉื”ื›ื“ื•ืจ ืคื’ืข ื‘ืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซื›ื“ื•ืจ ื ื–ืจืง ืžื”ืจืืฉื™ืช ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื”ืฆื•ืคื” ืžื“ื“ ืžืจื—ืง โ€ช dโ€ฌืžื”ืจืืฉื™ืชโ€ช .โ€ฌืžืฆืื• ืืช โ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื• ืขื‘ื•ืจื• โ€ช dโ€ฌืžืงืกื™ืžืœื™?โ€ฌ
โ€ซ?โ€ฌ
โ€ซื›ืžื” ืคืชืจื•ื ื•ืช ื™ืฉ ืขื‘ื•ืจโ€ฌ
โ€ซืžืฆืื• ืืช ืžืฉื•ื•ืืช ื”ืžืกืœื•ืœ ืฉืœ ื”ื›ื“ื•ืจโ€ฌ
โ€ซื ืชื•ืŸ ืชืื•ืฆืช ื”ื›ื•ื‘ื“ โ€ช gโ€ฌื›ืœืคื™ ืžื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_009.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื—ืœืงื™ืง ืžื•ื’ื‘ืœ ืœื ื•ืข ืขืœ ืžืขื’ืœ ื‘ืจื“ื™ื•ืก โ€ช.Rโ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืฉื’ื•ื“ืœ ื”ืžื”ื™ืจื•ืช ืฉืœ ื”ื—ืœืงื™ืงโ€ฌ
โ€ซืžืฆืื• ืืช ืžืฉื•ื•ืืช ื”ืžื™ืงื•ื ื•ื”ืชืื•ืฆื” ืฉืœ ื”ื—ืœืงื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_010.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซื—ืœืงื™ืง ืžื•ื’ื‘ืœ ืœื ื•ืข ืขืœ ืžืขื’ืœ ื‘ืจื“ื™ื•ืก โ€ช.Rโ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืฉื’ื•ื“ืœ ื”ืžื”ื™ืจื•ืช ืฉืœ ื”ื—ืœืงื™ืงโ€ฌ
โ€ซืžืฆืื• ืืช ืžืฉื•ื•ืืช ื”ืžื™ืงื•ื ืฉืœ ื”ื—ืœืงื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_011.htmlโ€ฌโ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืขืœ ืžืขื’ืœโ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืขืœ ืžืขื’ืœ ื‘ืจื“ื™ื•ืก โ€ช .3mโ€ฌื‘ื–ืžืŸ โ€ช t=0โ€ฌื”ื—ืœืงื™ืง ื—ื•ืœืฃ ื“ืจืš ื ืงื•ื“ื” )โ€ช (5,4โ€ฌื‘ื™ื—ืก ืœืจืืฉื™ืช ื”ืฆื™ืจื™ื โ€ช.Oโ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™โ€ช:โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ืžืจื›ื– ื”ืžืขื’ืœ ื ืžืฆื ื‘โ€ช ,(5,7)-โ€ฌื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ื”ื™ืโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซโ€ช.5โ€ฌโ€ฌ
โ€ซโ€ช.6โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืžื™ืงื•ื ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืžื”ื™ืจื•ืช ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืชืื•ืฆื” ืฉืœ ื”ื—ืœืงื™ืง ื‘ื›ืœ ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซืœโ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ื‘ื™ืŸโ€ฌ
โ€ซืชื—ื•ื ื”ื–ื•ื™ื•ืช ื‘ื• ื ืข ื•ืงื˜ื•ืจ ื”ืžืงื•ืโ€ช.โ€ฌโ€ฌ
โ€ซืชื—ื•ื ื”ื’ื“ืœื™ื ื‘ื• ื ืžืฆื ื•ืงื˜ื•ืจ ื”ืžืงื•ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_012.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื™ื—ืกื™ืชโ€ฌ
โ€ซืฉืœื’ ื™ื•ืจื“ ื‘ื›ื™ื•ื•ืŸ ืื ื›ื™ ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื ื”ื’ ื ื•ืกืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ ืื•ืคืงื™โ€ช,โ€ฌโ€ฌ
โ€ซื—ืฉื‘ื• ื‘ืื™ื–ื• ื–ื•ื•ื™ืช ื™ืจืื” ื”ื ื”ื’ ืืช ื”ืฉืœื’ ื”ื ื•ืคืœโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_013.htmlโ€ฌโ€ฌ
โ€ซื’ืฉื ืขืœ ืจื›ื‘ืชโ€ฌ
โ€ซืจื›ื‘ืช ื ื•ืกืขืช ืžื–ืจื—ื” ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 15โ€ฌืžื˜ืจ ืœืฉื ื™ื” โ€ช .โ€ฌื˜ื™ืคื•ืช ื’ืฉื ื”ื ื•ืคืœื•ืช ืื ื›ื™ืช ื™ื—ืกื™ืช ืœื›ื“ื•ืจ ื”ืืจืฅโ€ช ,โ€ฌืžืชื•ื•ืช ืขืœ ื–ื’ื•ื’ื™ืช ื—ืœื•ื ื•ืช ื”ืจื›ื‘ืชโ€ฌ
โ€ซืขืงื‘ื•ืชโ€ฌ
โ€ซื”ื ื˜ื•ื™ื•ืช ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30โ€ฌืžืขืœื•ืช ืœืื ืš )ื™ื—ืกื™ืช ืœื ื•ืกืข ื‘ืจื›ื‘ืช(โ€ช.โ€ฌืžืฆืโ€ฌ
โ€ซืโ€ช -โ€ฌืžื”ื• ื”ืจื›ื™ื‘ ื”ืื•ืคืงื™ ืฉืœ ืžื”ื™ืจื•ืช ื”ื˜ื™ืคื•ืช ื™ื—ืกื™ืช ืœืืจืฅ ? ื™ื—ืกื™ืช ืœืจื›ื‘ืช ?โ€ฌ
โ€ซื‘โ€ช -โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื˜ื™ืคื•ืช ื™ื—ืกื™ืช ืœืืจืฅ ? ื™ื—ืกื™ืช ืœืจื›ื‘ืช ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_014.htmlโ€ฌโ€ฌ
โ€ซื™ืจื™ ืงืœื™ืขโ€ฌ
โ€ซืงืœื™ืข ื ื•ืจื” ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืขืœ ืฆื™ืจ ื”โ€ฌ
โ€ซื•ื‘ื–ื•ื™ืชโ€ฌ
โ€ซโ€ช ,โ€ฌืžื˜ื ืง ื”ื ื•ืกืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ ื”ื”ืคื•ืš ืœื›ื™ื•ื•ืŸ ืชื ื•ืขืช ื”ืงืœื™ืขโ€ช.โ€ฌโ€ฌ
โ€ซืชืื•ืฆืช ื”ื›ื‘ื™ื“ื” ื”ื™ืโ€ฌ
โ€ซืโ€ช .โ€ฌื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ื”ืงืœื™ืข ืœื’ื•ื‘ื”ื• ื”ืžืงืกื™ืžืœื™โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื™ื”ื™ื” ืžืจื—ืง ื”ืคื’ื™ืขื” ืฉืœ ื”ืงืœื™ืข ื‘ื™ื—ืก ืœื ืงื•ื“ืช ื”ื™ืจื™โ€ช.โ€ฌโ€ฌ
โ€ซื˜ื ืง ืฉื ื™โ€ช ,โ€ฌื ื™ื™ื—โ€ช ,โ€ฌื™ื•ืจื” ืžืื•ืชื” ื ืงื•ื“ื” ื•ื‘ืื•ืชื” ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื–ื•ื™ืช ื”ื™ืจื™ื” ืฉืœื• ืขืœ ืžื ืช ืฉื˜ื•ื•ื— ื”ืคื’ื™ืขื” ื™ื”ื™ื” ื–ื”ื”?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_015.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื™ื—ืกื™ืชโ€ฌ
โ€ซื”ืขื•ืœื” ื‘ืžื”ื™ืจื•ืช ื‘ืžืขืœื” ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื›ื‘ืฆื™ื•ืจโ€ฌ
โ€ซืžืงืจื•ื ื™ืช ื‘ืื•ืจืšโ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ ืื ืš ืœืžื™ืฉื•ืจ ื”ืžืฉื•ืคืขโ€ช.โ€ฌโ€ฌ
โ€ซื–ื•ืจืงื™ื ื›ื“ื•ืจ ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืฆื ืืช ื’ื•ื“ืœื” ื”ืžืงืกื™ืžืœื™ ืฉืœ ื›ืš ืฉื”ื›ื“ื•ืจ ื™ืคื•ืœ ื—ื–ืจื” ื‘ืชื•ืš ื”ืงืจื•ื ื™ืชโ€ฌ
โ€ซื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_4_016.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื™ื—ืกื™ืชโ€ฌ
โ€ซื ื”ืจ ื–ื•ืจื ื‘ื–ื•ื•ื™ืช ืžืขืœ ืฆื™ืจ โ€ช xโ€ฌืžื”ื™ืจื•ืช ื”ื–ืจื™ืžื”โ€ฌ
โ€ซืฉื”ืกื™ืจื” ืขื•ื‘ืจืช ื‘ืฆื™ืจ โ€ช yโ€ฌื”ื•ื โ€ชLโ€ฌโ€ฌ
โ€ซืจื•ื—ื‘ ื”ื ื”ืจ โ€ช ,dโ€ฌืกื™ืจื” ืฉื˜ื” ืžืฆื“ ื™ืžื™ืŸ ืœืฉืžืืœ ืœืคื™ ื”ื—ืฅ ื”ืื“ื•ื ื ืชื•ืŸ ืฉื”ืžืจื—ืงโ€ฌ
โ€ซืžืฆืื• ืืช ื›ื™ื•ื•ืŸ ืžื”ื™ืจื•ืช ื”ืกื™ืจื” ื‘ื™ื—ืก ืœื ื”ืจ ืื ื ืชื•ื ื” ืžื”ื™ืจื•ืช ื”ืกื™ืจื”โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_021.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžื”ื™ืจื•ืชื• ืฉืœ ืจืฅ ื‘ืื™ืžื•ืŸ ื ืชื•ื ื” ื‘ื’ืจืฃ ื”ื‘ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ืื•ืชื• ืจืฅ ื”ืจืฅ ื‘ืื™ืžื•ื ื• ?โ€ฌ
โ€ซ)ืžื™ืงื•ื ื”ืจืฅ ื›ืชืœื•ืช ื‘ื–ืžืŸ(โ€ช ,โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ื’ืจืฃ ืฉืœโ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ืฉืœ ื”ื’ื•ืฃ ื‘โ€ช 10-โ€ฌื”ืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืช ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืžื” ืฆื‘ืข ื”ืื™ืžื•ื ื™ืช )" ื˜ืจืœื™ื ื’" ( ืฉืœ ื”ืจืฅ ?โ€ฌ
โ€ซื›ื ื’ื“ โ€ช.tโ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_022.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืื‘ืŸ ื ื•ืคืœืช ืžื’ื•ื‘ื” โ€ช 100โ€ฌืžื˜ืจื™ื )ืžืชื—ื™ืœื” ืžืžื ื•ื—ื”(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื” ื•ืžืจื—ืงื” ืžื”ืงืจืงืข ืื—ืจื™ ืฉื ื™ื” ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื–ืžืŸ ื”ื ืคื™ืœื” ื”ื›ื•ืœืœ ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื” ื‘ืจื’ืข ื”ืคื’ื™ืขื” ื‘ืงืจืงืข ?โ€ฌ
โ€ซื‘ืจื’ืข ืขื–ื™ื‘ืช ื”ืื‘ืŸ ื–ื•ืจืงื™ื ื›ืœืคื™ ืžืขืœื” ืกืœืขโ€ช ,โ€ฌื‘ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช ืฉืœ โ€ช 3โ€ฌืž\ืฉโ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื• ื”ื’ื•ื‘ื” ื”ืžื›ืกื™ืžืœื™ ืืœื™ื• ื™ื’ื™ืข ื”ืกืœืข ?โ€ฌ
โ€ซื”โ€ช .โ€ฌืžื™ ื™ื’ื™ืข ืœืงืจืงืข ืงื•ื“ื ื”ืื‘ืŸ ืื• ื”ืกืœืข ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_023.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžื™ืงื•ื ืฉืœ ื—ืœืงื™ืง ื ืชื•ืŸ ืข" ื™ ื•ืงื˜ื•ืจ ื”ืžืงื•ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืช ื”ื—ืœืงื™ืง ื›ืชืœื•ืช ื‘ื–ืžืŸโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื”ืชื—ืœืชื™ืช ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืชืื•ืฆืช ื”ื—ืœืงื™ืง ื›ืชืœื•ืช ื‘ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซื‘ื–ืžืŸโ€ฌ
โ€ซื•โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืขืจื›ื ืฉืœโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_024.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื›ื“ื•ืจ ืฉืœื’ ืžื—ืœื™ืง ื‘ืžื•ืจื“ ื’ื’ ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 400โ€ฌื›ืžื•ืจืื”โ€ช .โ€ฌื’ื•ื‘ื” ื”ื’ื’ โ€ช 14โ€ฌืžื˜ืจื™ื ื•ืžื”ื™ืจื•ืช ืขื–ื™ื‘ืช ื”ื›ื“ื•ืจ ืืช ื”ื’ื’ โ€ช 8โ€ฌืž\ืฉโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืžื”ืงื™ืจ ื™ืคื’ืข ื”ื›ื“ื•ืจ ื‘ืงืจืงืข ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืื“ื ืฉื’ื•ื‘ื”ื• โ€ช 2โ€ฌืžื˜ืจื™ื ืขื•ืžื“ ื‘ืžืจื—ืง โ€ช 6โ€ฌืžื˜ืจื™ื ืžื”ืงื™ืจโ€ช .โ€ฌื”ืื ื”ื•ื ื™ื—ื˜ื•ืฃ ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_025.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืื“ื ืขื•ืžื“ ืขืœ ืงืจื•ื ื™ืช ื”ื ืขื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ืฉืœ โ€ช 9.1โ€ฌืž\ืฉโ€ช .โ€ฌื”ื•ื ืžืขื•ื ื™ื™ืŸ ืœื–ืจื•ืง ื›ื“ื•ืจ ื›ืš ืฉื™ืขื‘ื•ืจ ื“ืจืš ื—ื™ืฉื•ืง ื”ื ืžืฆื โ€ช 4.9โ€ฌืžื˜ืจื™ื ืžืขืœโ€ฌ
โ€ซื”ื ืงื•ื“ื” ืžืžื ื” ื”ื›ื“ื•ืจ ืขื•ื–ื‘ ืืช ื™ื“ื•โ€ช .โ€ฌืœื ืจืง ื–ื”โ€ช ,โ€ฌื”ื•ื ื’ื ืžืขื•ื ื™ื™ืŸ ืฉื”ื›ื“ื•ืจ ื™ืขื‘ื•ืจ ืื•ืคืงื™ืช ื“ืจืš ื”ื—ื™ืฉื•ืงโ€ช .โ€ฌืžื”ื™ืจื•ืช ื–ืจื™ืงืช ื”ื›ื“ื•ืจ )ื™ื—ืกื™ืช ืœืื“ืโ€ฌ
โ€ซื”ื–ื•ืจืง( ื”ื™ื โ€ช 10.8โ€ฌืž\ืฉโ€ช.โ€ฌโ€ฌ
โ€ซื‘ืื™ื–ื” ืžืจื—ืง ืื•ืคืงื™ ืฆืจื™ืš ื”ืื“ื ืœืฉื—ืจืจ ืืช ื”ื›ื“ื•ืจ ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_026.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืฉื—ืงืŸ ื›ื“ื•ืจืกืœ ืžื—ื–ื™ืง ื‘ื™ื“ื• ื›ื“ื•ืจืกืœ ื•ื–ื•ืจืง ืื•ืชื• ืœืกืœ ืžืžืจื—ืง ืฉืœโ€ฌ
โ€ซโ€ช .โ€ฌื’ื•ื‘ื” ื”ืกืœ ื”ื•ืโ€ฌ
โ€ซืžืขืœ ื™ื“ื™ ื”ืฉื—ืงืŸ ื•ื–ื•ื•ื™ืช ื”ื–ืจื™ืงื” ื”ื™ืโ€ฌ
โ€ซ?ืโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื–ืจืงื” ืœืกืœโ€ฌ
โ€ซ)ื‘โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื›ื ื™ืกื” ืœืกืœ? )ื’ื•ื“ืœ ื•ื–ื•ื•ื™ืชโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_033.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ืœืื•ืจืš ืฆื™ืจ โ€ช xโ€ฌืขืœ ืคื™ ื”ื‘ื™ื˜ื•ื™โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืชืื•ืฆืชื• ืฉืœ ื”ื’ื•ืฃ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืช )ื‘โ€ช (t=0 -โ€ฌืฉืœ ื”ื’ื•ืฃโ€ช ,โ€ฌื•ืžื”ื• ืžื™ืงื•ืžื• ื”ื”ืชื—ืœืชื™?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ื›ืœืœื™ ืœืžื”ื™ืจื•ืชื• ืฉืœ ื”ื’ื•ืฃ ื›ืชืœื•ืช ื‘ื–ืžืŸ โ€ช.tโ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื‘ื–ืžืŸ โ€ช? t = 4 secโ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_034.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื‘ื›ื ื™ืกื” ืœื‘ืืจ ืฉื‘ืขโ€ช ,โ€ฌื‘ืจื—ื•ื‘ ืจื’ืจโ€ช ,โ€ฌืฉื ื™ ืจืžื–ื•ืจื™ื ื‘ืจืฆืฃโ€ช .โ€ฌื”ืžืจื—ืง ื‘ื™ื ื™ื”ื ื”ื•ื โ€ช 500โ€ฌืžื˜ืจื™ืโ€ช.โ€ฌโ€ฌ
โ€ซืžื›ื•ื ื™ืช ื ื•ืกืขืช ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ื•ื—ื•ืฆื” ืืช ื”ืจืžื–ื•ืจ ื”ืจืืฉื•ืŸ ื‘ืจื’ืข ืฉื‘ื• ื”ื•ื ืžืชื—ืœืฃ ืœื™ืจื•ืงโ€ช.โ€ฌโ€ฌ
โ€ซื–ืžืŸ ื”ืžื•ืคืข ืฉืœ ื›ืœ ืื—ื“ ืžื”ืจืžื–ื•ืจื™ื ื”ื•ื โ€ช 30โ€ฌืฉื ื™ื•ืชโ€ช) .โ€ฌื ื ื™ื— ืฉืœืจืžื–ื•ืจ ืจืง ืฉื ื™ ืžืฆื‘ื™ืโ€ช -โ€ฌืื“ื•ื ื•ื™ืจื•ืงโ€ช ,โ€ฌื ืชืขืœื ืžื”ืฆื”ื•ื‘ ืฉื‘ื™ื ื™ื”ืโ€ช(.โ€ฌโ€ฌ
โ€ซื”ืจืžื–ื•ืจ ื”ืฉื ื™ ื”ื•ืคืš ื™ืจื•ืง โ€ช 6โ€ฌืฉื ื™ื•ืช ืื—ืจื™ ืฉื”ืจืืฉื•ืŸ ื”ืคืš ื™ืจื•ืงโ€ช.โ€ฌโ€ฌ
โ€ซืžืฆืื• ืืช ืชื—ื•ื ื”ืžื”ื™ืจื•ื™ื•ืช ื”ืงื‘ื•ืขื•ืช ื‘ืงืž"ืฉ )ื‘ืชื—ื•ื ื”ืคื—ื•ืช ืื• ื™ื•ืชืจ ืกื‘ื™ืจ ืฉื‘ื™ืŸ โ€ช 40โ€ฌืœโ€ช 70โ€ฌืงืž"ืฉ( ืฉืขืœ ื”ืžื›ื•ื ื™ืช ืœื ืกื•ืข ื›ื“ื™ ืœืชืคื•ืก "ื’ืœโ€ฌ
โ€ซื™ืจื•ืง"โ€ช ,โ€ฌื›ืœื•ืžืจโ€ช ,โ€ฌืœืขื‘ื•ืจ ืืช ืฉื ื™ ื”ืจืžื–ื•ืจื™ื ื‘ืจืฆืฃโ€ช.โ€ฌโ€ฌ
โ€ซ)ื ื ื™ื— ื‘ืฉืืœื” ื–ื• ืฉื”ืžื›ื•ื ื™ืช ื ืžืฆืืช ืœื‘ื“ ืขืœ ื”ื›ื‘ื™ืฉโ€ช ,โ€ฌื•ื™ื›ื•ืœื” ืœื ืกื•ืข ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”(โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_5_035.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžื”ื™ืจื•ืชื• ืฉืœ ืจืฅ ื‘ืื™ืžื•ืŸ ื ืชื•ื ื” ื‘ื’ืจืฃ ื”ื‘ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ืื•ืชื• ืจืฅ ื”ืจืฅ ื‘ืื™ืžื•ื ื• ?โ€ฌ
โ€ซื›ื ื’ื“ โ€ช.tโ€ฌโ€ฌ
โ€ซ)ืžื™ืงื•ื ื”ืจืฅ ื›ืชืœื•ืช ื‘ื–ืžืŸ(โ€ช ,โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ื’ืจืฃ ืฉืœโ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ืฉืœ ื”ื’ื•ืฃ ื‘โ€ช 10-โ€ฌื”ืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืช ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืฉืจื˜ื˜ ื‘ืื•ืคืŸ ืกื›ืžื˜ื™ ื’ืจืฃ ื”ืžืชืืจ ืืช ืชืื•ืฆืช ื”ืจืฅ ื›ืคื•ื ืง' ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_6_001.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืืœืงื˜ืจื•ืŸ ื‘ืžื”ื™ืจื•ืช โ€ช Vโ€ฌื ื›ื ืก ืœืฉื“ื” ืžื’ื ื˜ื™ โ€ช Bโ€ฌื ืชื•ืŸ ืฉื”ืฉื“ื” ื’ื•ืจื ืœืืœืงื˜ืจื•ืŸ ืœืชืื•ืฆื” ื‘ื ื™ืฆื‘ ืœื›ื™ื•ื•ืŸ ื”ืชืงื“ืžื•ืชื• ื‘ื’ื•ื“ืœโ€ฌ
โ€ซื ืชื•ืŸ ืฉื”ืžืจื—ืง ื‘ืฆื™ืจ โ€ช xโ€ฌืฉื”ืืœืงื˜ืจื•ืŸ ื‘ืขืจ ื‘ืชื•ืš ื”ืฉื“ื” ื”ื•ื โ€ช.dโ€ฌโ€ฌ
โ€ซืžืฆืื• ืืช ื”ืžืจื—ืง ืฉื”ืืœืงื˜ืจื•ืŸ ืขื‘ืจ ื‘ืฆื™ืจ โ€ช.yโ€ฌโ€ฌ
โ€ซืžืฆืื• ืืช ื”ื”ื™ืกื˜ ื‘ื›ื™ื•ื•ืŸ ื”ืชืงื“ืžื•ืชื• ืฉืœ ื”ืืœืงื˜ืจื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_6_002.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื ืชื•ื ื” ืชืื•ืฆืช ื—ื™ื›ื•ืš ืฉืœ ื›ื“ื•ืจ ื‘ืื•ื•ื™ืจโ€ฌ
โ€ซื›ื“ื•ืจ ื ื–ืจืง ื‘ืžื”ื™ืจื•ืช ื‘ื–ื•ื•ื™ืช ืžืขืœ ืฆื™ืจ โ€ชxโ€ฌโ€ฌ
โ€ซืžืฆื• ืืชโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_008.htmlโ€ฌโ€ฌ
โ€ซโ€ชKinematicsโ€ฌโ€ฌ
โ€ซโ€ชA particle leaves the origin at t = 0 with an initial velocity vo = (3.6 m/s)i. It experiences a constant accelerationโ€ฌโ€ฌ
โ€ซโ€ชa = (-1.2 m/s2)i โ€“ (1.4 m/s2)j.โ€ฌโ€ฌ
โ€ซ?โ€ช(a) At what time does the particle reach its maximum x coordinateโ€ฌโ€ฌ
โ€ซ?โ€ช(b) What is the velocity of the particle at this timeโ€ฌโ€ฌ
โ€ซ?โ€ช(c) Where is the particle at this timeโ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_022.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืืฆืŸ ืื•ืœื™ืžืคื™ ืžืงื•ื•ื” ืœืกื™ื™ื ืืช ืจื™ืฆืช โ€ช 5000โ€ฌื”ืžื˜ืจื™ื ื‘ืคื—ื•ืช ืžโ€ช 13 -โ€ฌื“ืงื•ืชโ€ช .โ€ฌืื ืœืื—ืจ โ€ช 11โ€ฌื“ืงื•ืช ืฉืœ ืจื™ืฆื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ื ืฉืืจื• ืœื• โ€ช800โ€ฌโ€ฌ
โ€ซืžื˜ืจื™ื ืœืกื•ืฃ ื”ืžืกืœื•ืœโ€ช ,โ€ฌื•ืชืื•ืฆืชื• ื”ืžืงืกื™ืžืœื™ืช ื”ื™ื โ€ช [m/sec2] 0.2โ€ฌืžื”ื• ื”ื–ืžืŸ ื”ืžื™ื ื™ืžืœื™ ืฉืขืœื™ื• ืœื”ืื™ืฅ ืข" ืž ืœืขืžื•ื“ ื‘ื–ืžืŸ ืฉืœ โ€ช 13โ€ฌื“ืงื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_023.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื—ืœืงื™ืง " ืžืฉื•ื’ืข" ื ืข ืœืื•ืจืš ืฆื™ืจ โ€ช xโ€ฌืขืœ ืคื™ ื”ื‘ื™ื˜ื•ื™โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืชืื•ืฆืชื• ืฉืœ ื”ื’ื•ืฃ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืช )ื‘โ€ช (t=0 -โ€ฌืฉืœ ื”ื’ื•ืฃโ€ช ,โ€ฌื•ืžื”ื• ืžื™ืงื•ืžื• ื”ื”ืชื—ืœืชื™?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ื›ืœืœื™ ืœืžื”ื™ืจื•ืชื• ืฉืœ ื”ื’ื•ืฃ ื›ืชืœื•ืช ื‘ื–ืžืŸ โ€ช.tโ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื‘ื–ืžืŸ โ€ช? [t = 4 [secโ€ฌโ€ฌ
โ€ซื”โ€ช .โ€ฌื’ื•ืฃ ืื—ืจโ€ช ,โ€ฌืœื ืคื—ื•ืช ืžืฉื•ื’ืขโ€ช ,โ€ฌืžืื™ืฅ ืœืื•ืจืš ื”ืฆื™ืจ ืขืœ ืคื™ ื”ื‘ื™ื˜ื•ื™โ€ช:โ€ฌโ€ฌ
โ€ซโ€ช m/sec 1โ€ฌื•ื›ืŸ โ€ช. x(t = 0) = 5 mโ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ืœื”ืขืชืง ื”ื’ื•ืฃโ€ฌ
โ€ซื›ืืฉืจ = )โ€ชv(t = 0โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_024.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื ืชื•ืŸ ื’ืจืฃ ื”ืžืชืืจ ืืช ื”ืขืชืงื• ืฉืœ ื’ื•ืฃ ืœืื•ืจืš ืฆื™ืจ โ€ช.xโ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื”ืžืžื•ืฆืขืช ื‘โ€ช 9 -โ€ฌื”ืฉื ื™ื•ืช ื”ืจืืฉื•ื ื•ืช ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื• ื”ืจื’ืขื™ืช ื‘ื–ืžืŸ โ€ช? t = 3 secโ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ื—ื•ื–ืจ ื”ื’ื•ืฃ ืœื ืงื•ื“ืช ื”ืžื•ืฆื ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืžื•ืฆืขืช ื‘ื™ืŸ ื”ื–ืžืŸ ืฉืžืฆืืช ื‘ืกืขื™ืฃ ื’' ืœโ€ช? t = 15 sec -โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_025.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืกื˜ื•ื“ื ื˜ )ืจ'โ€ช ,โ€ฌืฉื ื‘ื“ื•ื™( ืงื ื‘ื‘ื•ืงืจ ื•ื™ื•ืฆื ืœืชื™ืจื’ื•ืœ ื‘ืคื™ืกื™ืงื” ื‘ืฉืขื” โ€ช) 7:45โ€ฌืฉืืœื” ืชื™ืื•ืจื˜ื™ืช ื‘ืœื‘ื“(โ€ช .โ€ฌื”ืžืจื—ืง ืžื‘ื™ืชื• ืœื›ื™ืชื” ื”ื•ื โ€ช.[m] 1200โ€ฌโ€ฌ
โ€ซื‘ืชื—ื™ืœืช ื“ืจื›ื• ืฆื•ืขื“ ื”ืกื˜ื•ื“ื ื˜ ื”ืขื™ื™ืฃ ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ืฉืœ โ€ช .[m/sec] 1โ€ฌืœืื—ืจ โ€ช [m] 500โ€ฌืžื‘ื™ืŸ ืจ' ื›ื™ ื”ื•ื ืขื•ืžื“ ืœืื—ืจ ื•ืžืื™ืฅ ื‘ืชืื•ืฆื”โ€ฌ
โ€ซืงื‘ื•ืขื” ื‘ืžืฉืš ื›โ€ช 20 -โ€ฌืฉื ื™ื•ืช ืœืžื”ื™ืจื•ืช ื’ื‘ื•ื”ื” ื™ื•ืชืจโ€ช .โ€ฌื‘ื”ื ื—ื” ื›ื™ ืจ' ืžืžืฉื™ืš ืœืœื›ืช ื‘ืื•ืชื” ืžื”ื™ืจื•ืช )ืœืื—ืจ ืฉื”ืื™ืฅ( ืขื“ ืœื›ื™ืชื” ื•ืžื’ื™ืข ื‘ื“ื™ื•ืงโ€ฌ
โ€ซื‘ืฉืขื” โ€ช:8:00โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื”ื™ื™ืชื” ื”ืชืื•ืฆื” ื‘ื” ื ืืœืฅ ืจ' ืœื”ืื™ืฅ ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ื™ื™ืชื” ืžื”ื™ืจื•ืชื• ืœืื—ืจ ืฉื”ืื™ืฅ ?โ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื”ืชื—ืฉื‘ ื‘ื ืกื™ื‘ื•ืชโ€ช ,โ€ฌื”ืขืจืšโ€ช/โ€ฌื™ ืžืชื™ ื”ืœืš ืจ' ืœื™ืฉื•ืŸ ื‘ืœื™ืœื” ืœืคื ื™โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_031.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืชื•ืชื— ืžื•ืคื ื” ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช o37โ€ฌืžืขืœ ื”ืื•ืคืง ื•ืžื”ื™ืจื•ืช ื”ืœื•ืข ืฉืœื• ] โ€ช.v0 = 750 [m/secโ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ื˜ื•ื•ื— ืฉืœ ื”ืชื•ืชื— ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื–ืžืŸ ื”ืžืขื•ืฃ ืฉืœ ื”ืคื’ื– ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื• ืฉืœ ื”ืคื’ื– ื‘ืขืช ื”ืคื’ื™ืขื” ื‘ืงืจืงืข ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื• ื”ื˜ื•ื•ื— ื”ืžืจื‘ื™ ืฉืœ ื”ืชื•ืชื— ื•ื‘ืื™ื–ื• ื–ื•ื•ื™ืช ื”ื•ื ืžื•ืฉื’ ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_032.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืžื˜ื•ืก ื“ื•ืืจ ื‘ืื•ืกื˜ืจืœื™ื” ืฆื•ืœืœ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30โ€ฌืžืขืœื•ืช ืžืชื—ืช ืœืื•ืคืง ื•ื‘ืžื”ื™ืจื•ืช ืฉืœ ] โ€ช .v0 = 100 [m/secโ€ฌื›ืืฉืจ ื”ืžื˜ื•ืก ื ืžืฆื ื‘ืžืจื—ืงโ€ฌ
โ€ซืื•ืคืงื™ ืฉืœ โ€ช [km] 2โ€ฌืžืชื™ื‘ืช ื”ื“ื•ืืจ ืฉืœ ื—ื•ื•ืช ื”ืงื ื’ืจื• ื”ื•ื ืžืคื™ืœ ืืช ื—ื‘ื™ืœืช ื”ื“ื•ืืจ ื”ื ื•ืคืœืช ื‘ื“ื™ื•ืง ื‘ืชื™ื‘ื”โ€ช .โ€ฌื‘ืื™ื–ื” ื’ื•ื‘ื” ืžืขืœ ืœืงืจืงืข ื”ืคื™ืœโ€ฌ
โ€ซื”ืžื˜ื•ืก ืืช ื”ื—ื‘ื™ืœื” ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_033.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื ื”ืจ ื–ื•ืจื ืฆืคื•ื ื” ื‘ืžื”ื™ืจื•ืช ืฉืœ ] โ€ช .v = 1.5[m/secโ€ฌืกื™ืจื” ืฉื˜ื” ื‘ืžื”ื™ืจื•ืช ]โ€ช v = 4[m/secโ€ฌื‘ื ื™ืฆื‘ ืœื ื”ืจ )ืžื–ืจื—ื”(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืกื™ืจื” ื™ื—ืกื™ืช ืœืืจืฅ ? )ืฉื™ืžื• ืœื‘โ€ช :โ€ฌื™ืฉ ืœื”ืฆื™ื’ ืืช ื”ืžื”ื™ืจื•ืช ื›ื•ื•ืงื˜ื•ืจ(โ€ฌ
โ€ซื‘โ€ช .โ€ฌืจื•ื—ื‘ ื”ื ื”ืจ ื”ื•ื โ€ช .[m] 1000โ€ฌื›ืžื” ื–ืžืŸ ื“ืจื•ืฉ ืœืกื™ืจื” ืœื—ืฆื•ืช ืื•ืชื• ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื™ื”ื™ื” ืžืจื—ืง ื”ืกื™ืจื”โ€ช ,โ€ฌื›ืืฉืจ ืชื’ื™ืข ืœื’ื“ื” ื”ืžื–ืจื—ื™ืชโ€ช ,โ€ฌืžื ืงื•ื“ืช ื”ื™ืฆื™ืื” ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืœืื—ืจ ื”ื’ื™ืขื• ืœื’ื“ื” ื”ืžื–ืจื—ื™ืช ืžื™ื”ืจ ืจื‘ ื”ื—ื•ื‘ืœ )ืจ'โ€ช ,โ€ฌืฉื ื‘ื“ื•ื™( ืœื—ื–ื•ืจ ื”ื‘ื™ืชื” ืœืืจื•ื—ืช ื”ืฆื”ืจื™ื™ืโ€ช .โ€ฌืœืฉื ื›ืš ืขืœื™ื• ืœื—ืฆื•ืช ืืช ื”ื ื”ืจ ื‘ื“ืจืšโ€ฌ
โ€ซื”ืงืฆืจื” ื‘ื™ื•ืชืจโ€ช .โ€ฌืœืื™ื–ื• ื–ื•ื•ื™ืช ืขืœื™ื• ืœื”ืคื ื•ืช ืืช ื—ืจื˜ื•ื ื”ืกื™ืจื” ื›ื“ื™ ืœื”ืกืคื™ืง ืœืื›ื•ืœ ืืจื•ื—ื” ื—ืžื” ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_034.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซื›ื“ื•ืจ ื ื–ืจืง ืื ื›ื™ืช ื‘ืžื”ื™ืจื•ืช โ€ชv0โ€ฌื›ืœืคื™ ืžืขืœื”โ€ช ,โ€ฌืžื’ื’ ืฉืœ ื‘ื ื™ื™ืŸ ื‘ื’ื•ื‘ื” ] โ€ช .h [mโ€ฌืœืื—ืจ ื›โ€ช [sec] 2 -โ€ฌืžืคื™ืœื™ื ืžื”ื’ื’ ื›ื“ื•ืจ ื ื•ืกืฃโ€ช .โ€ฌืฉื ื™ ื”ื›ื“ื•ืจื™ืโ€ฌ
โ€ซืžื’ื™ืขื™ื ื‘ื• ื–ืžื ื™ืช ืœืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืื ื™ื“ื•ืข ื›ื™ ] โ€ช v0 = 13 [m/secโ€ฌืžื”ื• ื’ื•ื‘ื” ื”ื‘ื ื™ื™ืŸ ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื”ืขืจืš ื”ืžืงืกื™ืžืœื™ ืฉืœ โ€ช v0โ€ฌืฉืขื‘ื•ืจื• ื™ื›ื•ืœื™ื ืฉื ื™ ื”ื›ื“ื•ืจื™ื ืœื”ื’ื™ืข ื™ื—ื“ ? )ืขืจืš ื–ื” ืื™ื ื• ืชืœื•ื™ ื‘โ€ช(h -โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื• ื”ืขืจืš ื”ืžื™ื ื™ืžืœื™ ืฉืœ โ€ช v0โ€ฌืฉืขื‘ื•ืจื• ื™ื›ื•ืœื™ื ืฉื ื™ ื”ื›ื“ื•ืจื™ื ืœื”ื’ื™ืข ื™ื—ื“ ?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_035.htmlโ€ฌโ€ฌ
โ€ซืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซืขื’ืœื” ืฉืื•ืจื›ื” โ€ช Lโ€ฌื ืขื” ื™ืžื™ื ื” ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” โ€ช .aโ€ฌื›ืืฉืจ ืžื”ื™ืจื•ืช ื”ืขื’ืœื” โ€ช uโ€ฌื ื–ืจืง ื›ื“ื•ืจ ืžื”ืงืฆื” ื”ื™ืžื ื™ ืฉืœ ื”ืขื’ืœื” ื›ืœืคื™ ืžืขืœื” ื‘ืžื”ื™ืจื•ืช โ€ช.voโ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืงืกื™ืžืœื™ืช โ€ช umโ€ฌื”ืžื•ืชืจืช ื›ืš ืฉื”ื›ื“ื•ืจ ื™ื™ืคื•ืœ ื‘ืชื•ืš ื”ืขื’ืœื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช โ€ช voโ€ฌื”ืžืงืกื™ืžืœื™ืช ื”ืžื•ืชืจืช ื›ืš ืฉื”ื›ื“ื•ืจ ื™ื™ืคื•ืœ ื‘ืชื•ืš ื”ืขื’ืœื”?โ€ฌ
โ€ซื’โ€ช .โ€ฌืขื‘ื•ืจ โ€ช voโ€ฌืžืงืกื™ืžืœื™โ€ช ,โ€ฌืžื”ื™ ื”ื“ืจืš ื”ืื•ืคืงื™ืช ืฉืขื•ื‘ืจ ื”ื›ื“ื•ืจ ื‘ื™ื—ืก ืœืืจืฅ ื‘ื–ืžืŸ ื”ืชืขื•ืคื”?โ€ฌ
โ€ซโ€ช07_Kinematics/e_07_8_036.htmlโ€ฌโ€ฌ
โ€ซโ€ชKinematicsโ€ฌโ€ฌ
โ€ซโ€ชYou drive on Interstate 10 from San Antonio to Houston, one-half the time at 35.0 mi/h (= 56.3 km/h) and theโ€ฌโ€ฌ
other half at 55.0 mi/h (= 88.5 km/h).
On the way back you travel one-half the distance at 35.0 mi/h and the other half at 55.0 mi/h. What is your
average speed
(a) from San Antonio to Houston?
(b) from Houston back to San Antonio?
and
(c) for the entire trip?
07_Kinematics/e_07_8_037.html
Kinematics
The position of an object moving in a straight line is given by x = At + Bt2 + Ct3,
where A = 3.0 m/s, B = -4.0 m/s2, and C = 1.0 m/s3.
(a) What is the position of the object at t = 0, 1, 2, 3, and 4 s?
(b) What is the object's displacement between t = 0 and t = 2 s? Between t = 0 and t = 4 s?
(c) What is the average velocity for the time interval from t = 2 to t = 4 s? From t = 0 to t = 3 s?
07_Kinematics/e_07_8_038.html
Kinematics
Two trains, each having a speed of 34 km/h, are headed toward each other on the same straight track. A bird
that can fly 58 km/h flies off the front of one train when they are 102 km apart and heads directly for the other
train. On reaching the other train it flies directly back to the first train, and so forth.
(a) How many trips can the bird make from one train to the other before the trains crash?
(b) What is the total distance the bird travels?
07_Kinematics/e_07_8_039.html
Kinematics
The position of a particle moving along the x axis is given by x = A + Bt3, where A = 9.75 cm and B = 1.50
cm/s3. Consider the time interval t = 2 to t = 3 s and calculate
(a) the average velocity;
(b) the instantaneous velocity at t = 2 s;
(c) the instantaneous velocity at t = 3 s;
(d) the instantaneous velocity at t = 2.5 s;
and
(e) the instantaneous velocity when the particle is midway between its positions at t = 2 and t = 3 s.
07_Kinematics/e_07_8_040.html
Kinematics
For each of the following situations, sketch a graph that is a possible description of position as a function of
time for a particle that moves along the x axis. At t = 1 s, the particle has
(a) zero velocity and positive acceleration;
(b) zero velocity and negative acceleration;
(c) negative velocity and positive acceleration;
(d) negative velocity and negative acceleration.
(e) For which of these situations is the speed of the particle increasing at t = 1 s?
07_Kinematics/e_07_8_041.html
Kinematics
If the position of an object is given by x = (2.0 m/s3)t3, find
(a) the average velocity and the average acceleration between t = 1 and t = 2 s and
(b) the instantaneous velocities and the instantaneous accelerations at t = 1 and t = 2 s.
(c) Compare the average and instantaneous quantities and in each case explain why the larger one is larger.
07_Kinematics/e_07_8_042.html
Kinematics
A train started from rest and moved with constant acceleration. At one time it was traveling at 33.0 m/s, and
160 m farther on it was traveling at 54.0 m/s. Calculate
(a) the acceleration,
(b) the time required to travel the 160 m,
(c) the time required to attain the speed of 33.0 m/s,
and
(d) the distance moved from rest to the time the train had a speed of 33.0 m/s.
07_Kinematics/e_07_8_043.html
Kinematics
At the instant the traffic light turns green, an automobile starts with a constant acceleration of 2.2 m/s2. At the
same instant a truck, traveling with a constant speed of 9.5 m/s, overtakes and passes the automobile.
(a) How far beyond the starting point will the automobile overtake the truck?
(b) How fast will the car be traveling at that instant?
(It is instructive to plot a qualitative graph of x versus t for each vehicle.)
07_Kinematics/e_07_8_044.html
Kinematics
The velocity of a particle moving in the xy plane is given by v = [(6.0m/s2)t - (4.0 m/s3)t2]i + (8.0 m/s)j.
Assume t > O.
(a) What is the acceleration when t = 3 s?
(b) When (if ever) is the acceleration zero?
(c) When (if ever) is the ve¬locity zero?
(d) When (if ever)does the speed equal 10 m/s?
07_Kinematics/e_07_8_045.html
Kinematics
A particle is moving in the xy plane with velocity v(t) = vx(t)i + vy(t)j and acceleration a(t) = ax(t)i + ay(t)j.
By taking the appropriate derivative, show that the magnitude of v can be constant only if axvx + ayvy = 0.
07_Kinematics/e_07_8_046.html
Kinematics
The legal speed limit on a highway is changed from 55 rni/h (= 88.5 km/h) to 65 rni/h (= 104.6 km/h).
How much time is thereby saved on a trip from the Buffalo entrance to the New York City exit of the New
York State
Thruway for someone traveling at the higher speed over this 435-mi (= 700-km) stretch of highway?
07_Kinematics/e_07_8_047.html
Kinematics
A car travels up a hill at the constant speed of 40 km/h and returns down the hill at the speed of 60 km/h.
Calculate the average speed for the round trip.
07_Kinematics/e_07_8_048.html
Kinematics
An iceboat sails across the surface of a frozen lake with constant acceleration produced by the wind.
At a certain instant its velocity is 6.30i - 8.42j in m/s.
Three seconds later the boat is instantaneously at rest.
What is its acceleration during this interval?
07_Kinematics/e_07_8_049.html
Kinematics
A particle moves so that its position as a function of time is
r(t) = Ai + Bt2j + Ctk
where A = 1.0 m, B = 4.0 m/s2, and C = 1.0 m/s.
Write expressions for
(a) its velocity and
(b) its acceleration as functions of time.
(c) What is the shape of the particle's trajectory?
07_Kinematics/e_07_8_050.html
Kinematics
A ball rolls off the edge of a horizontal tabletop, 4.23 ft high. It strikes the floor at a point 5.11 ft horizontally
away from the edge of the table.
(a) For how long was the ball in the air?
(b) What was its speed at the instant it left the table?
07_Kinematics/e_07_8_051.html
Kinematics
You throw a ball from a cliff with an initial velocity of 15 m/s at an angle of 20º below the horizontal. Find
(a) its horizontal displacement and
(b) its vertical displacement 2.3 s later.
07_Kinematics/e_07_8_052.html
Kinematics
Show that the maximum height reached by a projectile is ymax = (vo sin ฯ†)2/2g, where g = 9.8 m/s2 ฯ† is the
angle
between the initial trajectory and the plane of the ground.
07_Kinematics/e_07_8_053.html
Kinematics
A ball rolls off the top of a stairway with a horizontal velocity of magnitude 5.0 ft/so The steps are 8.0 in. high
and 8.0 in. wide.
Which step will the ball hit first?
07_Kinematics/e_07_8_054.html
Kinematics
A person walks up a stalled l5-m-long escalator in 90 s. When standing on the same escalator, now moving, the
person is carried up in 60 s.
How much time would it take that person to walk up the moving escalator?
Does the answer depend on the length of the escalator?
07_Kinematics/e_07_8_055.html
Kinematics
A transcontinental flight at 2700 mi is scheduled to take 50 min longer westward than eastward.
The air speed of the jet is 600 mi/h.
What assumptions about the jet-stream wind velocity, presumed to be east or west, are made in preparing the
schedule?
07_Kinematics/e_07_8_056.html
<!--[if !vml]-->Kinematics
A certain airplane has a speed of 180 milh and is diving at an angle of 27° below the horizontal when a radar
decoy is released.
The horizontal distance between the release point ilnd the point where the decoy strikes the ground is 2300 ft.
(a) How long was the decoy in the air?
(b) How high was the plane when the decoy was released? See the figure below.
<!--[endif]-->
07_Kinematics/e_07_8_057.html
Kinematics
A particle A moves along the line y = d (30 m) with a constant velocity v (v = 3.0 m/s) directed parallel to the positiv
in the figure below. A second particle B starts at the origin with zero speed and constant acceleration a (a = 0.40 m/s
at the same instant that particle A passes the y axis.
What angle ฮธ between a and the positive y axis would result in a collision between these two particles?
07_Kinematics/e_07_8_058.html
Kinematics
A ball is dropped from a height of 39.0 m. The wind is blowing horizontally and imparts a constant acceleration of 1
(a) Show that the path of the ball is a straight line and find the values of R and ฮธ in the figure below.
(b) How long does it take for the ball to reach the ground?
(c) With what speed does the ball hit the ground?
07_Kinematics/e_07_8_059.html
Kinematics
You throw a ball with a speed of 25.3 m/s at an angle of 42.0° above the horizontal directly toward a wall as
shown in the figure below.
The wall is 21.8 m from the release point of the ball.
(a) How long is the ball in the air before it hits the wall?
(b) How far above the release point does the ball hit the wall?
(c) What are the horizontal and vertical components of its velocity as it hits the wall?
(d) Has it passed the highest point on its trajectory when it hits?
07_Kinematics/e_07_8_060.html
Kinematics
A projectile is fired from the surface of level ground at an angle ฯ†o above the horizontal.
(a) Show that the elevation angle ฮธ of the highest point as seen from
the launch point is related to ฯ†o by tan ฮธ = (1/2) tan ฯ†o.
(b) Calculate ฮธ for ฯ†o = 45°.
07_Kinematics/e_07_8_061.html
โ€ซื—ืฅ ื•ืžื˜ืจื”โ€ฌ
โ€ซืงืฉืช ื™ื•ืจื” ื—ืฅ ืœืขื‘ืจ ืชืคื•ื— ื‘ืžืจื—ืง ื™ื“ื•ืข ื•ื‘ื’ื•ื‘ื” ื™ื“ื•ืขโ€ฌ.
โ€ซื‘ืื™ื–ื• ื–ื•ื™ืช ืขืœื™ื• ืœื›ื•ื•ืŸ ืืช ื”ืงืฉืช ืื ื‘ืจื’ืข ื”ื™ืจื™ื™ื” ื”ืชืคื•ื— ื”ื—ืœ ื‘ื ืคื™ืœื” ื—ื•ืคืฉื™ืชโ€ฌ
10_Newton/e_10_1_011.html
weight of a passenger
A plane takes off with the acceleration
passenger ?
10_Newton/e_10_1_012.html
at the angle
to the horizon. What is the weight of the 75 kg
Inertial and noninertial reference frames
What should be the length of the day on Earth to compensate the gravity at the equator ?
10_Newton/e_10_1_013.html
Inertial and noninertial reference frames
A body starts moving with the velocity from the center of the rotating disk (angular velocity
external forces. Describe the motion from the point of view of the rotating observer.
). There are no
10_Newton/e_10_1_014.html
Inertial and noninertial reference frames
What is the weight of a standing 1000 kg car on the equator ? What is its weight if it is moving in the east
direction with the velocity 300 km/hour ?
10_Newton/e_10_1_015.html
Inertial and noninertial reference frames
A biker enters a quarter-circle turn of the radius
body and the vertical ?
with the velocity . What is the angle between the biker's
10_Newton/e_10_1_016.html
Inertial and noninertial reference frames
A body hangs on a rope from the ceiling in a standing train. The train starts moving with the acceleration .
What is the angle between the rope and the vertical ?
10_Newton/e_10_1_017.html
Inertial and noninertial reference frames
A body hangs on a rope from the ceiling in a rotating (angular velocity
center is . What is the angle between the rope and the vertical ?
) cell. The distance from the rotation
10_Newton/e_10_1_018.html
Inertial and noninertial reference frames
A horizontal carousel rotates with the angular velocity
?
. What is the weight of a person who sits at the radius
10_Newton/e_10_1_019.html
Inertial and noninertial reference frames
A project of a space station suggests rotation in order to produce artificial gravity. If the diameter of the station
is 20 m, what should be the rotation period in order to produce the gravity equivalent to
?
10_Newton/e_10_1_020.html
Inertial and noninertial reference frames
A body is moving along
axis with constant velocity
in the inertial (standing) frame. Write down
and
in the rotating frame. What is the direction of acceleration as a function of time in the rotating
frame ?
10_Newton/e_10_1_021.html
Inertial and noninertial reference frames
A body falls with the velocity
Newton law in its frame.
(because of the air drag force). Write down the second
10_Newton/e_10_1_022.html
Inertial and noninertial reference frames
A river flows from the north to the south in the northern hemisphere at the latitude . The flow velocity is
the river width is . What is the difference of the water level at the western and eastern coasts ? (Hint:
Coriolis.)
and
10_Newton/e_10_1_031.html
Particle dynamics, Newton laws
A particle is moving so that
,
, where , , and
are constants. Find the force.
10_Newton/e_10_1_032.html
Particle dynamics, Newton laws
A body (mass
velocity. Find
) starts falling. The air friction force is
and
.
, where
and
is the body
) is thrown horizontally with the initial velocity . The air friction force is
and is the body velocity. Find
and
.
,
10_Newton/e_10_1_033.html
Particle dynamics, Newton laws
A body (mass
where
10_Newton/e_10_1_034.html
Particle dynamics, Newton laws
Force
acts on a particle (mass
) which is initially at rest. Find
and
.
10_Newton/e_10_1_035.html
Particle dynamics, Newton laws
At high speeds the air drag force (friction) is
velocity . Find
and
.
. A body is falling vertically in the air with the initial
10_Newton/e_10_1_036.html
Particle dynamics, Newton laws
A charged particle (charge , mass
) is accelerated by the electric field
. Find the trajectory.
10_Newton/e_10_1_037.html
Particle dynamics, Newton laws
A charged particle moves with constant velocity
(
- magnetic field). Find the electric field.
10_Newton/e_10_1_038.html
Particle dynamics, Newton laws
A charged particle (mass , charge , velocity ) enters a cylinder with the length . The entry point is at the
cylinder axis, and the particles enters at the angle
to the axis. There is a homogeneous magnetic field along
the axis inside the cylinder. At what distance from the axis the particle leaves the cylinder ?
10_Newton/e_10_1_039.html
Particle dynamics, Newton laws
A charged particle (mass
an electric field
where
, charge ) is at rest in an homogeneous magnetic field
. Suddenly, at
is switched on. The electric field is suddenly switched off at
,
. Describe the motion of the particle. What is its final energy ?
10_Newton/e_10_1_040.html
Particle Dynamics, Newton's Laws
A particle of mass
is at rest on top of a sphere. Suddenly it starts to slide.
(a) At which point it will leave the sphere?
(b) What is its velocity at that point?
10_Newton/e_10_1_041.html
Newton Laws
For the setting in the figure find the tension in each rope.
10_Newton/e_10_1_042.html
Newton Laws
The Atwood's Machine is composed of two unequal masses
from frictionless and massless pulley.
hunging on inflexible string
What is the tension in the string? If the string was hunging from the ceiling was is the mass that it can carry?
10_Newton/e_10_1_043.html
Particle Dynamics and Newton's Laws
A body free-fall in a presence of a drag force in the form
where
.
,
1. Draw the force diagram for the body during the free-fall.
2. Find the equation of motion of the body.
3. What is the terminal velocity, , of the body. The terminal velocity reached when the body is in
equilibrium.
4. Find the body velocity as a function of time. Assume
.
10_Newton/e_10_1_044.html
Particle Dynamics and Newton's Laws
On a circular hoop with radius there is a bead that can move freely without friction.
The hoop is rotated in a constant frequency
around the horizontal axis.
1. What are the forces that act on the bead in equilibrium.
2. Find the angle in that state as a function of .
3. What should be in order to move the bead to the center of the hoop? can the bead move beyond the center
of the hoop?
10_Newton/e_10_1_045.html
Particle Dynamics and Newton's Laws
A particle with mass
move in a magnetic field
Its initial velocity is
.
Find curvature radius.
10_Newton/e_10_1_046.html
Particle Dynamics and Newton's Laws
, where
.
โ€ซโ€ชA massโ€ฌโ€ฌ
โ€ซโ€ชis placed inside an upside down cone with an opening angle of .โ€ฌโ€ฌ
โ€ซโ€ชThe cone is rotated around it's symmetry axis in a constant angular velocity .โ€ฌโ€ฌ
โ€ซโ€ชThe friction coefficient between the mass and the cone isโ€ฌโ€ฌ
โ€ซโ€ชand the mass is at heightโ€ฌโ€ฌ
โ€ซโ€ชfrom the head of theโ€ฌโ€ฌ
โ€ซโ€ชcone.โ€ฌโ€ฌ
โ€ซ?โ€ชWhat are the maximal and minimal angular velocities in which the mass will stay in that heightโ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_047.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื›ื•ื— ืฉืœ โ€ช 20โ€ฌื ื™ื•ื˜ื•ืŸ ืคื•ืขืœ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30โ€ฌืžืขืœื•ืช ืžืขืœ ืฆื™ืจ โ€ช xโ€ฌืขืœ ื’ื•ืฃ ืฉืžืกืชื• โ€ช 4โ€ฌืง"ื’โ€ช .โ€ฌื”ื’ื•ืฃ ืžื•ื ื— ืขืœ ืžืฉื˜ื— ื—ืœืงโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ื’ื•ืฃโ€ฌ
โ€ซื‘โ€ช .โ€ฌืชื•ืš ื›ืžื” ื–ืžืŸ ื™ืขื‘ื•ืจ ื”ื’ื•ืฃ ืžืจื—ืง ืฉืœ โ€ช 10โ€ฌืžื˜ืจ ืื ื”ืชื—ื™ืœ ืžืžื ื•ื—ื”โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_048.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืฉืฉื•ืงืœ โ€ช 20โ€ฌืง"ื’ ื ืžืฆื ืขืœ ืžืฉื˜ื— ืžืฉื•ืคืข ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30โ€ฌืžืขืœื•ืช ื•ืœืœื ื—ื™ื›ื•ืšโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื’ื•ื“ืœื• ืฉืœ ื”ื›ื•ื— ืฉื”ืžื™ืฉื•ืจ ืžืคืขื™ืœ ืขืœ ื”ื’ื•ืฃโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_049.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืฉืชื™ ืžืฉืงื•ืœื•ืช ืชืœื•ื™ื•ืช ืข"ื™ ื—ื‘ืœ ืขืœ ื’ืœื’ืœืช ื—ืกืจืช ื—ื™ื›ื•ืš ื•ืžืกื”โ€ช .โ€ฌืžืฉืงื•ืœืช ืื—ืช ืฉื•ืงืœืช โ€ช 10โ€ฌืง"ื’ ื•ื”ืฉื ื™ื™ื” โ€ช 20โ€ฌืง"ื’โ€ช .โ€ฌื‘ื–ืžืŸ โ€ช t = 0โ€ฌืžืงื ื™ืโ€ฌ
โ€ซืœืžืฉืงื•ืœืช ืฉืœ ื”โ€ช 10 -โ€ฌืง"ื’ ืžื”ื™ืจื•ืช ืฉืœ โ€ช 5โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื” ื›ืœืคื™ ืžื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืชื™ ืชื—ื–ื•ืจ ื”ืžืฉืงื•ืœืช ืฉืœ ื”โ€ช 10 -โ€ฌืง"ื’ ืœื ืงื•ื“ืช ื”ื”ืชื—ืœื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื” ื‘ืจื’ืข ื”ื—ื–ืจื”?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ืžืจื—ืง ื”ืžืงืกื™ืžืœื™ ืฉื”ืžืฉืงื•ืœืช ืชื’ื™ืข ืžืชื—ืช ืœื ืงื•ื“ืช ื”ื”ืชื—ืœื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_050.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืฉื ื™ ืื ืฉื™ื ืจื•ืฆื™ื ืœืžืชื•ื— ื—ื‘ืœ ืฉืื•ืจื›ื• โ€ช 20โ€ฌืžื˜ืจ ื›ื“ื™ ืœื”ืจื™ื ืžืฉื ืฉืœ โ€ช 1โ€ฌืง"ื’ ื”ืชืœื•ื™ ื‘ืžืจื›ื–โ€ช.โ€ฌโ€ฌ
โ€ซื”ืื ืฉื™ื ื™ื›ื•ืœื™ื ืœื”ืคืขื™ืœ ื›ื•ื— ืžืฉื™ื›ื” ืžืงืกื™ืžืœื™ ืฉืœ โ€ช 300โ€ฌื ื™ื•ื˜ื•ืŸ ื•ื’ื•ื‘ื” ื”ื™ื“ื™ื™ื ืฉืœื”ื ืžื”ืจื™ืฆืคื” ื”ื™ื ื• โ€ช 1โ€ฌืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ื™ื”ื™ื” ื’ื•ื‘ื” ื”ืžืฉื ืžื”ืจื™ืฆืคื”? ืžืฆืื• ื“ืจื›ื™ื ื™ื•ืชืจ ืืคืงื˜ื™ื‘ื™ื•ืช )ื‘ืขื–ืจืช ืื•ืชื ื”ืืžืฆืขื™ื(?โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_051.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื˜ื•ืŸโ€ฌ
โ€ซืงืจื•ืŸ ื ื•ืกืข ื‘ืžื”ื™ืจื•ืช โ€ช 40โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื”โ€ช .โ€ฌืขืœ ื”ืงืจื•ืŸ ืžื•ื ื—ืช ืงื•ืคืกืโ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ืงื•ืคืกื ืœืงืจื•ืŸ ื”ื™ื ื•โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ื”ืžืจื—ืง ื”ืžื™ื ืžืœื™ ืœืคื ื™ ื ืงื•ื“ืช ื”ืขืฆื™ืจื” ืฉื”ืงืจื•ืŸ ื—ื™ื™ื‘ ืœื”ืชื—ื™ืœ ื•ืœื”ืื˜ )ื‘ืชืื•ืฆื” ืงื‘ื•ืขื”( ืื ื‘ืจืฆื•ื ื• ืœืขืฆื•ืจ ืžื‘ืœื™ ืฉื”ืงื•ืคืกื ืชื—ืœื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_052.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ื ืžืฆื ืขืœ ืžื™ืฉื•ืจ ื‘ืขืœ ื–ื•ื•ื™ืช ืžืฉืชื ื” โ€ช .โ€ฌื‘ื”ืชื—ืœื”โ€ฌ
โ€ซืโ€ช .โ€ฌื‘ืื™ื–ื” ื–ื•ื•ื™ืช ื”ื’ื•ืฃ ื™ืชื—ื™ืœ ืœื”ื—ืœื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซืžื” ืชื”ื™ื” ืชืื•ืฆืชื• ?โ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืขื‘ื•ืจ ื”ื–ื•ื•ื™ื•ืชโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซื ืชื•ืŸโ€ฌ
โ€ซื•ืœืื˜ ืœืื˜ ืžื’ื“ื™ืœื™ื ืืช ื”ื–ื•ื•ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_053.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื˜ื•ืŸโ€ฌ
โ€ซืงืจื•ื ื™ืช ื ื•ืกืขืช ื™ืžื™ื ื” ื‘ืชืื•ืฆื” โ€ช .aโ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื”ืชืื•ืฆื” โ€ช aโ€ฌื›ื“ื™ ืฉื’ื•ืฃ ืฉืฆืžื•ื“ ืœืงืจื•ื ื™ืช ืžื™ืžื™ืŸ )ืจืื” ืฆื™ื•ืจ( ืœื ื™ื—ืœื™ืงโ€ช .โ€ฌื‘ื˜ื ืืช ื”ืชืฉื•ื‘ื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘ืขื–ืจืชโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_054.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื˜ื•ืŸโ€ฌ
โ€ซืงืœื™ืข ืจื•ื‘ื” ืฉืžืกืชื• โ€ช 5โ€ฌื’ืจื ื™ื•ืฆื ืžืœื•ืข ืฉืœ ืงื ื” ืจื•ื‘ื” ืฉืื•ืจื›ื• โ€ช 50โ€ฌืก"ืž ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 800โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื”โ€ช .โ€ฌืื ื”ืชืื•ืฆื” ื‘ืงื ื” ืงื‘ื•ืขื” ืžื”โ€ฌ
โ€ซื’ื•ื“ืœื• ืฉืœ ื”ื›ื•ื— ืฉืคื•ืขืœ ืขืœ ื”ืงืœื™ืข ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_055.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžืคื™ืœื™ื ื’ื•ืฃ ืฉืžืชื—ื™ืœ ืžืžืฆื‘ ืžื ื•ื—ื”โ€ช .โ€ฌื›ื•ื— ื”ืชื ื’ื“ื•ืช ื”ืื•ื™ืจ ื”ื™ื ื• โ€ช:โ€ฌโ€ฌ
โ€ซืžืฆื ืืช ื”ืžื”ื™ืจื•ืช )โ€ช ,v(tโ€ฌื”ืชืื•ืฆื” )โ€ช ,a(tโ€ฌื•ืืช ื”ื›ื•ื— )โ€ช .F(tโ€ฌืžื” ื™ืงืจื” ืœื›ื•ื— ื›ืืฉืจโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_056.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืžื—ืœื™ืง ืขืœ ืคื ื™ ืžืฉื˜ื— ื‘ืงื• ื™ืฉืจ ื‘ื›ื™ื•ื•ืŸ ืฆื™ืจ โ€ช .xโ€ฌืžื™ืงื•ืžื• ื”ื”ืชื—ืœืชื™ )ื‘ื–ืžืŸ โ€ช (t = 0โ€ฌื”ื•ื โ€ช x = 0โ€ฌื•ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืช ื”ื™ืโ€ฌ
โ€ซื ืขืฆืจ ืœืื—ืจ ืฉืขื‘ืจ ืžืจื—ืง โ€ช Lโ€ฌื›ืชื•ืฆืื” ืžื”ื—ื™ื›ื•ืš ืขื ื”ืžืฉื˜ื—โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื›ืžื” ื–ืžืŸ ืœืงื— ืœื’ื•ืฃ ืœืขื‘ื•ืจ ืืช ื”ืžืจื—ืง โ€ช Lโ€ฌืขื“ ืฉื ืขืฆืจ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช.โ€ฌื”ืจืื™ ืฉืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื ืชื•ืŸ ืขืœ ื™ื“ื™ ื”ื‘ื™ื˜ื•ื™โ€ฌ
โ€ซโ€ช .โ€ฌื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_1_057.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซโ€ช.โ€ฌืœืจื•ืข ืžื–ืœื• ืฉืœ ื”ื ื”ื’โ€ฌ
โ€ซืžื›ื•ื ื™ืช ื ื˜ื•ืกืขืช ื‘ืœื™ืœื” ื—ืจืคื™ ื•ืžื’ื™ืข ืœืขื™ืงื•ืœ ื‘ื›ื‘ื™ืฉืฅ ืจื“ื™ื•ืก ื”ืขื™ืงื•ืœ ื”ื•ื โ€ช 320โ€ฌืžื˜ืจ ื•ื–ื•ื•ื™ืช ื”ื˜ื™ื™ืช ื”ื›ื‘ื™ืฉ ื”ื™ืโ€ฌ
โ€ซื”ื›ื‘ื™ืฉ ืžื›ื•ืกื” ืฉื›ื‘ืช ืงืจื—โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื”ื ื—ื” ืฉืื™ืŸ ื—ื™ื›ื•ืš ื‘ื™ืŸ ืฆืžื™ื’ื™ ื”ื’ืœื’ืœื™ื ื•ื”ื›ื‘ื™ืฉโ€ช ,โ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ืžื”ื™ืจื•ืช ื”ืžื›ื•ื ื™ืช ืขืœ ืžื ืช ืฉืœื ืชื—ืœื™ืง ื‘ืขื™ืงื•ืœ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื™ืงืจื” ืื ืžื”ื™ืจื•ืช ื”ืžื›ื•ื ื™ืช ืชื”ื™ื” ื’ื‘ื•ื”ื” ืžื”ืขืจืš ื”ืžื—ื•ืฉื‘ ื‘ืกืขืฃ ื'โ€ช ,โ€ฌื•ืžื” ื™ืงืจื” ืื ื”ืžื”ื™ืจื•ืช ืชื”ื™ื” ื ืžื•ื›ื” ืžืขืจืš ื–ื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_058.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซืจื•ื˜ื•ืจ ืฉืœ ืžืกื•ืง ืจื“ื™ื•ืกื• โ€ช 10โ€ฌืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ืฉืœื• ื›ื“ื™ ืฉืงืฆื” ื”ืจื•ื˜ื•ืจ ื™ื’ื™ืข ืœืžื”ื™ืจื•ืช ื”ืงื•ืœ )โ€ช 1224โ€ฌืง"ืž ืœืฉืขื”(โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžืฆื ืืช ื”ืชื“ื™ืจื•ืช ื•ื–ืžืŸ ื”ืžื—ื–ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืžื”ื™ืจื•ืชื” ื”ืงื•ื•ื™ืช ืฉืœ ื ืงื•ื“ื” ื”ื ืžืฆืืช ื‘ืืžืฆืข ืœื”ื‘ ื”ืจื•ื˜ื•ืจ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_1_059.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซืจื•ืฆื™ื ืœืกื•ื‘ื‘ ืื‘ืŸ ืขืœ ื—ื•ื˜ โ€ช .โ€ฌืžืกืช ื”ืื‘ืŸ โ€ช 10โ€ฌื’ืจื ื•ืื•ืจืš ื”ื—ื•ื˜ โ€ช 40โ€ฌืก"ืžโ€ช .โ€ฌื”ืžืชื™ื—ื•ืช ื”ืžืงืกื™ืžืœื™ืช ื”ืžื•ืชืจืช ืœืคื ื™ ืงืจื™ืขื” ื”ื™ื ื” โ€ช 160โ€ฌื ื™ื•ื˜ื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืงืกื™ืžืœื™ืช ื”ืžื•ืชืจืช ? ืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_001.htmlโ€ฌโ€ฌ
โ€ซื ืคื™ืœื” ืžื›ื“ื•ืจ ืคื•ืจื—โ€ฌ
โ€ซื˜ื•ื ื•ื’'ืจื™ ื ืžืฆืื™ื ื‘ื›ื“ื•ืจ ืคื•ืจื— ื”ื™ื•ืจื“ ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ืฉืœ โ€ช .1.88m/sโ€ฌืกื”"ื› ืžืกืชื ื•ืžืกืช ื”ื›ื“ื•ืจ ืคื•ืจื— ื”ื™ื โ€ช .kg 1080โ€ฌืขืœ ื”ื›ื“ื•ืจ ืคื•ืขืœโ€ฌ
โ€ซโ€ช .โ€ฌื‘ืจื’ืข ืžืกื•ื™ื ื–ื•ืจืง ื˜ื•ื ืืช ื’'ืจื™ ืžื”ื›ื“ื•ืจ )ืžืกื™ื‘ื•ืชื™ื•โ€ฌ
โ€ซื›ื•ื— ืขื™ืœื•ื™ ืงื‘ื•ืข ืฉืœ โ€ช 10.3kNโ€ฌื›ืœืคื™ ืžืขืœื”โ€ช ,โ€ฌื‘ื ื•ืกืฃ ืžืคืขื™ืœ ื”ืื•ื•ื™ืจ ื›ื•ื— ื’ืจืจโ€ฌ
โ€ซื”ื•ืโ€ช ,(...โ€ฌื›ืืฉืจ ืžืกืชื• ื”ื™ื โ€ช .72.5kgโ€ฌื‘ืจื’ืข ื–ื” ืžืฉืชื ื” ืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจโ€ช ,โ€ฌืื•ืœื ื‘ื’ืœืœ ื”ื’ืจืจ ืžืชื™ื™ืฆื‘ืช ื”ืžื”ื™ืจื•ืช )ืœืื—ืจ ื–ืžืŸ ืžืกื•ื™ื( ืขืœ ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืงื‘ื•ืขื” ื—ื“ืฉื”โ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจ ื›ืืฉืจ ื”ื•ื ืžื’ื™ืข ืœืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_002.htmlโ€ฌโ€ฌ
โ€ซืคื•ืขืœ ื‘ืงืจื•ื ื™ืชโ€ฌ
โ€ซื ืชื•ืŸ ืคื•ืขืœ ื”ื™ื•ืฉื‘ ื‘ืงืจื•ื ื™ืช ื”ืชืœื•ื™ื” ืขืœ ื’ื‘ื™ ื—ื‘ืœ ื”ืขื•ื‘ืจ ื“ืจืš ื’ืœื’ืœืช )ื—ืกืจืช ื—ื™ื›ื•ืš ืฉืžื•ืชืงื ืช ื‘ืชืงืจื”( ื•ื™ื•ืจื“ ื—ื–ืจื” ืœื™ื“ื™ื• ืฉืœ ื”ืคื•ืขืœโ€ช .โ€ฌืžืกืชโ€ฌ
โ€ซื”ืงืจื•ื ื™ืช ื•ื”ืคื•ืขืœ ื”ื™ื โ€ช.95kgโ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžืฆืโ€ช/โ€ฌื™ ื‘ืื™ื–ื” ื›ื•ื— ืฆืจื™ืš ืœืžืฉื•ืš ื”ืคื•ืขืœ ืืช ื”ื—ื‘ืœ ื›ื“ื™ ืœืขืœื•ืช ืœืžืขืœื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืžืฆืโ€ช/โ€ฌื™ ื‘ืื™ื–ื” ื›ื•ื— ืฆืจื™ืš ืœืžืฉื•ืš ื”ืคื•ืขืœ ืืช ื”ื—ื‘ืœ ื›ื“ื™ ืœืขืœื•ืช ืœืžืขืœื” ื‘ืชืื•ืฆื” ืฉืœ โ€ช.1.3m/s2โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_003.htmlโ€ฌโ€ฌ
โ€ซืžืฆื™ืืช ืžืกื™ืœืช ืชื ื•ืขืชื• ืฉืœ ื”ื’ื•ืฃ ื“ืจืš ืฉื“ื” ื›ื•ื— ื ืชื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ื ืข ื“ืจืš ืฉื“ื” ื›ื•ื— ืฉืžืฉื•ื•ืืชื• ื”ื™ืโ€ฌ
โ€ซื”ื’ื•ืฃ ื ื›ื ืก ืœืฉื“ื” ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืžืกื™ืœืช ืชื ื•ืขืชื• ืฉืœ ื”ื’ื•ืฃ ื“ืจืš ื”ืฉื“ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ืžืกื™ืœื” ื–ื• )โ€ช.(.Mathematica , Matlab etcโ€ฌโ€ฌ
โ€ซโ€ช .3โ€ฌืžื” ื™ื”ื™ื” ืžื™ืงื•ื ื”ื’ื•ืฃ ืœืื—ืจ ื—ืฆื™ ื“ืงื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_004.htmlโ€ฌโ€ฌ
โ€ซื—ื‘ืœ ืžืฉืชืœืฉืœโ€ฌ
โ€ซื—ื‘ืœ ืื—ื™ื“ ืฉืื•ืจื›ื• โ€ช L=1.5mโ€ฌืžื•ื ื— ืขืœ ื’ื‘ื™ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื—ืœืงโ€ช .โ€ฌืงื˜ืข ื‘ืื•ืจืš โ€ช b=15cmโ€ฌืžืฉืชืœืฉืœ ื›ืžืชื•ืืจ ื‘ืชืจืฉื™ืโ€ช .โ€ฌื‘ืจื’ืข ืžืกื•ื™ื ืžืจืคื™ืโ€ฌ
โ€ซืžื”ื—ื‘ืœ ืžืžื ื•ื—ื” ื•ื”ื•ื ื’ื•ืœืฉ ื›ืœืคื™ ืžื˜ื” ื‘ื”ืฉืคืขืช ื›ื•ื— ื”ื›ื•ื‘ื“โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืช ื”ื—ื‘ืœ ื‘ืจื’ืข ื‘ื• ืงืฆื”ื• ื ื™ืชืง ืžื”ืฉื•ืœื—ืŸื›ืžื” ื–ืžืŸ ื™ืขื‘ื•ืจ ืขื“ ืจื’ืข ื”ืชื ืชืงื•ืช ื”ื—ื‘ืœ ืžื”ืฉื•ืœื—ืŸ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_005.htmlโ€ฌโ€ฌ
โ€ซื”ื—ืœืงื” ืขืœ ืžื™ืฉื•ืจ ืžืฉื•ืคืขโ€ฌ
โ€ซื’ื•ืฃ ื”ื ืžืฆื ืขืœ ืคื ื™ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื‘ื–ื•ื•ื™ืช โ€ช ฮฑโ€ฌื ื”ื“ืฃ ื‘ืžื”ื™ืจื•ืช ืื•ืคืงื™ืช ื”ืชื—ืœืชื™ืช โ€ช .V0โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืฃ ืœื‘ื™ืŸ ื”ืžื™ืฉื•ืจ ื”ื•ืโ€ฌ
โ€ซื”ืžื™ืฉื•ืจ ื ื‘ื ื” ื›ืš ืฉโ€ช-โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžื” ื™ื”ื™ื• ืจื›ื™ื‘ื™ ื”ืชืื•ืฆื” ื‘ื›ื™ื•ื•ื ื™ื โ€ช xโ€ฌื•โ€ช ,y -โ€ฌื›ืฉื›ื™ื•ื•ืŸ ื”ืžื”ื™ืจื•ืช ื™ืฆื•ืจ ื–ื•ื•ื™ืช โ€ช ฮธโ€ฌืขื ื›ื™ื•ื•ืŸ ืฆื™ืจ ื”โ€ช?x-โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืžื” ื™ื”ื™ื” ืจื›ื™ื‘ ื”ืชืื•ืฆื” ื”ืžืฉื™ืงื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ โ€ช? ฮธโ€ฌโ€ฌ
โ€ซโ€ช .3โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ โ€ช ,ฮธโ€ฌื•ืžื”ื™ ื”ืžื”ื™ืจื•ืช ื›ืืฉืจโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_006.htmlโ€ฌโ€ฌ
โ€ซืื—ื™ื–ืช ื’ืœื’ืœื™ื ื‘ื›ื‘ื™ืฉโ€ฌ
โ€ซโ€ช .1โ€ฌืžื” ื”ื•ื ืžืจื—ืง ื”ื‘ืœื™ืžื” ืฉืœ ืจื›ื‘ ื”ื ื•ืกืข ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 100โ€ฌืงืž"ืฉโ€ช ,โ€ฌื›ืืฉืจ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ื’ืœื’ืœื™ื ืœื›ื‘ื™ืฉ ื”ื•ื โ€ช ? 0.8โ€ฌื›ืžื• ื›ืŸโ€ฌ
โ€ซื™ื“ื•ืข ื›ื™ ื–ืžืŸ ื”ืชื’ื•ื‘ื” ื”ืžืžื•ืฆืข ื”ื•ื โ€ช 0.65โ€ฌืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืคื™ ื›ืžื” ืขื•ืœื” ืžืจื—ืง ื”ื‘ืœื™ืžื” ื›ืืฉืจ ื™ื•ืจื“ ื’ืฉื ื•ื”ื›ื‘ื™ืฉ ืจื˜ื•ื‘?)ืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ื’ืœื’ืœื™ื ืœืืกืคืœื˜ ื”ืจื˜ื•ื‘ ื”ื•ื โ€ช(0.25โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_007.htmlโ€ฌโ€ฌ
โ€ซืžืกื” ืžื•ื ื—ืช ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื”ื ืข ื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืชโ€ฌ
โ€ซืกื‘ื™ื‘ ืฆื™ืจื•โ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ โ€ช m1โ€ฌืœืฉื•ืœื—ืŸ ื”ื•ืโ€ฌ
โ€ซืžืกื” โ€ช m1=0.5 kgโ€ฌืžื•ื ื—ืช ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื”ื ืข ื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืชโ€ฌ
โ€ซโ€ช .0.6โ€ฌื—ื•ื˜ ื—ืกืจ ืžืกื” ืžื—ื‘ืจ ืืช โ€ช m1โ€ฌืขื โ€ช) m2=3.2 kgโ€ฌื”ืชืœื•ื™ื” ื‘ืžืจื›ื– ื”ืฉื•ืœื—ืŸ( ื“ืจืš ื’ืœื’ืœืช ื—ืกืจืช ืžืกื” ื•ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช .1โ€ฌืžืฆืโ€ช/โ€ฌื™ ืชื—ื•ื ืœื’ื•ื“ืœื• ืฉืœ ื”ื—ืœืง ืฉืœ ื”ื—ื•ื˜ ื”ื ืžืฆื ื‘ืžืฆื‘ ืื•ืคืงื™ )โ€ช ,(Rโ€ฌืขื‘ื•ืจื• ืชื™ืฉืืจ ืžืกื” โ€ช m1โ€ฌื‘ืžื ื•ื—ื” ื‘ื™ื—ืก ืœืฉื•ืœื—ืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืช ืžื™ื ื™ืžืœื™ืช ื™ืฉ ืœืกื•ื‘ื‘ ืืช ื”ืฉื•ืœื—ืŸ ื›ืืฉืจ ื”ืžืกื” โ€ช m1โ€ฌื ืžืฆืืช ื‘ืžืจื—ืง โ€ช R0โ€ฌืžืžืจื›ื–ื• โ€ช ,โ€ฌื›ื“ื™ ืฉื”ื™ื ืชืชื—ื™ืœ ืœื ื•ืขโ€ฌ
โ€ซื”ื—ื•ืฆื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_008.htmlโ€ฌโ€ฌ
โ€ซื ืกื™ืขืช ืจื›ื‘ืช ืœืื•ืจืš ืขื™ืงื•ืœโ€ฌ
โ€ซืจื›ื‘ืช ืžื”ื™ืจื” ื™ื›ื•ืœื” ืœื ืกื•ืข ื‘ืžื”ื™ืจื•ืช ืžืงืกื™ืžืœื™ ืฉืœ โ€ช 310โ€ฌืงืž"ืฉโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืื ื”ืจื›ื‘ืช ื ื•ืกืขืช ื‘ืžื”ื™ืจื•ืชื” ื”ืžืงืกื™ืžืœื™ืช ืœืื•ืจืš ืขื™ืงื•ืœโ€ช ,โ€ฌืžื” ื”ื•ื ืจื“ื™ื•ืก ื”ืขื™ืงื•ืœ ื”ืžื™ื ื™ืžืœื™ ื›ืš ืฉื”ื ื•ืกืขื™ื ืœื ื™ืจื’ื™ืฉื• ืชืื•ืฆื”โ€ฌ
โ€ซื”ื’ื“ื•ืœื” โ€ช-โ€ฌืžโ€ช? 0.05gโ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื ืชื•ืŸ ืฉืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื’ืœื’ืœื™ ื”ืจื›ื‘ืช ืœืคืกื™ ื”ื‘ืจื–ืœ ื”ื•ื โ€ช .0.6โ€ฌืื™ื–ื” ืฉื™ืคื•ืข ื™ืฉ ืœืชืช ืœืžืกื™ืœื” ื›ืš ืฉื”ืจื›ื‘ืช ืœื ืชืชื”ืคืš ื‘ืขื™ืงื•ืœโ€ฌ
โ€ซื‘ืจื“ื™ื•ืก โ€ช 3โ€ฌืง"ืž?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_009.htmlโ€ฌโ€ฌ
โ€ซืกื—ืจื•ืจ ืื‘ืŸ ืขืœ ื—ื•ื˜โ€ฌ
โ€ซื ืขืจ ื”ืขื•ืžื“ ืขืœ ื’ื“ืจ ืžืกื—ืจืจ ื‘ืžืขื’ืœ ืื ื›ื™ ืื‘ืŸ ืฉืžืกืชื” โ€ช mโ€ฌื”ืงืฉื•ืจื” ื‘ื—ื•ื˜โ€ช .โ€ฌืจื“ื™ื•ืก ืžืขื’ืœ ื”ืกื™ื‘ื•ื‘ ื”ื•ื โ€ช ,Rโ€ฌื•ื”ืžืชื™ื—ื•ืช ื”ืžืงืกื™ืžืœื™ืช ืฉื”ื—ื•ื˜ ื™ื›ื•ืœโ€ฌ
โ€ซืœืฉืืช ื”ื™ื โ€ช.Tโ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌื‘ื›ืžื” ืกื™ื‘ื•ื‘ื™ื ืœื“ืงื” ื—ื™ื™ื‘ ื”ื ืขืจ ืœืกื—ืจืจ ืืช ื”ืื‘ืŸ ื›ื“ื™ ืฉื”ื—ื•ื˜ ื™ืงืจืข?โ€ฌ
โ€ซโ€ช .2โ€ฌื ืชื•ืŸ ื›ื™ ื’ื•ื‘ื” ืžืจื›ื– ื”ืžืขื’ืœ ืžืขืœ ืœืงืจืงืข ื”ื•ื โ€ช .hโ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืื•ืคืงื™ ืžืŸ ืžื”ื ืขืจ ืชืคื’ืข ื”ืื‘ืŸ ื‘ืงืจืงืข?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_012.htmlโ€ฌโ€ฌ
โ€ซื›ื•ื—ื•ืช ืžื“ื•ืžื™ืโ€ฌ
โ€ซืžืกื” ืžื•ื ื—ืช ืขืœ ื’ื‘ื™ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื‘ื–ื•ื™ืช โ€ช ,โ€ฌื”ืžืื™ืฅ ื™ืžื™ื ื” ื‘ืชืื•ืฆื” ืงื‘ื•ืขื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ืžืกื” ื•ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข ื”ื•ืโ€ฌ
โ€ซืžื”ื™ ืชืื•ืฆืช ื”ืžืกื” ื‘ื™ื—ืก ืœืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข?โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_013.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื”โ€ฌ
โ€ซืžืกื”โ€ฌ
โ€ซื ืขื” ื‘ืžื™ืฉื•ืจ ื•ืžื™ืงื•ืžื” ื‘ื–ืžืŸ ื ืชื•ืŸ ื‘ื‘ื™ื˜ื•ื™โ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืฆื•ืจืช ืžืกืœื•ืœื• ืฉืœ ื”ื—ืœืงื™ืง ื”ื™ื ืืœื™ืคืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ื”ื›ื•ื— ื”ืคื•ืขืœ ืขืœ ื”ื—ืœืงื™ืง ืžื•ืคื ื” ืชืžื™ื“ ืœื›ื™ื•ื•ืŸ ืžืจื›ื– ื”ืืœื™ืคืกื”โ€ช.โ€ฌโ€ฌ
โ€ซืœื ืงื•ื“ื”โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื”ืžื‘ื•ืฆืขืช ืขืœ ื”ื—ืœืงื™ืง ื›ืืฉืจ ื”ื•ื ื ืข ืžื”ื ืงื•ื“ื”โ€ฌ
โ€ซ? ื”ืกื‘ืจโ€ช/โ€ฌื™!โ€ฌ
โ€ซื“โ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื‘ืžืงืจื” ื”ืคืจื˜ื™โ€ฌ
โ€ซ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_014.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซืโ€ช .โ€ฌืื™ื–ื” ืžื”ื›ื•ื—ื•ืช ื”ื‘ืื™ื ื”ื ื›ื•ื—ื•ืช ืžืฉืžืจื™ื )ืจืžื–โ€ช :โ€ฌืœื ืžืกืคื™ืง ืœื‘ื“ื•ืง ืฉื”ืจื•ื˜ื•ืจ ืžืชืืคืก(โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื”ื ืขืฉื™ืช ืขืœ ื›ื•ื—ื•ืช ืืœื” ืขืœโ€ช-โ€ฌืคื ื™ ืžืกืœื•ืœ ืžืขื’ืœื™ ืกื‘ื™ื‘ ื”ืจืืฉื™ืชโ€ช ,โ€ฌื›ื ื’ื“ ื›ื™ื•ื•ืŸ ื”ืฉืขื•ืŸ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_015.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืฉืœื•ืฉ ืžืกื•ืช ืžื—ื•ื‘ืจื•ืช ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช .โ€ฌื›ืœ ื”ืžืฉื˜ื—ื™ื ื—ืœืงื™ื ื•ื”ืžืกื•ืช ื ื•ื’ืขื•ืช ื–ื• ื‘ื–ื•โ€ช .โ€ฌืžืกืช ื”ื—ื•ื˜ ื•ื”ื’ืœื’ืœืช ื–ื ื™ื—ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื’ื•ื“ืœื• ื•ื›ื™ื•ื•ื ื• ืฉืœ ื”ื›ื— โ€ช Fโ€ฌื›ื“ื™ ืฉื”ืžืกื” ื”ืฉื ื™ื” ืชืฉืืจ ื‘ืžื ื•ื—ื” ื‘ื™ื—ืก ืœืจืืฉื•ื ื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ืŸ ืชืื•ืฆื•ืช ื”ืžืกื•ืช ื›ืืฉืจ โ€ช?F=0โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌืžื”ื ื”ื›ื•ื—ื•ืช ื”ืคื•ืขืœื™ื ืขืœ ื”ื’ืœื’ืœืช? ืžื”ื• ื”ื›ื•ื— ื”ืฉืงื•ืœ ื”ืคื•ืขืœ ืขืœ ื”ื’ืœื’ืœืช?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_108.htmlโ€ฌโ€ฌ
โ€ซื›ื•ื— ื—ื™ื›ื•ืš ื“ื™ื ืžื™โ€ฌ
โ€ซื—ืœืงื™ืง ื ืข ื‘ืงื• ื™ืฉืจ ืชื—ืช ืคืขื•ืœืช ื›ื•ื— ื—ื™ื›ื•ืš ื“ื™ื ืืžื™โ€ช ,โ€ฌื›ืœื•ืžืจโ€ฌ
โ€ซืฉืœ ื”ื–ืžืŸ ื•ืฉืจื˜ื˜ื• ื’ืจืฃ ืฉืœ ื”ืžื”ื™ืจื•ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช kโ€ฌื”ื•ื ืงื‘ื•ืข ื•โ€ช v -โ€ฌื–ื• ืžื”ื™ืจื•ืชโ€ช .โ€ฌืžืฆืื• ืืช ืžื”ื™ืจื•ืช ื›ืคื•ื ืงืฆื™ื”โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_109.htmlโ€ฌโ€ฌ
โ€ซื›ื•ื— ืžืฉืชื ื”โ€ฌ
โ€ซืขื‘ื•ืจ ื”ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืฆืื• ืืช ื”ื›ื•ื—โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_201.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซื—ืœืงื™ืง ื‘ืขืœ ืžืกื” โ€ช mโ€ฌื ืžืฆื ื‘ืžื ื•ื—ื” ื‘ โ€ช .x=0โ€ฌืขืœ ื”ื—ืœืงื™ืง ืคื•ืขืœ ื›ื•ื— ื‘ื›ื™ื•ื•ืŸ ืฆื™ืจ ื” โ€ช .xโ€ฌื‘ื–ืžืŸ โ€ช t=Tโ€ฌืžืคืกื™ืง ื”ื›ื— ืœืคืขื•ืœ ืขืœ ื”ื—ืœืงื™ืงโ€ช .โ€ฌื ืชื•ืŸโ€ฌ
โ€ซื›ืืฉืจ โ€ช F0โ€ฌื• โ€ช Tโ€ฌืงื‘ื•ืขื™ืโ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื ื”ืžื™ืงื•ื ื•ื”ืžื”ื™ืจื•ืช ืฉืœ ื”ื—ืœืงื™ืง ื›ืชืœื•ืช ื‘ื–ืžืŸ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_202.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซืžืขืœื™ืช ืฉืžืกืชื” โ€ช 150โ€ฌืง"ื’ ืžื—ื•ื‘ืจืช ื‘ื›ื‘ืœ ื”ืžืกื•ื’ืœ ืœืขืžื•ื“ ื‘ืžืชื™ื—ื•ืช ืฉืœ โ€ช .5000Nโ€ฌืชืื•ืฆืชื” ื”ืžืงืกื™ืžืœื™ืช ืฉืœ ื”ืžืขืœื™ืช ื”ื™ื โ€ช 2โ€ฌืž"ืฉโ€ช .โ€ฌื‘ื”ื ื—ื” ื›ื™โ€ฌ
โ€ซืžืกืช ืื“ื ืžื‘ื•ื’ืจ ื”ื™ื โ€ช 80โ€ฌืง"ื’โ€ช ,โ€ฌืžื” ืžืกืคืจ ื”ืื ืฉื™ื ื”ืžืงืกื™ืžืœื™ ื”ื™ื›ื•ืœื™ื ืœื”ื›ื ืก ืœืžืขืœื™ืช?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_203.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซื ืชื•ื ื” ืžืขืจื›ืช ืฉืœ ืฉื ื™ ื’ื•ืคื™ืโ€ช .โ€ฌื”ืžืกื” ื”ืจืืฉื•ื ื” ื”ื™ื โ€ช 15โ€ฌืง"ื’ ื•ื”ืฉื ื™ื” โ€ช 30โ€ฌืง"ื’โ€ช .โ€ฌืžื”ื• ื”ื›ื— โ€ช Fโ€ฌื”ื“ืจื•ืฉ ืขืœ ืžื ืช ืฉโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืžืขืจื›ืช ืชื ื•ืขื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืžืขืจื›ืช ืชืื™ืฅ ื™ืžื™ื ื” ื‘ โ€ช 2โ€ฌืž"ืฉ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_204.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืขืจื›ืช ื‘ืื™ื•ืจโ€ช .โ€ฌื”ืžืข' ืžื•ืจื›ื‘ืช ืžื’ืœื’ืœื•ืช ื•ืžื—ื‘ืœื™ื ื—ืกืจื™ ืžืกื”โ€ช .โ€ฌืฆื™ืจื™ ื”ื’ืœื’ืœืช ื—ืกืจื™ ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซื—ืฉื‘ ืืช ื”ืชืื•ืฆื” ืฉืœ ื›ืœ ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_205.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซืกืคื™ื ื” ื ืžืฆืืช ื‘ืžืขืจื‘ื•ืœืช ื”ืžืกืชื•ื‘ื‘ืช ื‘ืชื“ื™ืจื•ืช ืื—ื™ื“ื” โ€ช) wโ€ฌื”ื›ื•ื•ื ื” ื”ื™ื ืฉื›ืœ ืืœืžื ื˜ ืžื™ื ืžืกืชื•ื‘ื‘ ื‘ืชื“ื™ืจื•ืช โ€ช wโ€ฌืกื‘ื™ื‘ ื”ืจืืฉื™ืชโ€ช (.โ€ฌืžื”ื• ื”ื›ื•ื—โ€ฌ
โ€ซืฉืขืœ ื”ืžื“ื—ืฃ ืœื”ืคืขื™ืœ ื›ืš ืฉื”ืกืคื™ื ื” ืชื ื•ืข ื‘ืงื• ื™ืฉืจ ื‘ืชืื•ืฆื” ืื—ื™ื“ื” )ื‘ื™ื—ืก ืœืžื™ื( ืžืžืจื›ื– ื”ืžืขืจื‘ื•ืœืช? ืจืžื–โ€ช :โ€ฌื”ื’ื“ืจ ืžืข' ืฆื™ืจื™ื ืขืœโ€ฌ
โ€ซื”ืžืขืจื‘ื•ืœืชโ€ช ,โ€ฌื•ืจืฉื•ื ืืช ืชื ื•ืขืช ื”ืกืคื™ื ื” ื‘ืžืข' ืฆื™ืจื™ื ื–ืืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_2_206.htmlโ€ฌโ€ฌ
โ€ซืชืจื’ื™ืœโ€ฌ
โ€ซืชื™ืœ ืงืฉื™ื— ื•ื—ืœืง ืžื•ื ื— ื‘ืžื™ืฉื•ืจ )โ€ช (x,yโ€ฌื›ืš ืฉืงืฆื” ืื—ื“ ืฉืœื• ื‘ืจืืฉื™ืชโ€ช .โ€ฌืฆื•ืจืช ื”ืชื™ืœ ืžืชื•ืืจืช ืข"ื™ โ€ช . (y(xโ€ฌืžืกื•ื‘ื‘ื™ื ืืช ื”ืชื™ืœ ืกื‘ื™ื‘ ืฆื™ืจ ื” โ€ช.yโ€ฌโ€ฌ
โ€ซืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ืฆื•ืจืช ื”ืชื™ืœ ืขืœโ€ช-โ€ฌืžื ืช ืฉืœื—ืจื•ื– ื‘ืขืœ ืžืกื” โ€ช mโ€ฌื”ืžื•ืฉื—ืœ ืขืœื™ื• ื™ื”ื™ื• ืฉืชื™ ื ืงื•ื“ื•ืช ืฉื™ื•ื•ื™ ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_2_207.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื‘ืžืขืจื›ืช ื”ืžืชื•ืืจืช ื‘ืื™ื•ืจ ืื™ืŸ ื—ื™ื›ื•ืš ื•ื”ื’ืœื’ืœื•ืช ื•ื”ื—ื•ื˜ื™ื ื—ืกืจื™ ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸโ€ช.M1=M2=5Kg :โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘ื• ืืช ื”ืชืื•ืฆื” ืฉืœ ื›ืœ ืื—ืช ืžืŸ ื”ืžืกื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื—ืฉื‘ื• ืืช ื”ืžืชื™ื—ื•ืช ื‘ื›ืœ ืื—ื“ ืžืŸ ื”ื—ื•ื˜ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_3_001.htmlโ€ฌโ€ฌ
โ€ซื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžื” ื”ืžืชื™ื—ื•ืช ื‘ื—ื•ื˜ื™ื?โ€ฌ
โ€ซโ€ช10_Newton/e_10_3_002.htmlโ€ฌโ€ฌ
โ€ซื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื ืชื•ื ื•ืช ืฉืชื™ ืžืกื•ืชโ€ฌ
โ€ซื‘ืชืฆื•ืจื” ื”ื‘ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื( ืžื”ื• ืžืงื“ื ื”ื—ื™ื›ื•ืš ื›ืš ืฉโ€ช m1โ€ฌืชืขืœื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”?โ€ฌ
โ€ซื‘( ืžื”ื• ืžืงื“ื ื”ื—ื™ื›ื•ืš ื›ืš ืฉโ€ช m1โ€ฌืชืจื“ ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_3_003.htmlโ€ฌโ€ฌ
โ€ซื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื ืชื•ืŸ ืฉื—ื™ื™ืŸ ื‘ื‘ืจื™ื›ื” ื”ืฉื•ื—ื” ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 5โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื”โ€ช .โ€ฌื‘ืžืจื—ืง ืžืกื•ื™ื ืžื”ื“ื•ืคืŸโ€ช ,โ€ฌืžืคืกื™ืง ื”ืฉื—ื™ื™ืŸ ืœื“ื—ื•ืฃ ืืช ืขืฆืžื• ื‘ืžื™ื ื•ื’ื•ืœืฉ ืขื“ ืœื“ื•ืคืŸโ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืชื• ื”ืกื•ืคื™ืช ื”ื™ื โ€ช 0.4โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื”โ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืžื”ื“ื•ืคืŸ ื”ืชื—ื™ืœ ื”ืฉื—ื™ื™ืŸ ื‘ื’ืœื™ืฉื”?โ€ฌ
โ€ซโ€ช .โ€ฌื›ื•ื— ื’ืจืจโ€ช:โ€ฌโ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_3_004.htmlโ€ฌโ€ฌ
โ€ซื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื( ืžื”ื™ ืžื”ื™ืจื•ืชื” ื”ืกื•ืคื™ืช ืฉืœ ื˜ื™ืคืช ื’ืฉื ื‘ื”ื’ื™ืขื” ืœืงืจืงืขโ€ช ,โ€ฌื‘ื”ื ื—ื” ืฉื˜ื™ืคืช ื’ืฉื ื ื•ืคืœืช ื ืคื™ืœื” ื—ื•ืคืฉื™ืช ืžืขื ืŸ ื‘ื’ื•ื‘ื” โ€ช 600โ€ฌืžื˜ืจ? ื”ืื ื”ื”ื ื—ื”โ€ฌ
โ€ซืกื‘ื™ืจื”?โ€ฌ
โ€ซื‘( ืžื”ื™ ืžื”ื™ืจื•ืชื” ื”ืกื•ืคื™ืช ืฉืœ ื˜ื™ืคืช ื’ืฉื ื‘ื”ื’ื™ืขื” ืœืงืจืงืขโ€ช ,โ€ฌื›ืืฉืจ ืœื•ืงื—ื™ื ื‘ื—ืฉื‘ื•ืŸ ืืช ื›ื•ื— ื”ื’ืจืจ ืฉืžืคืขื™ืœ ืขืœื™ื” ื”ืื•ื•ื™ืจ?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช :โ€ฌื›ื•ื— ื”ื’ืจืจ ื‘ืื•ื•ื™ืจโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_3_005.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸ โ€ช -โ€ฌืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซื( ื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื ื™ืชืŸ ืœืขื‘ื•ืจ ืกื™ื‘ื•ื‘ ืžืขื’ืœื™ ื‘ืขืœ ืจื“ื™ื•ืก ืฉืœ โ€ช 50โ€ฌืžื˜ืจื™ื ืขื ืจื›ื‘โ€ช ,โ€ฌืืฉืจ ื‘ื™ื ื• ืœื‘ื™ืŸ ื”ืืกืคืœื˜ ื™ืฉ ืžืงื“ื ื—ื™ื›ื•ืš ืฉืœโ€ฌ
โ€ซ?โ€ฌ
โ€ซื‘( ื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื ื™ืชืŸ ื™ื”ื™ื” ืœืขื‘ื•ืจ ืืช ื”ืกื™ื‘ื•ื‘ ืื ื”ื›ื‘ื™ืฉ ื™ื•ื’ื‘ื” ื‘โ€ช 15-โ€ฌืžืขืœื•ืช?โ€ฌ
โ€ซโ€ช10_Newton/e_10_3_006.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸ โ€ช -โ€ฌืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซื”ืื ืžืื‘ื“ื™ื ืžืฉืงืœ ื›ืืฉืจ ืขื•ื‘ืจื™ื ืžื™ืฉืจืืœ ืœืงื• ื”ืžืฉื•ื•ื”? ื›ืžื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_3_007.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸ โ€ช -โ€ฌืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซื’ื•ืฃ ืงื˜ืŸ ื ืข ื‘ืชื•ืš ืฆื™ื ื•ืจ ืžืขื’ืœื™ ื‘ืขืœ ืจื“ื™ื•ืก โ€ช Rโ€ฌื”ื ืžืฆื ื‘ืžื™ืฉื•ืจ ื”ืื•ืคืงื™โ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืฃ ืœืฆื™ื ื•ืจ ื”ื•ืโ€ฌ
โ€ซื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ื’ื•ืฃ ื”ื™ื โ€ช .โ€ฌืœืื—ืจ ืื™ื–ื” ืžืจื—ืง ื™ืขืฆืจ ื”ื’ื•ืฃ ื‘ืื•ืคืŸ ืžืขืฉื™?โ€ฌ
โ€ซโ€ช .โ€ฌืžื”ื™ืจื•ืชื•โ€ฌ
โ€ซโ€ช10_Newton/e_10_4_001.htmlโ€ฌโ€ฌ
โ€ซื–ื‘ื•ื‘ื™ื ื•ืžืื–ื ื™ื™ืโ€ฌ
โ€ซื–ื‘ื•ื‘ ืขื•ืžื“ ืขืœ ืงืจืงืขื™ืชื” ืฉืœ ืฆื ืฆื ืช ืกื’ื•ืจื” ื”ืžื•ืฆื‘ืช ืขืœ ืžืื–ื ื™ื™ื ืจื’ื™ืฉื™ื ื”ื ืžืฆืื™ื ื‘ืžืฆื‘ ืžืื•ื–ืŸโ€ช .โ€ฌืœืคืชืข ืžืชืจื•ืžื ื”ื–ื‘ื•ื‘โ€ช ,โ€ฌืžืจื—ืฃ ื‘ืžืงื•ืโ€ฌ
โ€ซืžืกืคืจ ืฉื ื™ื•ืช ื•ืฉื‘ ื•ื ื•ื—ืช ืขืœ ืงืจืงืขื™ืช ื”ืฆื ืฆื ืชโ€ช.โ€ฌโ€ฌ
โ€ซื”ืžืฉืคื˜ื™ื ื”ื‘ืื™ื ืžืชืืจื™ื ืืช ืžืฆื‘ ื›ืคื•ืช ื”ืžืื–ื ื™ื™ื ืœืื•ืจืš ื”ืชื”ืœื™ืš ืฉืชื•ืืจ ืœืขื™ืœโ€ช .โ€ฌืกืžืŸโ€ช/โ€ฌื™ ืืช ื”ืžืฉืคื˜ ื”ื ื›ื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื( ื”ืžืื–ื ื™ื™ื ื™ื™ืฉืืจื• ืžืื•ื–ื ื™ื ื›ืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื”ื›ืฃ ื”ืฉืžืืœื™ืช ืชืจื“ ื›ืืฉืจ ื”ื–ื‘ื•ื‘ ื™ืขืœื”โ€ช ,โ€ฌืชื™ืฉืืจ ืžืื•ื–ื ืช ื›ืืฉืจ ื”ื–ื‘ื•ื‘ ื™ืจื—ืฃ ื•ืชืขืœื” ื›ืฉื”ื–ื‘ื•ื‘ ื™ืจื“โ€ช.โ€ฌโ€ฌ
โ€ซื’( ื”ื›ืฃ ื”ืฉืžืืœื™ืช ืชืขืœื” ื›ืืฉืจ ื”ื–ื‘ื•ื‘ ื™ืขืœื”โ€ช ,โ€ฌืชื™ืฉืืจ ืžืื•ื–ื ืช ื›ืืฉืจ ื”ื–ื‘ื•ื‘ ื™ืจื—ืฃ ื•ืชืจื“' ื›ืฉื”ื–ื‘ื•ื‘ ื™ืจื“โ€ช.โ€ฌโ€ฌ
โ€ซื“( ื”ื›ืฃ ืขื ื”ื–ื‘ื•ื‘ ืชืจื“ ื›ืœ ืขื•ื“ ื”ื–ื‘ื•ื‘ ื‘ืื•ื•ื™ืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_002.htmlโ€ฌโ€ฌ
โ€ซืงื•ื ื•ืกโ€ฌ
โ€ซื”ื ืžืฆื ื‘ืชื•ืš ืงื•ื ื•ืก )ื”ืžืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจื•( ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซโ€ช .โ€ฌื–ื•ื•ื™ืช ื”ืคืชื™ื—ื” ืฉืœ ื”ืงื•ื ื•ืก ื”ื™ืโ€ฌ
โ€ซื ืชื•ืŸ ื’ื•ืฃโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื”ื•ืโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื–ืžืŸ ื”ืžื—ื–ื•ืจ ื”ืžื™ื ื™ืžืืœื™ ื•ื”ืžืงืกื™ืžืืœื™ ืฉืœ ื”ืงื•ื ื•ืก ื›ืš ืฉื”ื’ื•ืฃ ืœื ื™ื™ืคื•ืœโ€ช.โ€ฌโ€ฌ
โ€ซื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ื’ื•ืฃ ืœืงื•ื ื•ืกโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_003.htmlโ€ฌโ€ฌ
โ€ซื–ืจื™ืงืช ืื‘ืŸโ€ฌ
โ€ซืžื‘ื ื™ื™ืŸ ืฉื’ื•ื‘ื”ื•โ€ฌ
โ€ซื ื–ืจืงืช ืื‘ืŸ ื‘ื›ื™ื•ื•ืŸ ืื•ืคืงื™ ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืจื•ื— ื ื’ื“ื™ืช ืžืื™ื˜ื” ืืช ื”ืื‘ืŸ ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ื•ื›ืชื•ืฆืื” ืžื›ืš ืคื•ื’ืขืช ื”ืื‘ืŸ ื‘ืจื’ืœื™ ื”ื‘ื ื™ื™ืŸ ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืชืื•ืฆื” ื”ืื•ืคืงื™ืช ืฉืœ ื”ืื‘ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืžื”ื™ ืฆื•ืจืช ื”ืžืกืœื•ืœ ืฉืœ ื”ืื‘ืŸ?โ€ฌ
โ€ซ)ืจืžื–โ€ช :โ€ฌื‘ื—ื™ืจื” ื ื›ื•ื ื” ืฉืœ ืžืขืจื›ืช ื”ืฆื™ืจื™ืืชืคืฉื˜ ืืช ื”ืคืชืจื•ืŸ!(โ€ฌ
โ€ซโ€ช10_Newton/e_10_4_004.htmlโ€ฌโ€ฌ
โ€ซื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช .โ€ฌื‘ืจื’ืข ืžืกื•ื™ื ืžืชื—ื™ืœโ€ฌ
โ€ซโ€ช ,โ€ฌื”ืžื•ื ื—ื™ื ื–ื” ืขืœ ื’ื‘ื™ ื–ื” ื›ืคื™ ืฉืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช .โ€ฌื‘ื™ืŸ ืฉื ื™ ื”ื’ื•ืคื™ื ื™ืฉ ืžืงื“ื ื—ื™ื›ื•ืšโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื’ื•ืคื™ืโ€ฌ
โ€ซืขื“ ืืฉืจ ื”ืžื”ื™ืจื•ืช ื”ื™ื—ืกื™ืช ื‘ื ื™ื”ื ืชืชืืคืกโ€ช.โ€ฌโ€ฌ
โ€ซื™ื ื•ืข ืขืœ ื’ื‘ื™โ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืœืื™ื–ื” ืžืจื—ืง ื”ื’ื•ืฃโ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืœื ื•ืข ืขืœ ื’ื‘ื™โ€ฌ
โ€ซื’ื•ืฃโ€ฌ
โ€ซื( ื‘ืžืขืจื›ืช ืฆื™ืจื™ื ืื™ื ืจืฆื™ืืœื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื‘ืžืขืจื›ืช ืฆื™ืจื™ื ืœื ืื™ื ืจืฆื™ืืœื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_005.htmlโ€ฌโ€ฌ
โ€ซืฉื ื™ ื’ื•ืคื™ืโ€ฌ
โ€ซื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื’ื•ืฃ ื‘ืขืœ ืžืกื” ื”ืžื•ื ื— ืข"ื’ ื˜ืจื™ื– ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื( ืžื”ื™ ืžื”ื™ืจื•ืช ื”ื˜ืจื™ื– ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซโ€ช ? ,โ€ฌื”ืกื‘ืจโ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืžื” ื™ืงืจื” ื‘ื’ื‘ื•ืœ ืฉโ€ช-โ€ฌโ€ฌ
โ€ซื‘ืžืขืจื›ืช ื”ื˜ืจื™ื–?โ€ฌ
โ€ซื’( ืžื”ื™ ืžื”ื™ืจื•ืช ื”ืžืกื”โ€ฌ
โ€ซื“( ืžื”ื• ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืžื™ื ื™ืžืœื™ ื‘ื™ืŸ ื”ืžืกื” ืœื˜ืจื™ื– ื›ืš ืฉืœื ืชื™ื•ื•ืฆืจ ืชื ื•ืขื” ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_4_006.htmlโ€ฌโ€ฌ
โ€ซื—ืจื•ื– ืžืกืชื—ืจืจโ€ฌ
โ€ซื ืชื•ืŸ ื—ืจื•ื– ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื‘ื” ื™ืชืžืงื ื”ื—ืจื•ื–โ€ช.โ€ฌโ€ฌ
โ€ซื”ืžื•ืฉื—ืœ ืขืœ ื˜ื‘ืขืช ืื ื›ื™ืช ื—ืกืจืช ื—ื™ื›ื•ืš ื‘ืขืœืช ืจื“ื™ื•ืกโ€ฌ
โ€ซโ€ช ,โ€ฌื”ืžืกืชื•ื‘ื‘ืช ื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืชโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ื”ื–ื•ื•ื™ืชโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_007.htmlโ€ฌโ€ฌ
โ€ซืฉืืœื” โ€ช2โ€ฌโ€ฌ
โ€ซื•ืงืฉื•ืจื” ืœืžื™ืฉื•ืจโ€ฌ
โ€ซืžื•ื ื—ืช ืขืœ ื’ื‘ื™ ืžืกื”โ€ฌ
โ€ซืžื•ื ื—ืช ืขืœ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื—ืกืจ ื—ื™ื›ื•ืš ื‘ืขืœ ื–ื•ื™ืช )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช .โ€ฌืžืกื”โ€ฌ
โ€ซืžืกื”โ€ฌ
โ€ซืœืื•ืจืšโ€ฌ
โ€ซื ืžืฆืืช ืขืœ ื’ื‘ื™ ืžืกื”โ€ฌ
โ€ซื”ืžืฉื•ืคืข ื‘ื—ื•ื˜ ื—ืกืจ ืžืกื” ืฉื›ื™ื•ื•ื ื• ืžืงื‘ื™ืœ ืœืื•ืคืงโ€ช .โ€ฌื‘ื—ืจโ€ช/โ€ฌื™ ืžืขืจื›ืช ืฆื™ืจื™ื ืœื ื•ื—ื™ื•ืชืšโ€ช .โ€ฌื”ื ื—โ€ช/โ€ฌื™ ื›ื™ ื”ืžืกื”โ€ฌ
โ€ซื›ืœ ืฉืœื‘ื™ ื”ืชื ื•ืขื”โ€ช .โ€ฌื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“ โ€ช.โ€ฌโ€ฌ
โ€ซื›ืืฉืจ ื‘ื™ืŸ ืฉืชื™ ื”ืžืกื•ืช ืื™ืŸ ื—ื™ื›ื•ืšโ€ช:โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ืžืกื”โ€ฌ
โ€ซโ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจ ืงื™ื™ื ื‘ื™ืŸ ืฉืชื™ ื”ืžืกื•ืช ื—ื™ื›ื•ืš ืกื˜ื˜ื™โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช ,โ€ฌื•โ€ช-โ€ฌโ€ฌ
โ€ซื›ืš ืฉื”ืžืขืจื›ืช ืชื™ืฉืืจ ื‘ืžื ื•ื—ื”? ื‘ื˜ืโ€ช/โ€ฌื™ ืืช ืชืฉื•ื‘ืชืš ื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื’ื•ื“ืœื” ื”ืžืงืกื™ืžืœื™ ืฉืœโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_008.htmlโ€ฌโ€ฌ
โ€ซื ืคื™ืœื” ืขื ื—ื™ื›ื•ืšโ€ฌ
โ€ซื›ื“ื•ืจ ื ื•ืคืœ ืžืžื˜ื•ืก ืฉื ืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื‘ื’ื•ื‘ื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ื—ื™ื›ื•ืš ื”ืื•ื™ืจ ื”ื•ืโ€ฌ
โ€ซื”ื•ื ืžืงื“ื ื”ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซื›ืืฉืจโ€ฌ
โ€ซื•ื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“ โ€ช.โ€ฌโ€ฌ
โ€ซืžืฆื ืืช ื•ืงื˜ื•ืจ ื”ืžื™ืงื•ื ืฉืœ ื”ื›ื“ื•ืจ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_4_009.htmlโ€ฌโ€ฌ
โ€ซื ืคื™ืœื” ืขื ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื›ื“ื•ืจ ื ื•ืคืœ ืžืžื’ื“ืœ ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ื—ื™ื›ื•ืš ื”ืื•ื™ืจ ื”ื•ืโ€ฌ
โ€ซื”ื•ื ืžืงื“ื ื”ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช.โ€ฌื›ืืฉืจโ€ฌ
โ€ซโ€ช .โ€ฌื•ื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“โ€ฌ
โ€ซืžื”ื• ื”ื’ื•ื‘ื” ืฉืœ ื”ื›ื“ื•ืจ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_010.htmlโ€ฌโ€ฌ
โ€ซืžืฆื™ืืช ืžืกื™ืœืช ืชื ื•ืขืชื• ืฉืœ ื”ื’ื•ืฃ ื“ืจืš ืฉื“ื” ื›ื•ื— ื ืชื•ืŸโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื”ื’ื•ืฃ ื ื›ื ืก ืœืฉื“ื” ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซื’ื•ืฃ ื ืข ื“ืจืš ืฉื“ื” ื›ื•ื— ืฉืžืฉื•ื•ืืชื• ื”ื™ืโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆื ืืช ื•ืงื˜ื•ืจ ื”ืžืงื•ื ืฉืœ ื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช10_Newton/e_10_4_011.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื™ื—ืกื™ืชโ€ฌ
โ€ซืงืจื•ื ื™ืช ื ืขื” ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” โ€ช .aโ€ฌืžืกื” โ€ช Mโ€ฌื”ืงืฉื•ืจื” ื‘ื—ื•ื˜ ืœื’ื’ ื”ืคื ื™ืžื™ ืฉืœ ื”ืงืจื•ื ื™ืช ื™ื•ืฆืจืช ื–ื•ื•ื™ืช ืฉืœโ€ฌ
โ€ซโ€ชaโ€ฌื•โ€ช.gโ€ฌโ€ฌ
โ€ซืขื ื”ืื ืšโ€ช .โ€ฌื‘ื˜ื ืืชโ€ฌ
โ€ซื‘ืืžืฆืขื•ืช โ€ชMโ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_041.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืื“ื ื”ืฉื•ืงืœ โ€ช 700โ€ฌื ื™ื•ื˜ื•ืŸ ืžื•ืฉืš ืืช ืขืฆืžื• ื›ืœืคื™ ืžืขืœื” ื‘ืขื–ืจืช ืžืขืจื›ืช ื’ืœื’ืœื•ืช ื›ืžื•ืจืื” ื‘ืื™ื•ืจโ€ช .โ€ฌืžืกืช ื”ืžืฉื˜ื— โ€ช .200Nโ€ฌืžืกื•ืช ื”ื’ืœื’ืœื•ืชโ€ฌ
โ€ซื•ื”ื—ื‘ืœื™ื ื–ื ื™ื—ื•ืชโ€ช .โ€ฌื”ื’ืœื’ืœื•ืช ื—ืกืจื•ืช ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื‘ืื™ื–ื” ื›ื— ืขืœื™ื• ืœืžืฉื•ืš ืืช ื”ื—ื‘ืœ ื‘ื›ื“ื™ ืฉื™ืขืœื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื‘ืื™ื–ื” ื›ื— ืขืœ ื‘ืŸโ€ช-โ€ฌืื“ื ืขืœ ื”ืงืจืงืข ืœืžืฉื•ืš ืืช ื”ื—ื‘ืœ ืขืœ ืžื ืช ืฉื™ืขืœื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_042.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื“ื•ื—ืคื™ื ืงื•ื‘ื™ื” ื‘ืขืœืช ืžืกื” โ€ช mโ€ฌื‘ื›ื™ื•ื•ืŸ ื”ืื•ืคืงื™ ืขืœ ืžืฉื˜ื— ื‘ืขืœ ืžืงื“ื ื—ื™ื›ื•ืš ืกื˜ื˜ื™ โ€ช msโ€ฌื•ืžืงื“ื ื—ื™ื›ื•ืš ื“ื™ื ืžื™ โ€ช .mkโ€ฌื ืชื•ืŸ ื›ื™ โ€ช. ms0.5= mkโ€ฌโ€ฌ
โ€ซื‘ โ€ช t=0โ€ฌื”ืงื•ื‘ื™ื” ื ืขื” ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซ?โ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื•โ€ฌ
โ€ซโ€ช .โ€ฌื‘ืื™ื–ื” ืžื”ื™ืจื•ืช ื•ืชืื•ืฆื” ื”ืงื•ื‘ื™ื” ืชื ื•ืข?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื‘ืฉืœื‘ ื›ืœืฉื”ื• ืขื•ืฆืจื™ื ืืช ื”ืงื•ื‘ื™ื” ื•ืžืคืขื™ืœื™ื ืขืœื™ื” ื›ื•ื—โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื• ื”ื›ื•ื— ื”ืžื™ื ื™ืžืœื™ ืฉื™ืฉ ืœื”ืคืขื™ืœ ืขืœ ื”ืงื•ื‘ื™ื” ื‘ื›ื“ื™ ืฉืชืชื—ื™ืœ ืœื ื•ืข?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_043.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื”ืชืžื•ื ื” ืžืจืื” ื—ืชืš ืฉืœ ื›ื‘ื™ืฉ ืฉื—ืฆื•ื‘ ื‘ื”ืจโ€ช .โ€ฌื”ืงื• โ€ช Aโ€™Aโ€ฌืžืกืžืœ ืžื™ืฉื•ืจ ืขืœื™ื• ื”ื—ืœืงื” ื”ื™ื ืืคืฉืจื™ืชโ€ช .โ€ฌืกืœืข โ€ช Bโ€ฌื ืžืฆื ื™ืฉื™ืจื•ืช ืžืขืœ ื”ื›ื‘ื™ืฉโ€ฌ
โ€ซโ€ช ,โ€ฌื–ื•ื•ื™ืช ื”ื”ื˜ื™ื” ืฉืœโ€ฌ
โ€ซื•ืžื•ืคืจื“ ืžืฉืืจ ื”ื”ืจ ืข" ื™ ืกื“ืง ื’ื“ื•ืœโ€ช ,โ€ฌื›ืš ืฉืจืง ื›ื•ื— ื”ื—ื™ื›ื•ืš ืžื•ื ืข ืžื”ืกืœืข ืœื”ื—ืœื™ืงโ€ช .โ€ฌืžืกืช ื”ืกืœืข ื”ื™ืโ€ฌ
โ€ซื”ืฉื™ืคื•ืข ื”ื™ื โ€ช 24โ€ฌืžืขืœื•ืช โ€ช ,โ€ฌื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื”ื•ื โ€ช .0.63โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™โ€ฌ
โ€ซืโ€ช.โ€ฌื”ืกืœืข ืœื ื™ื—ืœื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืื ืžื™ื ืงื•ืคืื™ื ื‘ืกื“ืง ื•ืžืจื—ื™ื‘ื™ื ืื•ืชื• ื›ืš ืฉื”ื ืžืคืขื™ืœื™ื ื›ื•ื— โ€ช Fโ€ฌืขืœ ื”ืกืœืข )ื‘ืžืงื‘ื™ืœ ืœืžื™ืฉื•ืจ โ€ช ,(Aโ€™Aโ€ฌืžื”ื• ื”ืขืจืš ื”ืžื™ื ื™ืžืœื™ ืฉืœ โ€ชFโ€ฌโ€ฌ
โ€ซืฉื™ื’ืจื•ื ืœืกืœืข ืœื”ื—ืœื™ืง?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_044.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื‘ื“ื™ืื’ืจืžื” ืœโ€ช A -โ€ฌืžืฉืงืœ ืฉืœ โ€ช 44Nโ€ฌื•ืœโ€ช B -โ€ฌืžืฉืงืœ ืฉืœ โ€ช .22Nโ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ โ€ช Aโ€ฌืœืฉื•ืœื—ืŸ ื”ื•ื โ€ช 0.2โ€ฌื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ื“ื™ื ืžื™ ื‘ื™ืŸ โ€ชAโ€ฌโ€ฌ
โ€ซืœืฉื•ืœื—ืŸ ื”ื•ื โ€ช.0.15โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ืžืฉืงืœ ื”ืžื™ื ื™ืžืœื™ ืฉืœ โ€ช Cโ€ฌืขืœโ€ช-โ€ฌืžื ืช ืฉ โ€ช Aโ€ฌืœื ื™ื—ืœื™ืง?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื ื ื™ื— ื›ื™ ื‘ื‘ืช ืื—ืช ืžืจื™ืžื™ื ืืช โ€ช .Cโ€ฌืžื” ืชื”ื™ื” ืชืื•ืฆืช โ€ช? Aโ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_071.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช mโ€ฌืžืชื—ื™ืœ ืœื”ื—ืœื™ืง ืžืžื ื•ื—ื” ื‘ืฉื™ื ื”ื’ื•ื‘ื” ืฉืœ ืžืกื™ืœื” ื—ืฆื™ ืžืขื’ืœื™ืช ื—ืœืงื” ืฉืจื“ื™ื•ืกื” โ€ช R=1โ€ฌืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆืื• ื‘ื™ื˜ื•ื™ ืœื›ื•ื— ื”ื ื•ืจืžืœื™ โ€ช Nโ€ฌื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ื•ื•ื™ืช โ€ช .jโ€ฌื‘ืื™ื–ื• ื–ื•ื•ื™ืช ื”ื’ื•ืฃ ื™ืชื ืชืง ืžื”ืžืกื™ืœื” ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ื’ื•ืฃ ื‘ืจื’ืข ื”ื ื™ืชื•ืง ืžื”ืžืกื™ืœื” ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_072.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื™ืœื“ ืฉืžืฉืงืœื• โ€ช 40โ€ฌืงื’" ื› ื”ื•ืœืš ืขืœ ืงื•ืจื” ืฉืื•ืจื›ื” โ€ช 2โ€ฌืžื˜ืจื™ื ื•ืžืกืชื” โ€ช 20โ€ฌืงื’" ื›โ€ช .โ€ฌื”ืงื•ืจื” ืชืœื•ื™ื” ื‘ืฉื ื™ ื—ื•ื˜ื™ืโ€ช .โ€ฌื—ื•ื–ืง ื”ื—ื•ื˜ ื”ื™ืžื ื™ ืœืงืจื™ืขื” ื”ื•ืโ€ฌ
โ€ซโ€ช 350โ€ฌื ื™ื•ื˜ื•ืŸ ื•ื—ื•ื–ืง ื”ื—ื•ื˜ ื”ืฉืžืืœื™ ื”ื•ื โ€ช 400โ€ฌื ื™ื•ื˜ื•ืŸโ€ช .โ€ฌืžื”ื• ื”ืชื—ื•ื ื‘ื• ื™ื›ื•ืœ ื”ื™ืœื“ ืœืฉื”ื•ืช ืžื‘ืœื™ ืฉื”ื—ื•ื˜ ื™ืงืจืข ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_073.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžื•ื˜ ืฉืžืกืชื• โ€ช mโ€ฌื•ืื•ืจื›ื• โ€ช Lโ€ฌืžื—ื•ื‘ืจ ื‘ืขื–ืจืช ื—ื•ื˜ ืœืงื™ืจ ืœื ื—ืœืง ื›ืžื•ืจืื”โ€ช .โ€ฌื ืชื•ืŸ โ€ช . b=700โ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ืžืงื“ื ื”ื—ื™ื›ื•ืš ื›ื“ื™ ืฉื”ืžืงืœ ืœืโ€ฌ
โ€ซื™ื—ืœื™ืง )ืชืฉื•ื‘ื” ืžืกืคืจื™ืช!(?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_074.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 5โ€ฌืง" ื’ ืชืœื•ื™ ืขืœ ืžื•ื˜ ืื—ื™ื“ ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช โ€ช 600โ€ฌืžื”ืงื™ืจโ€ช .โ€ฌื”ืžื•ื˜โ€ช ,โ€ฌืฉืžืกืชื• โ€ช ,mโ€ฌืžื•ื—ื–ืง ืข" ื™ ื—ื‘ืœ ื”ื™ื•ืฆืจ ืขื ื”ืงื™ืจ ื–ื•ื•ื™ืช ื‘ืช โ€ช.450โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ื”ืžืชื™ื—ื•ืช โ€ช Tโ€ฌื‘ื—ื‘ืœ ื–ื”โ€ช ,โ€ฌืื ื™ื“ื•ืข ืฉโ€ช? m=2kg-โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌื”ืจืื• ืฉื›ืืฉืจ โ€ช ,mà 0โ€ฌื”ื›ื•ื— ืฉืžืคืขื™ืœ ื”ืžื•ื˜ ืขืœ ื ืงื•ื“ืช ื—ื™ื‘ื•ืจ ื”ื—ื•ื˜ื™ื ืžื›ื•ื•ืŸ ืœืื•ืจืš ื”ืžื•ื˜โ€ช) .โ€ฌื”ืขืจื” โ€ช :โ€ฌื›ื•ื— ื–ื” ืžืืคืก ืืช ื”ืฉืงื•ืœ ืฉืœ โ€ช Tโ€ฌื•โ€ชT -โ€ฌโ€ฌ
โ€ซ'(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_075.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื—ื•ื˜ )ื—ืกืจ ืžืกื” ื•ืขื•ื‘ื™( ืžืœื•ืคืฃ ืขืœ ื“ื™ืกืงื” ืื—ื™ื“ื” ืฉืžืกืชื” โ€ช 100โ€ฌื’ืจืโ€ช .โ€ฌืžื—ื–ื™ืงื™ื ื‘ืงืฆื” ื”ื—ื•ื˜ ื•ืขื•ื–ื‘ื™ืโ€ช .โ€ฌืžื”ื™ ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ื“ื™ืกืงื” ?โ€ฌ
โ€ซืชื•ืš ื›ืžื” ื–ืžืŸ ืชื™ืกืชื•ื‘ื‘ ื”ื“ื™ืกืงื” โ€ช 10โ€ฌืจื“ื™ืื ื™ื )ื›ืžื” ืกื™ื‘ื•ื‘ื™ื ื–ื” ?(โ€ช ,โ€ฌืื ื”ืชื—ื™ืœื” ืžืžื ื•ื—ื” ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_076.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื ืชื•ื ื” ื”ืžืขืจื›ืช ื”ื‘ืื”โ€ช .โ€ฌื”ืืœืžื ื˜ื™ื ื”ื™ื—ื™ื“ื™ื ื”ืžืชื—ื›ื›ื™ื ื”ื ื”ื—ื•ื˜ ื•ื”ื’ืœื’ืœืช ื”ืชื—ืชื•ื ื”โ€ช .โ€ฌื’ืœื’ืœืช ื–ื• ืžืชื’ืœื’ืœืช ืœืœื ื”ื—ืœืงื”โ€ช .โ€ฌื ืชื•ืŸ โ€ชm1=1 kg, :โ€ฌโ€ฌ
โ€ซโ€ช .m2=2 kg , R=10cm, m3=3 kgโ€ฌืžื”ื™ ืชืื•ืฆืช โ€ช?m1โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_077.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืงื•ืจื” ืฉืžืกืชื” โ€ช m=10kgโ€ฌื•ืื•ืจื›ื” โ€ช L=0.5 mโ€ฌืžื•ื—ื–ืงืช ืœืงื™ืจ ืข" ื™ ืฆื™ืจ ืžืฉื—ืจืจื™ื ืืช ื”ืงื•ืจื” ืžืžื ื•ื—ื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ืžื•ืžื ื˜ ื”ืื™ื ืจืฆื™ื” ืฉืœ ื”ืงื•ืจื” ื™ื—ืกื™ืช ืœืฆื™ืจ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžืฆืื• ื‘ื™ื˜ื•ื™ ืœืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ื•ื•ื™ืช โ€ช.bโ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ืŸ ื”ืชืื•ืฆื•ืช ื”ืžืฉื™ืงื™ื•ืช โ€ช aA, aBโ€ฌื‘ื ืงื•ื“ื•ืช โ€ช Aโ€ฌื•โ€ช B-โ€ฌื›ืืฉืจ โ€ช? b=500โ€ฌื ืชื•ืŸ โ€ช? OA=0.15m, OB=0.4m :โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_078.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซโ€ช.โ€ฌืฉื ื™ ื‘ื•ืœื™ื ืžื—ื•ื‘ืจื™ื ื‘ื—ื•ื˜ ืฉืžืกืชื• ื–ื ื™ื—ื”โ€ฌ
โ€ซโ€ช.โ€ฌื‘ื•ืœ ื ืžื•ื ื— ืขืœ ืฉื•ืœื—ืŸ ืขื ืžืงื“ื ื—ื™ื›ื•ืš โ€ช 0.4โ€ฌื›ืฉื›ื•ื— ื—ื™ืฆื•ื ื™ ืคื•ืขืœ ืขืœ ื”ืžืขืจื›ืช ื”ื™ื ื ืžืฆืืช ื‘ืฉโ€ช.โ€ฌืžโ€ฌ
โ€ซืžื” ื’ื•ื“ืœื• ื•ื›ื™ื•ื•ื ื• ืฉืœ ื›ื•ื— ื”ื—ื™ื›ื•ืš ื”ืคื•ืขืœ ืขืœ ื”ื‘ื•ืœ ื›ืืฉืจ ื’ื•ื“ืœื• ืฉืœ ื”ื›ื•ื— ื”ื—ื™ืฆื•ื ื™ ื”ื•ื โ€ช 250โ€ฌื ื™ื˜ื•ืŸ )โ€ช1โ€ฌโ€ฌ
โ€ซืžื”ื• ื’ื•ื“ืœื• ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ื›ื•ื— ื”ื—ื™ืฆื•ื ื™ ื›ืš ืฉื”ืžืขืจื›ืช ืชืฉืืจ ืขื“ื™ื™ืŸ ื‘ืžื ื•ื—ื” )โ€ช2โ€ฌโ€ฌ
โ€ซื”ื›ื•ื— ื—ื“ืœ ืœืคืขื•ืœโ€ช ,โ€ฌื‘ืื™ื–ื• ืชืื•ืฆื” ื™ื ื•ืข ื”ื‘ื•ืœ )โ€ช3โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_079.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช Mโ€ฌืžื•ื ื— ืขืœ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื—ืœืง ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช โ€ช.โ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช mโ€ฌืžื•ื ื— ืขืœื™ื• ื›ืฉื”ื•ื ืงืฉื•ืจ ืืœื™ื• ืข"ื™ ื—ื•ื˜ ื”ืขื•ื‘ืจ ืกื‘ื™ื‘ ื’ืœื’ืœืช ื‘ืขืœืช ืžืกื” ื–ื ื™ื—ื”โ€ฌ
โ€ซืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืคื™ื ื”ื•ืโ€ฌ
โ€ซืžืฉื—ืจืจื™ื ืืช ื”ืžืขืจื›ืช ืžืžื ื•ื—ื” ื•ื”ื™ื ืžืชื—ื™ืœื” ืœื”ื—ืœื™ืงโ€ฌ
โ€ซื‘ื˜ื ืืช ืชืฉื•ื‘ื•ืชื™ืš ื‘ืขื–ืจืช โ€ชm ,M, g , ,โ€ฌโ€ฌ
โ€ซโ€ช (1โ€ฌืฉืจื˜ื˜ ืืช ืชืจืฉื™ืžื™ ื”ื›ื•ื—ื•ืช ืขื‘ื•ืจ ื›ืœ ืื—ื“ ืžื”ื’ื•ืคื™ืโ€ฌ
โ€ซโ€ช(2โ€ฌื—ืฉื‘ ืืช ืชืื•ืฆืช ื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช(3โ€ฌื›ืžื” ื–ืžืŸ ื™ื—ืœื™ืง ื”ื’ื•ืฃ โ€ชmโ€ฌืขืœ โ€ช Mโ€ฌืขื“ ืฉืงืฆื•ืชื™ื”ื ื”ื™ืžื ื™ื™ื ื™ืชืœื›ื“ื•โ€ช,โ€ฌโ€ฌ
โ€ซื‘ื”ื ื—ื” ืฉื‘ืจื’ืข ืฉื—ืจื•ืจ ื”ืžืขืจื›ืช ื”ื’ื•ืคื™ื ื ืžืฆืื™ื ื‘ืžืจื—ืง โ€ชLโ€ฌื–ื” ืžื–ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_080.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซื›ื“ื•ืจ ืงื˜ืŸ ืฉืžืกืชื• โ€ช mโ€ฌืงืฉื•ืจ ืœืงืฆื”ื• ืฉืœ ื—ื•ื˜ ืฉืื•ืจื›ื• โ€ช.lโ€ฌโ€ฌ
โ€ซืงืฆื”ื• ื”ืฉื ื™ ืฉืœ ื”ื—ื•ื˜ ืžื—ื•ื‘ืจ ืืœ ืฆื™ืจ ืกื™ื‘ื•ื‘ ืื•ืงืคื™ ื—ืœืง ื”ืขื•ื‘ืจ ื‘ื’ื•ื‘ื” โ€ช3lโ€ฌโ€ฌ
โ€ซืžืขืœ ื”ืจืฆืคื”โ€ช .โ€ฌืื•ื—ื–ื™ื ื‘ื›ื“ื•ืจ ื›ืืฉืจ ื”ื—ื•ื˜ ืžืชื•ื— ื‘ืื•ืจืš โ€ช lโ€ฌื•ื”ื›ื“ื•ืจ ื ืžืฆื ืื ื›ื™ืช ืžืขืœ ืฆื™ืจ ื”ืกื™ื‘ื•ื‘โ€ช.โ€ฌโ€ฌ
โ€ซื‘ืžืฆื‘ ื–ื” ืžืขื ื™ืงื™ื ืœื›ื“ื•ืจ ืžื”ื™ืจื•ืช ืื•ืคืงื™ืช โ€ช V0โ€ฌืขืœ ืžื ืช ืฉื”ื›ื“ื•ืจ ื™ื‘ืฆืข ืชื ื•ืขื” ื‘ืžืขื’ืœ ื–ืงื•ืฃโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื™ื โ€ชm,l,gโ€ฌโ€ฌ
โ€ซื(ืžื” ื”ืžื”ื™ืจื•ืช ื”ืžื™ื ื™ืžืœื™ืช โ€ชV0โ€ฌืฉื”ื›ื“ื•ืจ ืื›ืŸ ื™ื‘ืฆืข ืชื ื•ืขื” ืžืขื’ืœื™ืช ื–ืงื•ืคื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื‘(ืžืขื ืงื™ื ืœื›ื“ื•ืจ ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืชโ€ฌ
โ€ซื‘ื”ื ื—ื” ืฉื”ื—ื•ื˜ ื ืงืจืข ื‘ืจื’ืข ืฉืžืชื™ื—ื•ืชื• ืขื•ืœื” ืขืœโ€ฌ
โ€ซื’( ืžื” ืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจ ื‘ืจื’ืข ืฉื”ื—ื•ื˜ ื ืงืจืข?โ€ฌ
โ€ซืžืฆื ืืชโ€ฌ
โ€ซื“( ืชื•ืš ื›ืžื” ื–ืžืŸ ืžืจื’ืข ืงืจื™ืขืช ื”ื—ื•ื˜ ื™ืคื’ืข ื”ื›ื“ื•ืจ ื‘ืจืฆืคื”โ€ช .โ€ฌื‘ื—ืœืง ื–ื” ื”ื ื—โ€ฌ
โ€ซื‘ื” ื ืžืฆื ื”ื›ื“ื•ืจ ื‘ืจื’ืข ืฉื”ื—ื•ื˜ ื ืงืจืขโ€ช.โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_5_081.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
โ€ซืžืกื” ื ืงื•ื“ืชื™ืช ืžืชื—ื™ืœื” ืืช ืชื ื•ืขืชื” ืžืžืฆื‘ ืžื ื•ื—ื” ืžืคืกื’ืช ืžืฉื˜ื— ื›ื“ื•ืจื™ ื‘ืขืœ ืจื“ื™ื•ืก โ€ชRโ€ฌโ€ฌ
โ€ซื”ืžืฉื˜ื— ื—ืœืง ื•ืงื‘ื•ืขโ€ฌ
โ€ซื(ื—ืฉื‘ื• ืืช ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ืฉืœ ื”ืžืกื” ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ื•ื•ื™ืชโ€ฌ
โ€ซื‘(ื—ืฉื‘ื• ืืช ื”ืชืื•ืฆื” ื”ืจื“ื™ืืœื™ืช ื•ื”ืžืฉื™ืงื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ื•ื•ื™ืชโ€ฌ
โ€ซื’( ื‘ืื™ื–ื” ื–ื•ื•ื™ืช ืชืฉืชื—ืจืจ ื”ืžืกื” ืžื”ืžืฉื˜ื— ื”ื›ื“ื•ืจื™โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_082.htmlโ€ฌโ€ฌ
โ€ซืชืงืœื™ื˜ ืžืกืชื•ื‘ื‘ ื‘ืงืฆื‘ ืฉืœ โ€ช 33โ€ฌืกื™ื‘ื•ื‘ื™ื ื‘ื“ืงื”โ€ฌ
โ€ซื(ืžื”ื™ ืชื“ื™ืจื•ืช ื”ืกื™ื‘ื•ื‘ ืฉืœ ื”ืชืงืœื™ื˜?โ€ฌ
โ€ซื‘(ืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœ ื”ืกื™ื‘ื•ื‘?โ€ฌ
โ€ซื’( ืชื•ืš ื›ืžื” ื–ืžืŸ ืžืชืงื“ืžืช ื ืงื•ื“ื” ืขืœ ื”ืชืงืœื™ื˜ ืœื–ื•ื•ื™ืช ืฉืœ โ€ช 90โ€ฌืžืขืœื•ืช?โ€ฌ
โ€ซโ€ช10_Newton/e_10_5_083.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืช ื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื‘ื•ืœ ืงื˜ืŸ ืฉืžืกืชื• โ€ช mโ€ฌืžื—ืœื™ืง ืขืœ ืžืกื™ืœื” ื”ืžืกืชื™ื™ืžืช ื‘ืžืกื™ืœื” ืžืขื’ืœื™ืช ืื ื›ื™ืชโ€ช ,โ€ฌื›ืžืชื•ืืจ ื‘ืฆื™ื•ืจโ€ช .โ€ฌื”ืžืกื™ืœื” ื›ื•ืœื” ื ื˜ื•ืœืช ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซื( ืžื”ื™ ื”ืžื”ื™ืจื•ืช ืฉืœ ื”ื‘ื•ืœ )ื’ื•ื“ืœ ื•ื›ื™ื•ื•ืŸ( ื‘ื”ื’ื™ืขื• ืœื ืงื•ื“ื” โ€ช?Tโ€ฌโ€ฌ
โ€ซื‘(ืžื”ื™ ื”ืชืื•ืฆื” ื”ืจื“ื™ืืœื™ืช ื•ื”ืชืื•ืฆื” ื”ืžืฉื™ืงื™ืช ื‘ื ืงื•ื“ื” โ€ช?Tโ€ฌโ€ฌ
โ€ซื’( ืžื”ื• ื’ื•ื“ืœ ื”ื›ื•ื— ื”ืฉืงื•ืœ ื”ืคื•ืขืœ ืขืœ ื”ื‘ื•ืœ ื‘ื ืงื•ื“ื” โ€ช?Tโ€ฌโ€ฌ
โ€ซ ืชืคืขื™ืœ ืขืœื™ื• ื”ืžืกื™ืœื” ื›ื•ื— ื ื•ืจืžืœื™ ื”ืฉื•ื•ื”โ€ฌQ โ€ซื“( ื‘ืื™ื–ื” ื’ื•ื‘ื” ืžืขืœ ืชื—ืชื™ืช ื”ืžืกื™ืœื” ื”ืžืขื’ืœื™ืช ื™ืฉ ืœืฉื—ืจืจ ืืช ื”ื‘ื•ืœ ื›ื“ื™ ืฉื‘ื”ื’ื™ืขื• ืืœ ื”ื ืงื•ื“ื”โ€ฌ
?โ€ซืœืžืฉืงืœื•โ€ฌ
10_Newton/e_10_8_010.html
Friction
Block m1 in the figure below has a mass of 4.20 kg and block m2 has a mass of 2.30 kg. The coefficient of
kinetic friction between m2 and the horizontal plane is 0.47. The inclined plane is frictionless. Find (a) the
acceleration of the blocks and (b) the tension in the string.
10_Newton/e_10_8_026.html
Circular motion
A car moves at a constant speed on a straight but hilly road. One section has a crest (peak) and dip of the same
250-m radius, as shown in the figure below. (a) As the car passes over the crest, the normal force on the car is
one-half the 16-kN weight of the car. What will be the normal force on the car as it passes through the bottom
of the dip? (b) What is the greatest speed at which the car can move without leaving the road at the top of the
hill? (c) Moving at the speed found in (b), what will be the normal force on the car as it moves through the
bottom of the dip?
10_Newton/e_10_8_040.html
(โ€ซื—ื™ื›ื•ืš )ื‘ืขื™ื” ื‘ืชืœืช ืžื™ืžื“โ€ฌ
.โ€ซ ื”ืืจื’ื– ืžืชื—ื›ืš ืขื ืฉืชื™ ื”ืฆืœืขื•ืชโ€ฌ.โ€ซืืจื’ื– ื‘ืฆื•ืจืช ืงื•ื‘ื™ื” ืžื—ืœื™ืง ื‘ืชื•ืš ืชืขืœื” ืžืฉื•ืคืขืช ื”ืžื•ืฆื’ืช ื‘ืฆื™ื•ืจ ื”ืฉืžืืœื™ ื•ื—ืชื›ื” ืžื•ืฆื’ ื‘ืฆื™ื•ืจ ื”ื™ืžื ื™โ€ฌ
.
โ€ซ ื—ืฉื‘ื• ืืช ืชืื•ืฆืช ื”ืืจื’ื– ื‘ืืžืฆืขื•ืชโ€ฌ.
โ€ซืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืฃ ื•ื”ืฆืœืขื•ืช ื”ื•ืโ€ฌ
10_Newton/e_10_8_041.html
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
:โ€ซื ืชื•ื ื” ื”ืžืขืจื›ืช ื”ื‘ืื”โ€ฌ
T1, T2, T3 โ€ซื™ ืืช ื”ืžืชื™ื—ื•ื™ื•ืชโ€ฌ/โ€ซืžืฆืโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_042.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืงื•ืคืกื ืžื•ื ื—ืช ืข" ื’ ืจืฆืคืช ืงืจื•ืŸ ื”ื ืข ื‘ืžื”ื™ืจื•ืช โ€ช .[m/sec] 40โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ืงื•ืคืกื ืœืจืฆืคืช ื”ืงืจื•ืŸ ื”ื•ื โ€ช .ms = 0.3โ€ฌืื ื”ืงืจื•ืŸโ€ฌ
โ€ซืžืชื—ื™ืœ ืœื‘ืœื•ื ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ืขื“ ืœืขืฆื™ืจื” ืžื•ื—ืœื˜ืชโ€ช ,โ€ฌืžื”ื• ื”ืžืจื—ืง ื”ืžื™ื ื™ืžืœื™ ืฉืขืœื™ื• ืœืขื‘ื•ืจ ืขื“ ืฉื™ื™ืขืฆืจ ืข" ืž ืฉื”ืงื•ืคืกื ืœื ืชื—ืœื™ืง ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_043.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžืขืœื™ืช ืฉืžืกืชื” โ€ช [kg] 150โ€ฌืžื—ื•ื‘ืจืช ื‘ื›ื‘ืœ ื”ืžืกื•ื’ืœ ืœืขืžื•ื“ ื‘ืžืชื™ื—ื•ืช ืฉืœ ืขื“ โ€ช .[N] 5000โ€ฌืชืื•ืฆืชื” ื”ืžืงืกื™ืžืœื™ืช ืฉืœ ื”ืžืขืœื™ืช ื”ื™ื โ€ช] 2โ€ฌโ€ฌ
โ€ซโ€ช .[m/secโ€ฌื‘ื”ื ื—ื” ื›ื™ ืžืกืชื• ืฉืœ ืื“ื ื‘ื•ื’ืจ ื”ื™ื โ€ช [kg] 80โ€ฌืžื” ืžืกืคืจ ื”ืื ืฉื™ื ื”ืžืงืกื™ืžืœื™ ืฉื™ื›ื•ืœื™ื ืœืขืœื•ืช ื‘ืžืขืœื™ืช ื‘ื‘ื˜ื—ื” ? ืžื” ื”ื™ื” ืžืกืคืจโ€ฌ
โ€ซื”ืื ืฉื™ื ืื™ืœื• ื”ื™ืชื” ื”ืžืขืœื™ืช ื™ื•ืจื“ืช ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_044.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช [kg] 5โ€ฌืชืœื•ื™ ืขืœ ืžื•ื˜ ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 60oโ€ฌืžื”ืงื™ืจโ€ช .โ€ฌื”ืžื•ื˜ ืžื•ื—ื–ืง ืขืœ ื™ื“ื™ ื—ื‘ืœ ื”ื™ื•ืฆืจ ืขื ื”ืงื™ืจ ื–ื•ื•ื™ืช ืฉืœ โ€ช)45oโ€ฌืจืื”โ€ช/โ€ฌื™ ืฆื™ื•ืจ(โ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื™ ื”ืžืชื™ื—ื•ืช ื‘ื—ื•ื˜ ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_045.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื ืชื•ื ื” ืžืขืจื›ืช ืฉืœ ืฉื ื™ ื’ื•ืคื™ื ) ืจืื”โ€ช/โ€ฌื™ ืฆื™ื•ืจ(โ€ช .โ€ฌื”ื’ืœื’ืœืช ื•ื”ืžืฉื˜ื—ื™ื ื—ืกืจื™ ื—ื™ื›ื•ืšโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ] โ€ช m1 = 15 [kgโ€ฌื•โ€ช .m2 = 30 [kg ] -โ€ฌืžื” ืฆืจื™ืšโ€ฌ
โ€ซืœื”ื™ื•ืช ื”ื›ื•ื— โ€ช fโ€ฌื›ื“ื™ ืฉโ€ช: -โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืžืขืจื›ืช ืชื ื•ืข ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืžืขืจื›ืช ืชืื™ืฅ ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ืฉืœ โ€ช [m/sec2] 2โ€ฌืฉืžืืœื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_046.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžื›ื•ื ื™ืช ืžืฉื•ืœืฉืช ืžืื™ืฆื” ืฉืžืืœื” ื‘ืชืื•ืฆื” ืฉืœ โ€ช .[m/sec2] 10โ€ฌืžื–ื•ื•ื“ื” ืฉืžืกืชื” โ€ช [kg] 20โ€ฌื ืฉื›ื—ื” ืขืœ " ื’ื’" ื”ืžื›ื•ื ื™ืช )ื”ืฆื“ ื”ืžืฉื•ืคืข(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืื ืชื™ืคื•ืœ ื”ืžื–ื•ื•ื“ื” ืžื”ืžื›ื•ื ื™ืช ื‘ืžื”ืœืš ื”ื ืกื™ืขื”โ€ช ,โ€ฌื›ืœื•ืžืจ ืชื ื•ืข ื›ืœืคื™ ืžื˜ื” ื‘ื™ื—ืก ืœื’ื’ ?โ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌื™ืฉ ืœื—ืฉื‘ ืžื”ื• ื”ื›ื•ื— ื”ื ื•ืจืžืœื™โ€ฌ
โ€ซื‘โ€ช.โ€ฌื™ืฉ ืœื—ืฉื‘ ืืช ื”ืชืื•ืฆื” ืฉืœ ื”ืžื–ื•ื•ื“ื”โ€ฌ
โ€ซื’โ€ช.โ€ฌื™ืฉ ืœืฉืื•ืœ ืžื” ื›ื™ื•ื•ืŸ ืชืื•ืฆืช ื”ืžื–ื•ื•ื“ื” ื‘ื™ื—ืก ืœืžื›ื•ื ื™ืชโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_051.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืจ'โ€ช ,โ€ฌืกื˜ื•ื“ื ื˜ ืฉืžืกืชื• โ€ช [kg] 70โ€ฌื ืกืข ื‘ื—ื•ืคืฉืช ื”ืคืกื— ืœืขืฉื•ืช ืกืงื™ ืžื™ืโ€ช .โ€ฌืœืื—ืจ ืฉืœื‘ืฉ ืืช ื”ืžื’ืœืฉื™ื™ื ื•ืชืคืก ืืช ื”ื—ื‘ืœ ื‘ื—ื•ื–ืงื” ืกื™ืจืช ื”ืžืจื•ืฅโ€ฌ
โ€ซื”ืื™ืฆื” ื‘ืชืื•ืฆื” ืงื‘ื•ืขื” ืฉืœ โ€ช .[a = 5 [m/sec2โ€ฌื™ื“ื•ืข ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ืžื’ืœืฉื™ื™ื ืœืžื™ื ื”ื•ืโ€ช 0.2โ€ฌื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸโ€ฌ
โ€ซื”ืกื™ืจื”โ€ช ,โ€ฌืฉืžืกืชื” โ€ช ,[kg] 800โ€ฌืœืžื™ื ื”ื•ื โ€ช.0.35โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ื”ืžืชื™ื—ื•ืช ื‘ื—ื‘ืœ ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื”ื›ื•ื— ื”ื›ื•ืœืœ ืฉืžืคืขื™ืœ ื”ืžื ื•ืข ืขืœ ื”ืกื™ืจื” ืขืœ ืžื ืช ืฉืชืื™ืฅ ื‘ืชืื•ืฆื” ื–ื• ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืœืื—ืจ ื–ืžืŸ ืžื” ื”ืชื—ื™ืœ ื–ืจื ื—ื–ืง ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช [m/sec] 3โ€ฌื‘ื›ื™ื•ื•ืŸ ื”ืคื•ืš ืœื›ื™ื•ื•ืŸ ืชื ื•ืขืช ื”ืกื™ืจื” ื•ื”ื’ื•ืœืฉโ€ช ,โ€ฌืžื”ื• ื”ื›ื•ื— ืฉืฆืจื™ืš ืœื”ืคืขื™ืœ ื”ืžื ื•ืขโ€ฌ
โ€ซืข" ืž ืœืืคืฉืจ ืืช ื”ืชื ื•ืขื” ื”ืžื•ืืฆืช ื‘ืชื ืื™ื ืืœื” ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืœืื—ืจ ื”ืคืœื’ื” ืงืฆืจื” ื”ืชืขื™ื™ืฃ ื”ืงื™ื™ื˜ืŸ ื•ืขืœื” ืœืกื™ืจื”โ€ช ,โ€ฌืืš ืื•ื™ื”โ€ช ,โ€ฌืžื ื•ืข ื”ืกื™ืจื” ืื™ื ื• ืคื•ืขืœ ื•ื”ืกื™ืจื” ื”ื—ืœื” ืœื”ื™ืกื—ืฃ ืขื ื”ื–ืจื )ื›ืœื•ืžืจโ€ช ,โ€ฌืžื”ื™ืจื•ืชโ€ฌ
โ€ซื”ืกื™ืจื” ื‘ื™ื—ืก ืœืžื™ื ื”ื™ื โ€ช .(0โ€ฌืจ' ืžื™ื“ ื”ืชื ื“ื‘ ืœืคืชื•ืจ ืืช ื”ื‘ืขื™ื” ื•ืื—ื– ื‘ืžืฉื•ื˜ื™ืโ€ช .โ€ฌืื ื™ื“ื•ืข ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ืกื™ืจื” ืœืžื™ื ื”ื•ื โ€ช0.5โ€ฌโ€ฌ
โ€ซืžื”ื• ื”ื›ื•ื— ื”ืžื™ื ื™ืžืœื™ ืฉืขืœื™ื• ืœื”ืคืขื™ืœ ืข" ืž ืœื”ืื™ืฅ ืืช ื”ืกื™ืจื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_052.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื›ื“ื™ ืœืžืžืŸ ืืช ื—ื•ืคืฉืช ื”ืกืงื™ ื ืืœืฅ ืจ' )ืฉืžืกืชื• ืขืœืชื” ืœโ€ช ([kg] 72 -โ€ฌืœืขื‘ื•ื“ ื‘ื ื™ืงื•ื™ ื—ืœื•ื ื•ืช ื‘ื‘ื ื™ื™ื ื™ื ื’ื‘ื•ื”ื™ืโ€ช .โ€ฌื‘ืžื”ืœืš ืขื‘ื•ื“ืชื• ืขืžื“ ืขืœ ืœื•ื—โ€ฌ
โ€ซืฉืžืกืชื• โ€ช [kg] 40โ€ฌื”ืžื—ื•ื‘ืจ ืœื’ืœื’ืœืช ื—ืกืจืช ื—ื™ื›ื•ืš ื•ืžืกื” ื‘ืื•ืคืŸ ื”ื‘ืโ€ช:โ€ฌโ€ฌ
โ€ซื›ื“ื™ ืœืขืœื•ืช ื•ืœืจื“ืช ืฆืจื™ืš ืจ' ืœืžืฉื•ืš ื‘ื—ื‘ืœโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืื™ื–ื” ื›ื•ื— ืฆืจื™ืš ืจ' ืœื”ืคืขื™ืœ ืขืœ ื”ื—ื‘ืœ ื›ื“ื™ ืœืขืœื•ืช ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ื”ื›ื•ื— ืื•ืชื• ื”ื•ื ืžืคืขื™ืœ ืขืœ ื”ืœื•ื— ื‘ืžืงืจื” ื–ื” ?โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื• ื”ื›ื•ื— ืื•ืชื• ื”ื•ื ืžืคืขื™ืœ ืขืœ ืžื ืช ืœืจื“ืช ื‘ืชืืฆื” ืงื‘ื•ืขื” ืฉืœ โ€ช? [m/sec] 1โ€ฌโ€ฌ
โ€ซื“โ€ช.โ€ฌืœืื—ืจ ืฉื”ืชืขื™ื™ืฃ ื”ื—ืœื™ื˜ ืจ' ืœื™ื™ืขืœ ืืช ื”ืžืขืจื›ืช ื•ื”ื•ืกื™ืฃ ื’ืœื’ืœืช ื ื•ืกืคืชโ€ฌ
โ€ซืคืชื•ืจโ€ช/โ€ฌื™ ืืช ืฉืœื•ืฉืช ื”ืกืขื™ืคื™ื ืโ€ช,โ€ฌื‘โ€ช,โ€ฌื’ ืขื‘ื•ืจ ื”ืžืขืจื›ืช ื”ื—ื“ืฉื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_053.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื›ื•ืฉืจ ื”ื”ืคืจื“ื” ืฉืœ ืžื›ืฉื™ืจ ื”ืฆื ื˜ืจื™ืคื•ื’ื” )ืžื›ืฉื™ืจ ืœื”ืคืจื“ืช ืžืกื•ืช( ืžื•ื’ื“ืจ ื›ืชืื•ืฆื” ื”ืจื“ื™ืืœื™ืช )ืฆื ื˜ืจื™ืคื˜ืœื™ืช( ื‘ืกื™ื‘ื•ื‘โ€ช ,โ€ฌื‘ื™ื—ื™ื“ื•ืช ืฉืœ โ€ช.gโ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืžื›ืฉื™ืจ ืฆื ื˜ืจื™ืคื•ื’ื” ื”ืžืกืชื•ื‘ื‘ ื‘ืชื“ื™ืจื•ืช ืฉืœ ]โ€ช f = 50 [Hzโ€ฌื•ืจื“ื™ื•ืกื• ] โ€ช .R = 15 [cmโ€ฌืžื”ื• ื›ื•ืฉืจ ื”ื”ืคืจื“ื” ืฉืœ ื”ืžื›ืฉื™ืจ ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_054.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžื›ื•ื ื™ืช ื ื•ืกืขืช ื‘ื›ื‘ื™ืฉ ืžืขื’ืœื™ ืฉืจื“ื™ื•ืกื• โ€ช ,[m] 100โ€ฌื”ืžื•ื’ื‘ื” ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช.10°โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ืžื›ื•ื ื™ืช ืœื›ื‘ื™ืฉ ื”ื•ื โ€ช.0.3โ€ฌโ€ฌ
โ€ซืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืฉื™ืงื™ืช ื”ืžืงืกื™ืžืœื™ืช ื”ืืคืฉืจื™ืช ื›ืš ืฉื”ืžื›ื•ื ื™ืช ืชืžืฉื™ืš ื‘ืžืกืœื•ืœ ื”ืžืขื’ืœื™ ? ืžื” ื™ืงืจื” ืื ืชืขื‘ื•ืจ ื”ืžื›ื•ื ื™ืช ืžื”ื™ืจื•ืช ื–ื• ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_055.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืžืกื•ื‘ื‘ื™ื ื“ืœื™ ืžื™ื ืงืฉื•ืจ ื‘ื—ื‘ืœ ืฉืื•ืจื›ื• ] โ€ช R = 1.2 [mโ€ฌื‘ืฆื•ืจื” ืื ื›ื™ืชโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžื™ื ื™ืžืœื™ืช ืฉืœ ื”ื“ืœื™ ื‘ื ืงื•ื“ืช ื”ืคืกื’ื” ืข" ืž ืฉื”ืžื™ืโ€ฌ
โ€ซืœื ื™ื™ืฉืคื›ื• ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_061.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื ืชื•ื ื•ืช ืฉืชื™ ืžืกื•ืช ื”ืงืฉื•ืจื•ืช ื‘ื—ื‘ืœื™ื ืขืœ ืคื™ ื”ืฆื™ื•ืจโ€ช .โ€ฌืžืกื” โ€ช Bโ€ฌืžืฉืงืœื” โ€ช [N] 711โ€ฌื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ื ื” ื•ื‘ื™ืŸ ื”ืฉื•ืœื—ืŸ ื”ื•ื โ€ช.µ s = 0.25โ€ฌโ€ฌ
โ€ซื”ื ื—โ€ช/โ€ฌื™ ื›ื™ ื”ื—ื‘ืœ ื‘ื™ืŸ ืžืกื” โ€ช Bโ€ฌื•ื”ืงืฉืจ ืžืชื•ื— ืื•ืคืงื™ืชโ€ช .โ€ฌืžืฆื ืืช ื”ืžืกื” ื”ืžืงืกื™ืžืœื™ืช ืฉืœ โ€ช Aโ€ฌื›ืš ืฉื”ืžืขืจื›ืช ืชื™ืฉืืจ ื‘ืžื ื•ื—ื”โ€ช .โ€ฌืžื” ื™ืงืจื” ืื ืžืกื”โ€ฌ
โ€ซโ€ช Aโ€ฌืชื”ื™ื” ื›ื‘ื™ื“ื” ื™ื•ืชืจ ?โ€ฌ
โ€ซโ€ช10_Newton/e_10_8_062.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื‘ืฆื™ื•ืจ ื™ืฉื ืŸ ืฉืœื•ืฉ ืžืฉืงื•ืœื•ืชโ€ช.A, B, C ,โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ืžืกืชืŸ ืฉืœ ื”ืžืฉืงื•ืœื•ืช โ€ช Aโ€ฌื•โ€ช B -โ€ฌื”ืŸ ] โ€ช MA= 4 [kgโ€ฌื•โ€ช Mb = 2 [kg] -โ€ฌื•ื›ื™ ืžืงื“ืžื™ ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื•ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ืžืฉืงื•ืœืช โ€ช Aโ€ฌืœืฉื•ืœื—ืŸโ€ฌ
โ€ซื”ื โ€ช µ s = 0.2โ€ฌื•โ€ช.µ k = 0.15 -โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื” ื”ืžืกื” ื”ืžื™ื ื™ืžืœื™ืช ืฉืœ ืžืฉืงื•ืœืช โ€ช Cโ€ฌืขืœ ืžื ืช ืฉื”ืžืขืจื›ืช ืชื™ืฉืืจ ื‘ืžื ื•ื—ื” ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžืจื™ืžื™ื ื‘ืคืชืื•ืžื™ื•ืช ืืช ืžืฉืงื•ืœืช โ€ช ,Cโ€ฌืžื” ืชื”ื™ื” ืชืื•ืฆืชื” ืฉืœ ืžืฉืงื•ืœืช โ€ช? Aโ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_063.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซืœื›ื‘ื•ื“ ื”ืื‘ื™ื‘ ื”ืžืœื‘ื‘ ืงื ื” ืจ' ืœืื™ืžื• ืฉืขื•ืŸ ืงื™ืจ ืขื ืฉื ื™ ืžื—ื•ื’ื™ืโ€ช ,โ€ฌืฉืขื•ืช ื•ื“ืงื•ืชโ€ช ,โ€ฌืฉืจื“ื™ื•ืกื ] โ€ช. Rh = 15 [cm], Rm = 30 [cmโ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื—ื•ื’ื™ื ? ืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžืฉื™ืงื™ืช ืฉืœ ืงืฆื•ืช ื”ืžื—ื•ื’ื™ื ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืจ' ืกื™ื™ื ืœื›ื•ื•ืŸ ืืช ื”ืฉืขื•ืŸ ื‘ืฉืขื” โ€ช , 14:50โ€ฌืžื” ื”ื–ื•ื•ื™ืช ืฉื™ืขื‘ื•ืจ ืžื—ื•ื’ ื”ื“ืงื•ืช ืขื“ ืฉื™ืฉื™ื’ ืืช ืžื—ื•ื’ ื”ืฉืขื•ืช ?โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื” ื”ืžืจื—ืง ืฉื™ืขื‘ื•ืจ ืงืฆื” ืžื—ื•ื’ ื”ื“ืงื•ืช ืžื ืงื•ื“ืช ื”ืžืคื’ืฉ ืขื“ ื”ืฉืขื” โ€ช? 15:45โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_064.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื“ื™ืกืงื” ืžื—ืœื™ืงื” ืข" ื’ ืฉื•ืœื—ืŸ ืœืœื ื—ื™ื›ื•ืšโ€ช .โ€ฌื”ื“ื™ืกืงื” ืžื—ื•ื‘ืจืช ืœืžืกื” ื”ืชืœื•ื™ื” ืžืชื—ืช ืœืžืจื›ื– ื“ืจืš ื—ื•ืจ ื‘ืฉื•ืœื—ืŸโ€ช .โ€ฌืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื”ืžื”ื™ืจื•ืชโ€ฌ
โ€ซ)ื–ื•ื•ื™ืชื™ืช ื•ืžืฉื™ืงื™ืช( ืฉืœ ื”ื“ื™ืกืงื” ืขืœ ืžื ืช ืฉื”ืžืกื” ืชื™ืฉืืจ ื‘ืžื ื•ื—ื” ืื ืื•ืจืš ื”ื—ื‘ืœ ืžื”ื—ื•ืจ ืœื“ื™ืกืงื” ื”ื•ื โ€ช ? [cm] 25โ€ฌื‘ื›ืžื” ืชืฉืชื ื”โ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ืื ืžืจื—ืง ื–ื” ื™ืชืงืฆืจ ืœโ€ช? [cm] 15 -โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_065.htmlโ€ฌโ€ฌ
โ€ซื—ื•ืงื™ ื ื™ื•ื˜ื•ืŸโ€ฌ
โ€ซื›ื“ื•ืจ ืฉืžืกืชื• ]โ€ช m = 1.4 [kgโ€ฌืžื—ื•ื‘ืจ ื‘ืขื–ืจืช ืฉื ื™ ืžื™ืชืจื™ื ืœืžื•ื˜ ื”ืžืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจื• โ€ช .โ€ฌื™ื“ื•ืข ื›ื™ ื”ืžืชื™ื—ื•ืช ื‘ืžื™ืชืจ ื”ืขืœื™ื•ืŸ ื”ื™ื ] โ€ชTu = 35โ€ฌโ€ฌ
โ€ซ]โ€ช Nโ€ฌื•ื›ื™ ืฉื ื™ ื”ืžื™ืชืจื™ื ืฉืื•ืจื›ื โ€ช [m] 1.7โ€ฌืžืชื•ื—ื™ืโ€ช ,โ€ฌื•ื›ื™ ื”ืžืจื—ืง ื‘ื™ืŸ ื ืงื•ื“ืช ื”ืงืฉื™ืจื” ืฉืœ ื”ืžื™ืชืจื™ื ืขืœ ื”ืžื•ื˜ ื”ื™ื โ€ช [m] 1.7โ€ฌื’ื ื›ืŸโ€ช .โ€ฌื—ืฉื‘โ€ช/โ€ฌื™โ€ฌ
โ€ซืืชโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืžืชื™ื—ื•ืช ื‘ืžื™ืชืจ ื”ืชื—ืชื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌื”ื›ื•ื— ื”ืคื•ืขืœ ืขืœ ื”ื›ื“ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจ )ืžืฉื™ืงื™ืช ื•ื–ื•ื•ื™ืชื™ืช(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช10_Newton/e_10_8_066.htmlโ€ฌโ€ฌ
โ€ซโ€ชNewton Lawsโ€ฌโ€ฌ
โ€ซโ€ชA 5.5-kg block is initially at rest on a frictionless horizontal surface.โ€ฌโ€ฌ
โ€ซโ€ชIt is pulled with a constant horizontal force of 3.8 N.โ€ฌโ€ฌ
(a) What is its acceleration?
(b) How long must it be pulled before its speed is 5.2 m/s?
(c) How far does it move in this time?
10_Newton/e_10_8_067.html
Newton Laws
An electron travels in a straight line from the cathode of a vacuum tube to its anode, which is 1.5 cm away.
It starts with zero speed and reaches the anode with a speed of 5.8 X 106 m/s. Assume constant acceleration
and compute the force on the electron.
This force is electrical in origin. The electron's mass is 9.11 X 10-31 kg.
10_Newton/e_10_8_068.html
Newton Laws
A car traveling at 53 km/h hits a bridge support. A passenger in the car moves forward a distance of 65 cm
(with respect to the road) while being brought to rest by an inflated air bag.
What force (assumed constant) acts on the passen¬ger's upper torso, which has a mass of 39 kg?
10_Newton/e_10_8_069.html
Newton Laws
A certain force gives object m1 an acceleration of 12.0 m/s2. The same force gives object m2 an acceleration of
3.30 m/s2.
What acceleration would the force give to an object whose mass is
(a) the difference between m1 and m2 and
(b) the sum of m1 and m2 ?
10_Newton/e_10_8_070.html
Newton Laws
Two blocks, with masses m1 = 4.6 kg and m2 = 3.8 kg, are connected by a light spring on a horizontal
frictionless table. At a certain instant, when m2 has an acceleration a2 = 2.6 m/s2,
(a) what is the force on m2 and
(b) what is the acceleration of m1 ?
10_Newton/e_10_8_071.html
Newton Laws
A space trave1er whose mass is 75.0 kg leaves Earth. Compute his weight
(a) on Earth,
(b) on Mars, where g = 3.72 m/s2, and
(c) in interplanetary space.
(d) What is his mass at each of these locations?
10_Newton/e_10_8_072.html
Newton Laws
A 12,000-kg airplane is in level flight at a speed of 870 km/h.
What is the upward-directed lift force exerted by the air on the airplane?
10_Newton/e_10_8_073.html
Newton Laws
A jet plane starts from rest on the runway and accelerates for takeoff at 2.30 m/s2 (= 7.55 ft/s2).
It has two jet engines, each of which exerts a thrust of 1.40 X 105 N (= 15.7 tons). What is the weight of the
plane?
10_Newton/e_10_8_074.html
Newton Laws
(a) Two l0-lb weights are attached to a spring scale as shown in the figure below. What is the reading of the
scale?
(b) A single l0-lb weight is attached to a spring scale which itself is at¬tached to a wall, as shown below.
What is the reading of the scale? (Ignore the weight of the scale.)
10_Newton/e_10_8_075.html
Newton Laws
A car moving initially at a speed of 50 mi/h (~80 km/h) and weighing 3000 Ib (~13,000 N) is brought to a stop
in a distance of 200 ft (~61 m). Find
(a) the braking force and
(b) the time required to stop.
Assuming the same braking force, find
(c) the distance and
(d) the time required to stop if the car were going 25 mi/h (~40 km/h) initially.
10_Newton/e_10_8_076.html
Newton Laws
An object is hung from a spring scale attached to the ceiling of an elevator.
The scale reads 65 N when the elevator is standing still.
(a) What is the reading when the elevator is moving upward with a constant speed of 7.6 m/s?
(b) What is the reading of the scale when the elevator is moving upward with a speed of 7.6 m/s and
decelerating at 2.4 m/s2 ?
10_Newton/e_10_8_077.html
Newton Laws
A 77-kg person is parachuting and experiencing a downward acceleration of 2.5 m/s2 shortly after opening the
parachute.
The mass of the parachute is 5.2 kg.
(a) Find the upward force exerted on the parachute by the air.
(b) Calculate the downward force exerted by the person on the parachute.
10_Newton/e_10_8_078.html
Newton Laws
A l5,000-kg helicopter is lifting a 4500-kg car with an up¬ward acceleration of 1.4 m/s2. Calculate
(a) the vertical force the air exerts on the helicopter blades and
(b) the tension in the upper supporting cable. See the picture below.
10_Newton/e_10_8_079.html
Newton Laws
A l400-kg jet engine (below) is fastened to the fuselage of a passenger jet by just three bolts (this is the usual
practice). Assume that each bolt supports one-third of the load.
(a) Calculate the force on each bolt as the plane waits in line for clearance to take off.
(b) During flight, the plane encounters turbulence, which suddenly imparts an upward vertical acceleration of
2.60 m/s2 to the plane.
Calculate the force on each bolt now. Why are only three bolts used?
10_Newton/e_10_8_080.html
Newton Laws
A child's toy consists of three cars that are pulled in tandem on small frictionless rollers as shown in the figure
below.
The cars have masses m1 = 3.1 kg, m2= 2.4 kg, and m3 = 1.2 kg. If they are pulled to the right with a
horizontal force P = 6.5 N, find
(a) the acceleration of the system,
(b) the force exerted by the second car on the third car, and
(c) the force exerted by the first car on the second car.
10_Newton/e_10_8_081.html
Newton Laws
Two blocks are in contact on a frictionless table, as shown below. A horizontal force is applied to one block, as
shown below.
(a) If m1 = 2.3 kg, m2 = 1.2 kg, and F = 3.2 N, find the force of contact between the two blocks.
(b) Show that if the same force F is applied to m2 rather than to m1, the force of contact between the blocks is
2.1 N, which is not the same value de¬rived in (a). Explain.
10_Newton/e_10_8_082.html
A body with mass m is acted on by two forces F1 and F2, as shown in the figure below.
If m = 5.2 kg, F1 = 3.7 N, and F2 = 4.3 N, find the vector acceleration of the body.
10_Newton/e_10_8_083.html
A 5.1-kg block is pulled along a frictionless floor by a cord that exerts a force P = 12 N at an angle = 25°
above the horizontal, as shown in the figure below.
(a) What is the acceleration of the block?
(b) The force P is slowly increased. What is the value of P just before the block is lifted off the floor?
(c) What is the acceleration of the block just before it is lifted off the floor?
10_Newton/e_10_8_084.html
A worker drags a crate across a factory floor by pulling on a rope tied to the crate.
The rope, which is inclined at 38.0° above the horizontal, exerts a force of 450 N on the crate.
The floor exerts a horizontal resistive force of 125 N, as shown in the figure below.
Calculate the acceleration of the crate
(a) if its mass is 96.0 kg, and
(b) if its weight is 96.0 N.
10_Newton/e_10_8_085.html
A 1200-kg car is being towed up an 18° incline by means of a rope attached to the rear of a truck. The rope
makes an angle of 27° with the incline.
What is the greatest distance that the car can be towed in the first 7.5 s starting from rest if the rope has a
breaking strength of 4.6 kN?
Ignore all resistive forces on the car. See the figure below:
10_Newton/e_10_8_086.html
A 110-kg crate is pushed at constant speed up a frictionless 34° ramp, as shown in the figure below.
What horizontal force F is required? (Hint: Resolve forces into components parallel to the ramp.)
10_Newton/e_10_8_087.html
An elevator weighing 6200 lb is pulled upward by a cable with an acceleration of 3.8 ft/s2.
(a) What is the tension in the cable?
(b) What is the tension when the elevator is accelerating downward at 3.8 ft/s2 but is still moving upward?
10_Newton/e_10_8_088.html
A lamp hangs vertically from a cord in a descending elevator.
The elevator has a deceleration of 2.4 m/s2 before coming to a stop.
(a) If the tension in the cord is 89 N, what is the mass of the lamp?
(b) What is tension in the cord when the elevator ascends (goes up) with an upward acceleration of 2.4 m/s2?
10_Newton/e_10_8_089.html
An 11-kg monkey is climbing a massless rope attached to a 15-kg log (piece of wood) over a frictionless tree
limb.
(a) With what mini¬mum acceleration must the monkey climb up the rope so that it can raise the 15-kg log off
the ground?
If, after the log has been raised off the ground, the monkey stops climbing and hangs on to the rope, what will
now be
(b) the monkey's acceleration and
(c) the tension in the rope?
10_Newton/e_10_8_090.html
The figure below shows a section of an alpine cable-car system. The maximum permitted mass of each car with
occupants is 2800 kg.
The cars, riding on a support cable, are pulled by a second cable attached to each pylon (support column).
What is the difference in tension between adjacent sections of pull cable if the cars are accelerated up at a 35°
incline at 0.81 m/s2, as shown below?
10_Newton/e_10_8_091.html
The man in the figure below weighs 180 lb; the platform and at¬tached frictionless pulley weigh a total of 43
lb. Ignore the weight of the rope.
With what force must the man pull up on the rope in order to lift himself and the platform upward at 1.2 ft/s2?
10_Newton/e_10_8_092.html
Block B in the figure below weighs 712 N. The coefficient of static friction between block B and the table is
0.25.
Find the maximum weight of block A for which block B will remain at rest.
10_Newton/e_10_8_093.html
Block m1 in the figure below has a mass of 4.20 kg and block m2 has a mass of 2.30 kg.
The coefficient of kinetic friction between m2 and the horizontal plane is 0.47.
The inclined plane is frictionless. Find
(a) the acceleration of the blocks and
(b) the tension in the string.
10_Newton/e_10_8_094.html
In the figure below, object B weighs 94.0 lb and object A weighs 29.0 lb.
Between object B and the plane the coefficient of sta¬tic friction is 0.56 and the coefficient of kinetic friction
is 0.25.
(a) Find the acceleration of the system if B is initially at rest.
(b) Find the acceleration if B is moving up the plane.
(c) What is the acceleration if B is moving down the plane? The plane is inclined by 42.0°.
10_Newton/e_10_8_095.html
During an Olympic bobsled run, a European team takes a turn of radius 25 ft at a speed of 60 mi/h.
What acceleration do the riders experience
(a) in ft/s2 and
(b) in units of g?
10_Newton/e_10_8_096.html
A 2400-lb (= 10.7-kN) car traveling at 30 mi/h (= 13.4 m/s) attempts to round an unbanked curve with a radius
of 200 ft (= 61.0 m).
(a) What force of friction is required to keep the car on its circular path?
(b) What minimum coefficient of static friction between the tires and road is required?
10_Newton/e_10_8_097.html
A circular curve of highway is designed for traffic moving at 60 km/h (= 37 mi/h).
(a) If the radius of .the curve is 150 m (= 490 ft), what is the correct angle of banking of the road?
(b) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that
would keep traffic from skidding (slipping) at this speed?
10_Newton/e_10_8_098.html
A conical pendulum is formed by attaching a 53-g pebble to a l.4-m string.
The pebble swings around in a circle of radius 25 cm.
(a) What is the speed of the pebble?
(b) What is its acceleration?
(c) What is the tension in the string?
10_Newton/e_10_8_099.html
A banked circular highway curve is designed for traffic moving at 95 km/h. The radius of the curve is 210 m.
Traffic is moving along the highway at 52 km/h on a stormy day.
(a) What is the minimum coefficient of friction between tires and road that will allow cars to turn without
sliding?
(b) With this value of the coefficient of friction, what is the greatest speed at which the cars can turn without
sliding?
10_Newton/e_10_8_100.html
A particle of mass m is subjected to a net force F(t) given by F(t)=Fo(1-t/T)i; that is, F(t) equals Fo at t = 0 and
decreases linearly to zero in time T. The particle passes the origin x = 0 with velocity voi. Show that at the
instant t = T that F(t) vanishes, the speed v and distance x traveled are given by v(T) = vo + aoT/2, and x(T) =
voT + aoT^2/3, where ao = Fo/m is the initial acceleration.
10_Newton/e_10_8_101.html
A horizontal force F of 12 lb pushes a block weighing 5.0 lb against a vertical wall as shown in the figure
below.
The coefficient of static friction between the wall and the block is 0.60 and the coeffi¬cient of kinetic friction is
0.40. Assume the block is not mov¬ing initially.
(a) Will the block start moving?
(b) What is the force exerted on the block by the wall?
10_Newton/e_10_8_102.html
7.96-kg block rests on a plane inclined at 22.0° to the horizontal, as shown in the figure below.
The coefficient of static friction is 0.25, while the coefficient of kinetic friction is 0.15.
(a) What is the minimum force F, parallel to the plane, that will prevent the block from slipping down the
plane?
(b) What is the minimum force F that will start the block moving up the plane?
(c) What force F is required to move the block up the plane at constant velocity?
10_Newton/e_10_8_103.html
The two blocks, m = 16 kg and M = 88 kg, shown in the figure below are free to move.
The coefficient of static friction between the blocks is 0.38, but the surface beneath M is frictionless.
What is the minimum horizontal force F required to hold m against M?
10_Newton/e_10_8_104.html
A massless rope is tossed over a wooden dowel (cylinder) of radius r in order to lift a heavy object of weight
W off of the floor, as shown in the figure below.
The coefficient of sliding friction between the rope and the dowel is mu. Show that the minimum downward
pull on the rope necessary to lift the object is Fdown=Wexp(pi mu).
(Hint: This problem requires techniques from integral calculus.)
10_Newton/e_10_8_105.html
A 4.40-kg block is put on top of a 5.50-kg block. In order to cause the top block to slip on the bottom one, held
fixed, a horizontal force of 12.0 N must be applied to the top block.
The assembly of blocks is now placed on a horizontal, frictionless table, as shown below Find
(a) the maximum horizon¬tal force F that can be applied to the lower block so that the blocks will move
together,
(b) the resulting acceleration of the blocks, and
(c) the coefficient of static friction between the blocks.
10_Newton/e_10_8_106.html
You are driving a car at a speed of 85 km/h when you notice a barrier across the road 62 m ahead.
(a) What is the minimum coefficient of static friction between tires and road that will allow you to stop without
striking the barrier?
(b) Suppose that you are driving at 85 km/h on a large empty parking lot. What is the minimum coefficient of
static friction that would allow you to turn the car in a
62-m radius circle and, in this way, avoid collision with a wall 62 m ahead?
10_Newton/e_10_8_107.html
A car moves at a constant speed on a straight but hilly road. One section has a crest (peak) and dip of the same
250-m radius, as shown in the figure below.
(a) As the car passes over the crest, the normal force on the car is one-half the 16-kN weight of the car. What
will be the normal force on the car as it passes through the bottom of the dip?
(b) What is the greatest speed at which the car can move without leaving the road at the top of the hill?
(c) Moving at the speed found in (b), what will be the normal force on the car as it moves through the bottom
of the dip?
10_Newton/e_10_8_108.html
A 1.34-kg ball is attached to a rigid vertical rod by means of two massless strings each 1.70 m long. The strings
are attached to the rod at points 1.70 m apart.
The system is rotating about the axis of the rod, both strings being taut and forming an equilateral triangle with
the rod, as shown below.
The tension in the upper string is 35.0 N.
(a) Find the tension in the lower string.
(b) Calculate the net force on the ball at the instant shown in the figure.
(c) What is the speed of the ball?
10_Newton/e_10_8_109.html
A very small cube of mass m is placed on the inside of a funnel (see figure below) rotating about a vertical axis
at a constant rate of w revolutions per second.
The wall of the funnel makes an angle theta with the horizontal. The coefficient of static friction between cube
and funnel is mu and the center of the cube is at a distance r from the axis of rotation.
Find the
(a) largest and
(b) smallest values of w for which the cube will not move with respect to the funnel.
13_Energy_momentum/e_13_1_011.html
conservative forces
For each of the forces given below check whether it is conservative and find the potential energy, if possible:
a)
,
,
b)
,
,
13_Energy_momentum/e_13_1_012.html
Potential energy, conservation laws
Potential energy is given by
13_Energy_momentum/e_13_1_013.html
. At what
aparticle is in equilibrium ?
period of the bound motion
Potential energy is given by
the particle with the energy
(one-dimensional motion). Find the period of the bound motion of
.
13_Energy_momentum/e_13_1_014.html
Potential energy, conservation laws
Force is given by
,
(cylindrical coordinates). Is the force conservative ? If yes,
find the potential. What is conserved in this force ?
13_Energy_momentum/e_13_1_015.html
Potential energy, conservation laws
Potential energy is given in polar coordinates by
conserved ? Find the torque at
.
. Find the force. Is angular momentum
13_Energy_momentum/e_13_1_016.html
Potential energy, conservation laws
A particle orbit is
. Find the central force.
13_Energy_momentum/e_13_1_017.html
Potential energy, conservation laws
A bead (mass ) is moving on a circularly shaped wire(
) without friction and is connected to the
two points,
and
, with identical springs (spring constant ) of initially zero
length (so that
where is the length of the spring). a) Write down the force vectors. b) Derive the
potential energy (if the forces are conservative). c) Find the velocity as a function of angle
(for a given
energy). c) Find the angular momentum relative to the coordinate origin as a function of .
13_Energy_momentum/e_13_1_018.html
Potential energy, conservation laws
A bead (mass ) is moving on an elliptically shaped (
) wire without friction. The bead is
attracted to the focus
by the force inversely proportional to the distance
squared between the bead and
the focus,
. The bead is attracted to the center of the ellipse by the force proportional to the
distance
between the bead and the center,
. a) Write down the force vectors. b) Derive the
potential energy (if the forces are conservative). c) Find the velocity as a function of angle
(for a given
energy). c) Find the angular momentum relative to the coordinate origin as a function of .
13_Energy_momentum/e_13_1_019.html
Potential energy, conservation laws
In a galaxy the gravitational potential (potential energy) is
,
. Find the relation between
the total energy and angular momentum for circular orbits. Find the dependence of the orbit period on the
radius.
13_Energy_momentum/e_13_1_020.html
Potential energy, conservation laws
A particle moves under the influence of the body
which is in the coordinate origin. In the beginning the
particle is at a very large distance from , moves with the velocity and would pass at the distance from
if there were no interaction (this is called \textit{impact parameter}). What is the minimal distance between the
particle and
for the potential energy is
(analyze
and
).
13_Energy_momentum/e_13_1_021.html
Potential energy, conservation laws
Find
for a particle with
.)
in the potential energy
. (Hint:
for
13_Energy_momentum/e_13_1_022.html
apogee and perigee
A satellite of the mass , moving in the Earth potential
, has the total energy
momentum . Find the maximum (apogee) and minimum (perigee) distance from the Earth.
and angular
13_Energy_momentum/e_13_1_023.html
Potential energy, conservation laws
A particle (mass ) is moving in the central field
angular momentum suddenly are changed by
and
from the attracting body on the new orbit ?
on a circular orbit
. The energy and
. What are the maximal and minimal distances
13_Energy_momentum/e_13_1_024.html
Energy Conservation and Work
A necklace is laying on a table without friction when a quarter of it is hunging from the edge.
The necklace's length is
and its mass is .
What is work needed in order to pull the necklace to the table?
Do it in two ways:
(a) Using the definition of work.
(b) Using energy conservation.
13_Energy_momentum/e_13_1_025.html
Momentum Conservation
The mass of a rocket at time is given by
( is constatnt)
due to gases that exhausting from it (in the opposite direction).
The gases velocity is
with respect to the rocket.
If the rocket take off from earth
1. What is the condition for the rocket to take off despite the gravity force?
2. Find the rocket velocity as a function of time.
3. A friction force of the form
( is constant) is also acting on the rocket, find
4. What is the rocket velocity after a long time (
)?
13_Energy_momentum/e_13_1_026.html
Potential Energy, Coservation Laws
A particle move in a potential energy
.
.
1. What is the force? draw the graph of U(x).
2. What is the direction of the force at every point?
3. What are the equilibrium points? are they stable?
4. For given energies, what are the possible trajectories?
13_Energy_momentum/e_13_1_027.html
Work
Two forces acting on a particle
N
N
The particle moves from the point
to the point
1. What is the work done on the particle?
2. What would be the work if the particle move from
to ?
.
13_Energy_momentum/e_13_1_028.html
Work
A particle move according to
1. Find the kinetic energy of the particle in the points
and
.
2. What is the work done on the particle by the centripetal force between the points
3. Show that in order to bring back the particle to point
and
?
the work needed is 0.
13_Energy_momentum/e_13_1_029.html
Momentum Conservation
A rocket is moving due to exhausting gases at high velocity in the opposite direction. The gases are products of
fuel burning so the mass of the rocket is reduced all the time. The rate of exhausting is constant and the gases
velocity is
with respect to the rocket.
1. Derive the equations of motion for a rocket moving in space (far from earth). Find the volcity as a function of
time where you can assume that
and
.
2. If the rocket takes off from earth what is its velocity as a function of time?
13_Energy_momentum/e_13_1_030.html
Potential energy, conservation laws
A particle's trajectory is
and it moves in a central force. Find
.
13_Energy_momentum/e_13_1_031.html
Potential energy, conservation laws
A particle (mass , energy
such that
.
Find the angular momentum.
) move in a potential
,
13_Energy_momentum/e_13_1_032.html
Potential energy, conservation laws
A particle (mass
) move in central field
in a circular orbit. Its angular momentum is .
Find its energy and the time that is needed to complete a cycle.
13_Energy_momentum/e_13_1_033.html
Collisions
A mass
is attached to a loose spring (spring constant ) and lies on a plane with kinetic friction coefficient
.
A bullet (mass ) is fired toward the mass in a velocity .
How much will the spring contract?
13_Energy_momentum/e_13_1_034.html
Momentum Conservation
A particle is at rest in a constant magnetic field
.
Suddenly it decays into 3 particles (with mass and charge)
,
and
.
The particles
and
make a circular motion with diameter of
and
respectively.
Assume that the angle between the initial velocities of m1 and m2 is .
What is the velocity of the third particle?
13_Energy_momentum/e_13_1_035.html
Many Particle System
Show that the kinetic energy of two particles with masses
center of mass and the kinetic energy of the relative motion.
and
splits into the kinetic energy of the
13_Energy_momentum/e_13_1_036.html
Many Particle System
Two bodies (
and
) are at distamce
from each other.
At
starts to "pull"
with a constant force .
1. Where will the masses meet?
2. What will be the velocities of the bodies before they collide?
13_Energy_momentum/e_13_1_037.html
โ€ซืื ืจื’ื™ื” ื•ืชื ื•ืขื” ืžืขื’ืœื™ืชโ€ฌ
. R=1m โ€ซ ืœืžืกื™ืœื” ืžืขื’ืœื™ืช ืื ื›ื™ืช ื‘ืขืœืช ืจื“ื™ื•ืกโ€ฌA โ€ซ ืขืœ ืžื™ืฉื•ืจ ืื•ืคืงื™ ื”ืžืชื—ื‘ืจ ื‘ื ืงื•ื“ื”โ€ฌV0 โ€ซื’ื•ืฃ ืžื—ืœื™ืง ืœืœื ื—ื™ื›ื•ืš ื‘ืžื”ื™ืจื•ืชโ€ฌ
.C โ€ซ ืžืชื ืชืง ื”ื’ื•ืฃ ืžื”ืžืกื™ืœื” ื•ืžืžืฉื™ืš ื‘ืžืขื•ืคื• ืขื“ ืœืคื’ื™ืขืชื• ื‘ืžื™ืฉื•ืจ ื”ืื•ืคืงื™ ื‘ื ืงื•ื“ื”โ€ฌh = 1.707R โ€ซ ื”ื ืžืฆืืช ื‘ื’ื•ื‘ื”โ€ฌB โ€ซื‘ื ืงื•ื“ื”โ€ฌ
? B โ€ซ ืžื”ื™ ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื‘ืขื•ื‘ืจื• ื‘ื ืงื•ื“ื”โ€ฌ.โ€ซืโ€ฌ
?โ€ซ ืฉืœ ื”ื’ื•ืฃโ€ฌV0 โ€ซ ืžื” ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืชโ€ฌ.โ€ซื‘โ€ฌ
? A-โ€ซ ื•โ€ฌC โ€ซ ืžื” ื”ืžืจื—ืง ื‘ื™ืŸ ื”ื ืงื•ื“ื•ืชโ€ฌ.โ€ซื’โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_038.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื”โ€ฌ
โ€ซื™ื“ื•ืข ืฉื”ื›ื— ืฉืคื•ืขืœ ืขืœ ื’ื•ืฃ ื”ื•ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื”ื“ืจื•ืฉื” ื‘ืฉื‘ื™ืœ ืœื”ื–ื™ื– ืืช ื”ื’ื•ืฃ ืžโ€ช-โ€ฌโ€ฌ
โ€ซืืœโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื”ื“ืจื•ืฉื” ื‘ืฉื‘ื™ืœ ืœื”ื–ื™ื– ืืช ื”ื’ื•ืฃ ืžโ€ช-โ€ฌโ€ฌ
โ€ซืืœโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_039.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ืžืขื’ืœื™ืช ื•ืฉื™ืžื•ืจ ืชื ืขโ€ฌ
โ€ซืงืœื™ืข ืฉืžืกืชื• โ€ช mโ€ฌืคื•ื’ืข ื‘ื‘ื•ืœ ืขืฅ ืฉืžืกืชื• โ€ช Mโ€ฌื‘ืžื”ื™ืจื•ืช โ€ช vโ€ฌื•ื™ื•ืฆื ื‘ืžื”ื™ืจื•ืช โ€ช. v/2โ€ฌโ€ฌ
โ€ซื‘ื•ืœ ื”ืขืฅ ื”ืงืฉื•ืจ ื‘ื—ื•ื˜ ืžื‘ืฆืข ืชื ื•ืขื” ืžืขื’ืœื™ืช ื‘ืจื“ื™ื•ืก โ€ช.Rโ€ฌโ€ฌ
โ€ซืžื” ืฆืจื™ื›ื” ืœื”ื™ื•ืช ื”ืžื”ื™ืจื•ืช ื”ืžื™ื ื™ืžืœื™ืช โ€ช vโ€ฌืฉืœ ื”ืงืœื™ืข )ืœืคื ื™ ื”ืคื’ื™ืขื”( ื›ื“ื™ ืฉื‘ื•ืœ ื”ืขืฅ ื™ื‘ืฆืข ืกื™ื‘ื•ื‘ ืฉืœื ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_040.htmlโ€ฌโ€ฌ
โ€ซืชื ืขโ€ฌ
โ€ซื’ื•ืฃ ื ื–ืจืง ื›ืœืคื™ ืžืขืœื” ื‘ืžื”ื™ืจื•ืช โ€ช 30โ€ฌืžื˜ืจ ืœืฉื ื™ื” ื‘ื”ืฉืคืขืช โ€ช .gโ€ฌื‘โ€ช 2-โ€ฌืฉื ื™ื•ืช ื”ืชื ืข ืฉืœื• ืžืฉืชื ื” ื‘ โ€ช] -20โ€ฌืง"ื’*ืžื˜ืจโ€ช/โ€ฌืฉื ื™ื”[โ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ืžืชืงืฃ ืฉืคืขืœ ืขืœื™ื• ื‘ืžืฉืš โ€ช 2โ€ฌื”ืฉื ื™ื•ืช?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ืžืกืชื•?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื• ื”ื›ื•ื— ืฉืคื•ืขืœ ืขืœื™ื•?โ€ฌ
โ€ซื“โ€ช .โ€ฌืžืฆื ืืช ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ืœืื—ืจ โ€ช 2โ€ฌืฉื ื™ื•ืชโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_041.htmlโ€ฌโ€ฌ
โ€ซื”ืชื ื’ืฉื•ืชโ€ฌ
โ€ซืงืœื™ืข ืฉืžืกืชื• โ€ช 5โ€ฌื’ืจื ื•ืžื”ื™ืจื•ืชื• โ€ช 400โ€ฌืžื˜ืจ ืœืฉื ื™ื” ื ื•ืจื” ืœืขื‘ืจ ืžื˜ื•ื˜ืœืช ื‘ืœื™ืกื˜ื™ืช ืฉืžืกืชื” โ€ช 4โ€ฌืง"ื’โ€ช.โ€ฌโ€ฌ
โ€ซื‘ื›ืžื” ืชืขืœื” ื”ืžื˜ื•ื˜ืœืช ืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช? ื›ืžื” ืื—ื•ื– ืžื”ืื ืจื’ื™ื” ื”ืœืš ืœืื™ื‘ื•ื“?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_042.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื”โ€ฌ
โ€ซืœื‘ื ื” ื‘ืžืฉืงืœ โ€ช 263โ€ฌื’ืจื ื ื•ืคืœืช ืขืœ ืงืคื™ืฅ ืื ื›ื™ ืขื ืงื‘ื•ืข ืงืคื™ืฅ ืฉืœโ€ฌ
โ€ซืขื“ ืฉื ืขืฆืจ ืœืจื’ืข ื‘ืžื ื•ื—ื”โ€ช .โ€ฌื‘ื–ืžืŸ ืฉื”ืงืคื™ืฅ ืžืชื›ื•ื•ืฅโ€ช ,โ€ฌืžืฆืื•โ€ช:โ€ฌโ€ฌ
โ€ซื( ื›ืžื” ืขื‘ื•ื“ื” ื ืขืฉื™ืช ืขืœ ื™ื“ื™ ื›ื•ื— ื”ื›ื‘ื™ื“ื” ?โ€ฌ
โ€ซื‘( ื›ืžื” ืขื‘ื•ื“ื” ื ืขืฉื™ืช ืขืœ ื™ื“ื™ ื›ื•ื— ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซื’( ื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ืคื’ืขื” ื”ืœื‘ื ื” ื‘ืงืคื™ืฅ ?โ€ฌ
โ€ซโ€ช .โ€ฌื”ืœื‘ื ื” ื ืฆืžื“ืช ืœืงืคื™ืฅโ€ช ,โ€ฌื•ื”ืงืคื™ืฅ ืžืชื›ื•ื•ืฅ โ€ช 11.8โ€ฌืก"ืžโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_043.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžืกื” ืžืฉืชื ื”โ€ฌ
โ€ซืขื’ืœื” ืขืžื•ืกืช ื—ื•ืœ ื ื’ืจืจืช ืขืœ ื™ื“ื™ ื›ื•ื— ืงื‘ื•ืขโ€ฌ
โ€ซื”ืชื ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื›ืชื‘ื• ื•ืคืชืจื• ืืช ืžืฉื•ื•ืื•ืชโ€ฌ
โ€ซโ€ช .โ€ฌืขืงื‘ ื—ื•ืจ ื‘ืงืจืงืขื™ืช ื ืฉืคืš ื—ื•ืœ ื‘ืงืฆื‘ ืื—ื™ื“โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_044.htmlโ€ฌโ€ฌ
โ€ซโ€ชDefinition of torqueโ€ฌโ€ฌ
โ€ซโ€ชA particle is located atโ€ฌโ€ฌ
โ€ซโ€ช. A constant force of magnitude 2.6N acts onโ€ฌโ€ฌ
โ€ซโ€ชthe particle. find the components of the torque about the origin when the force acts inโ€ฌโ€ฌ
โ€ซโ€ชa) the positiveโ€ฌโ€ฌ
โ€ซโ€ชdirection.โ€ฌโ€ฌ
โ€ซโ€ชb) the negativeโ€ฌโ€ฌ
โ€ซโ€ชdirection.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_045.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื‘ืžืกื” ืžืฉืชื ื”โ€ฌ
โ€ซืœื•ื•ื™ืŸ ื ืข ื“ืจืš ืฆื‘ื™ืจ ื’ื–ื™ ื“ืœื™ืœ ื‘ืื–ื•ืจ ื‘ื• ื›ื— ื”ื›ื•ื‘ื“ ื–ื ื™ื—โ€ช .โ€ฌืชื•ืš ื›ื“ื™ ืชื ื•ืขืชื•โ€ช ,โ€ฌืžืกืคื— ื”ืœื•ื•ื™ืŸ ื—ืœืงื™ืงื™ื ืžื”ืฆื‘ื™ืจ ื”ื’ื–ื™ ื›ืš ืฉืžืกืชื• ื’ื“ืœื” ื‘ืงืฆื‘โ€ฌ
โ€ซ)โ€ช - kโ€ฌืงื‘ื•ืข(โ€ช .โ€ฌืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸ ื‘ โ€ช t=0โ€ฌื”ื™ืโ€ฌ
โ€ซื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช ,โ€ฌื•ืžืกืชื• ืื–โ€ฌ
โ€ซโ€ช .โ€ฌื ื ื™ื— ื›ื™ ื”ืฆื‘ื™ืจ ื”ื’ื–ื™ ื‘ืžื ื•ื—ื”โ€ฌ
โ€ซโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_1_046.htmlโ€ฌโ€ฌ
โ€ซืคื•ื˜ื ืฆื™ืืœ ืžืจื›ื–ื™โ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื‘ืžืกืœื•ืœ ืืœื™ืคื˜ื™ ื›ืืฉืจโ€ฌ
โ€ซื’ื•ืฃ ื‘ืขืœ ืžืกื” โ€ช mโ€ฌื ืข ื‘ืคื•ื˜ื ืฆื™ืืœ ืžืจื›ื–ื™โ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ืฉืœื• ืžื™ื™ื“ื™ืช ื›ืš ืฉื”ื•ื ืžืžืฉื™ืš ื‘ืžืกืœื•ืœ ืžืขื’ืœื™โ€ช .โ€ฌืžื” ืฉื™ื ื•ื™ ื”ืชื ืข ื”ืžื™ื™ื“ื™ ื‘ืื•ืชื” ื ืงื•ื“ื” ?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช .โ€ฌื‘ื ืงื•ื“ื”โ€ฌ
โ€ซืžืฉื ื™ื ืืชโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_004.htmlโ€ฌโ€ฌ
โ€ซื›ื•ื—ื•ืช ืžืฉืžืจื™ืโ€ฌ
โ€ซืžื™ ืžื‘ื™ืŸ ื”ื›ื•ื—ื•ืช ื”ื‘ืื™ื ื”ื•ื ื›ื•ื— ืžืฉืžืจ?โ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื›ื›ื•ื— ื”ื›ื‘ื™ื“ื”โ€ฌ
โ€ซโ€ช .3โ€ฌื›ื•ื— ืฉืœ ืงืคื™ืฅโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_005.htmlโ€ฌโ€ฌ
โ€ซ"ืฉืจื™ืคืช ืงืœื•ืจื™ื•ืช"โ€ฌ
โ€ซื”ืื ืจื’ื™ื” ืฉื‘ืžื–ื•ืŸ ื ืžื“ื“ืช ื‘ื™ื—ื™ื“ื•ืช ืฉืœ ืงืœื•ืจื™ื”โ€ช .โ€ฌืงืœื•ืจื™ื” ื–ื• ื™ื—ื™ื“ืช ืื ืจื’ื™ื” ืชืจืžื™ืชโ€ช ,โ€ฌืฉื™ื—ืก ื”ืžืจืชื” ืœ โ€“โ€ช SIโ€ฌื”ื•ื โ€ช . cal = 4.2 J 1โ€ฌื‘ื”ื ื—ื”โ€ฌ
โ€ซืฉืืคืฉืจ ืœื”ืžื™ืจ ื›โ€ช 50% -โ€ฌืžื›ืžื•ืช ื”ืื ืจื’ื™ื” ืฉืื’ื•ืจื” ื‘ืžื–ื•ืŸ ืœืื ืจื’ื™ื” ืžื›ืื ื™ืชโ€ช ,โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืœืื™ื–ื” ื’ื•ื‘ื” ืืชโ€ช/โ€ฌื” ืฆืจื™ืšโ€ช/โ€ฌื” ืœื˜ืคืก )ืœืžืฉืœ ื‘ืกื˜ืคืจ ื‘ื—ื“ืจโ€ฌ
โ€ซื›ื•ืฉืจ( ื›ื“ื™ ืœืฉืจื•ืฃ ืืช ื”ืื ืจื’ื™ื” ื”ืžืชืงื‘ืœืช ืžืื›ื™ืœืช โ€ช gr 100โ€ฌืœื—ืโ€ช .โ€ฌื ืชื•ืŸ ืฉื‘ื›ืžื•ืช ื–ื• ืฉืœ ืœื—ื ื™ืฉ โ€ช.kcal 250โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_006.htmlโ€ฌโ€ฌ
โ€ซืžืกื” ื ืขื” ืžืคืกื’ืช ืžืฉื˜ื— ื›ื“ื•ืจื™โ€ฌ
โ€ซืžืกื” โ€ช mโ€ฌืžืชื—ื™ืœื” ืœื ื•ืข ืžืคืกื’ืช ืžืฉื˜ื— ื›ื“ื•ืจื™ ื‘ืขืœ ืจื“ื™ื•ืก โ€ช .Rโ€ฌื”ืžืฉื˜ื— ื”ื•ื ื—ืœืง ื•ืžืงื•ื‘ืข ืœืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ โ€ช.ฮธโ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ โ€ช.ฮธโ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ื”ืชืื•ืฆื” ื”ืจื“ื™ืืœื™ืช ื•ื”ืžืฉื™ืงื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ โ€ช.ฮธโ€ฌโ€ฌ
โ€ซื”ืจืื”โ€ช/โ€ฌื™ ืฉืื™ืœื• ื”ื™ื” ืงื™ื™ื ื—ื™ื›ื•ืšโ€ช ,โ€ฌื”ื™ื” ื”ื—ืœืงื™ืง ืขื•ื–ื‘ ืืช ื”ืžืฉื˜ื— ื‘ื–ื•ื•ื™ืช ื’ื“ื•ืœื” ื™ื•ืชืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_007.htmlโ€ฌโ€ฌ
โ€ซืคื’ื™ืขืช ืืกื˜ืจื•ืื™ื“ ื‘ื›ื“ื•ืจ ื”ืืจืฅโ€ฌ
โ€ซืืกื˜ืจื•ืื™ื“ โ€ช 1989FCโ€ฌื”ื ื• ื‘ืขืœ ืจื“ื™ื•ืก ืฉืœ โ€ช 800โ€ฌืžื˜ืจโ€ช,โ€ฌื•ืฆืคื™ืคื•ืช โ€ช) ฯ = 7 gr/cm3โ€ฌื‘ืขืจืš ืฉืœ ื‘ืจื–ืœ(โ€ช .โ€ฌืžื”ื™ืจื•ืชื• ืฉืœ ื”ืืกื˜ืจื•ืื™ื“ ื‘ื™ื—ืก ืœื›ื“ื•ืจโ€ฌ
โ€ซื”ืืจืฅ ื”ื™ื โ€ช .km/hr 74014โ€ฌื‘ื”ื ื—ื” ื•ื”ื•ื ื”ื™ื” ืคื•ื’ืข ื‘ื›ื“ื•ืจ ื”ืืจืฅโ€ช ,โ€ฌื•ื”ืคื’ื™ืขื” ื”ื™ื™ืชื” ืžืชืจื—ืฉืช ื‘ื™ื )ืกื™ื›ื•ื™ ืฉืœ โ€ช ,(70%โ€ฌืœืื™ื–ื” ื’ื•ื‘ื” ื”ื™ื”โ€ฌ
โ€ซืžืชืจื•ืžื ื”ื˜ืกื•ื ืžื™ )ื ื“ ืžื™ื( ื‘ืขืงื‘ื•ืช ื”ืคื’ื™ืขื”?โ€ฌ
โ€ซื”ื ื—ื•ืช ื•ื ืชื•ื ื™ื ื ื•ืกืคื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืฆื•ืจืช ื”ื˜ืกื•ื ืžื™ ื”ื™ื ื”ื™ื ืžืฉื•ืœืฉืชโ€ช ,โ€ฌืฉืœื™ืฉ ืžืื ืจื’ื™ื™ืช ื”ืคื’ื™ืขื” ืžื•ืฉืงืขืช ื‘ื™ืฆื™ืจืช ื”ื˜ืกื•ื ืžื™โ€ช ,โ€ฌืฆืคื™ืคื•ืช ื”ืžื™ื ื”ื™ื โ€ช ,ฯ = 1 gr/cm3โ€ฌืžื”ื™ืจื•ืชโ€ฌ
โ€ซื”ื”ืชืงื“ืžื•ืช ืฉืœ ื”ื˜ืกื•ื ืžื™ ื”ื™ื โ€ช ,km/hr 1700โ€ฌืื•ืจืš ืงื• ื”ื—ื•ืฃ ื‘ื• ืคื•ื’ืข ื”ื˜ืกื•ื ืžื™ ื”ื•ื โ€ช .km 10000โ€ฌื”ื ื— ืฉื”ื’ื•ื‘ื” ืฉืœ ืžืจื›ื– ื”ืžืกื” )ื‘ืฉื‘ื™ืœโ€ฌ
โ€ซื—ื™ืฉื•ื‘ ื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช( ื ืžืฆื ื‘ืฉืœื™ืฉ ื”ื’ื•ื‘ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_008.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขืช ื˜ื‘ืขืช ืขืœ ืงืฉืชโ€ฌ
โ€ซื˜ื‘ืขืช ื‘ืขืœืช ืžืกื” โ€ช m=5 kgโ€ฌื ืขื” ืขืœ ืžืกื™ืœื” ื—ืกืจืช ื—ื™ื›ื•ืš โ€ช ,ABCโ€ฌืฉืฆื•ืจืชื” ื—ืฆื™ ืžืขื’ืœ ื‘ืจื“ื™ื•ืก โ€ช .R=2 mโ€ฌืขืœ ื”ื˜ื‘ืขืช ืคื•ืขืœื™ื ืฉื ื™ ื›ื•ื—ื•ืชโ€ฌ
โ€ซื‘ืขืœื™ ื’ื•ื“ืœ โ€ช F = 40 Nโ€ฌื•โ€ช .F'= 150 N -โ€ฌื”ื›ื•ื— ืžืฉื™ืง ืœืžืขื’ืœ ื‘ืžืฉืš ื›ืœ ื–ืžืŸ ื”ืชื ื•ืขื”โ€ช .โ€ฌื”ื›ื•ื— 'โ€ช Fโ€ฌืฉื•ืžืจ ืขืœ ื›ื™ื•ื•ืŸ ืงื‘ื•ืข ืฉืœ โ€ช 30ºโ€ฌืžืขืœ ืœืื•ืคืงโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืขื‘ื•ื“ื” ื”ื›ื•ืœืœืช ื”ืžื‘ื•ืฆืขืช ืขืœ ื™ื“ื™ ืžืขืจื›ืช ื”ื›ื•ื—ื•ืช ื”ืคื•ืขืœื™ื ืขืœ ื”ื˜ื‘ืขืช ื›ืืฉืจ ื”ื™ื ื ืขื” ืžโ€ช A-โ€ฌืœโ€ช.B-โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืขื‘ื•ื“ื” ื”ื›ื•ืœืœืช ื”ืžื‘ื•ืฆืขืช ืขืœ ื™ื“ื™ ืžืขืจื›ืช ื”ื›ื•ื—ื•ืช ื”ืคื•ืขืœื™ื ืขืœ ื”ื˜ื‘ืขืช ื›ืืฉืจ ื”ื™ื ื ืขื” ืžโ€ช A-โ€ฌืœโ€ช.C-โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_009.htmlโ€ฌโ€ฌ
โ€ซืœื‘ ื”ืื“ืโ€ฌ
โ€ซืœื‘ ื”ืื“ื ื”ื•ื ืžืฉืื‘ื” ืจื‘ืช ื›ื•ื— ื•ืืžื™ื ื” ื‘ื™ื•ืชืจโ€ช .โ€ฌื‘ื›ืœ ื™ืžืžื” ื”ื•ื ืฉื•ืื‘ ื•ืคื•ืœื˜ ื›โ€ช 7500 -โ€ฌืœื™ื˜ืจื™ื ืฉืœ ื“ืโ€ช .โ€ฌืื ื”ืขื‘ื•ื“ื” ื”ื ืขืฉื™ืช ืขืœ ื™ื“ื™ ื”ืœื‘โ€ฌ
โ€ซืฉื•ื•ื” ืœืขื‘ื•ื“ื” ื”ื“ืจื•ืฉื” ืœื”ืขืœื•ืช ื›ืžื•ืช ื›ื–ื• ืฉืœ ื“ื ืœื’ื•ื‘ื” ืžืžื•ืฆืข ืฉืœ ืื™ืฉื” โ€“ โ€ช ,1.63mโ€ฌื•ืื ืฆืคื™ืคื•ืช ื”ื“ื ืฉื•ื•ื” ืœืฆืคื™ืคื•ืช ื”ืžื™ืโ€ช,โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌื›ืžื” ืขื‘ื•ื“ื” ืขื•ืฉื” ื”ืœื‘ ื‘ื™ืžืžื”?โ€ฌ
โ€ซโ€ช .2โ€ฌืžื” ืชืคื•ืงืช ื”ื”ืกืคืง ืฉืœื• ื‘ื•ืื˜ื™ื?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_010.htmlโ€ฌโ€ฌ
โ€ซืคืขืœื•ืœืŸโ€ฌ
โ€ซืžืกืช ืคืขืœื•ืœืŸ ื‘ืงืจืงืก โ€ช 60โ€ฌืง"ื’โ€ช ,โ€ฌื”ืคืขืœื•ืœืŸ ืžื‘ืฆืข ืชืจื’ื™ืœ ื‘ื• ื”ื•ื ื ื•ืจื” ืขืœ ื™ื“ื™ ืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืข ืฉืœ โ€ช 2000โ€ฌื 'โ€ช/โ€ฌืž'โ€ช .โ€ฌื”ืงืคื™ืฅ ืžื›ื•ื•ืฅ ืœืคื ื™ ื”ื™ืจื™ ื‘โ€ช1 -โ€ฌโ€ฌ
โ€ซืžื˜ืจ ื‘ื™ื—ืก ืœืžืฆื‘ ื”ืจืคื•ื™ ื•ืžืขืจื›ืช ื”ื™ืจื™ ืžื•ืฆื‘ืช ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช °30โ€ฌืžืขืœ ื”ืื•ืคืงโ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ื›ื“ืื™ ืœืขื•ื–ืจ ื”ืคืขืœื•ืœืŸ ืœื”ืฆื™ื‘ ืืช ืจืฉืช ื”ื‘ื™ื˜ื—ื•ืŸโ€ฌ
โ€ซื›ื“ื™ ืฉืœื ื™ืื‘ื“ ืืช ื”ืขื‘ื•ื“ื” )ื•ืืช ื”ื‘ื•ืก(โ€ช.โ€ฌโ€ฌ
โ€ซื”ื ื™ื—ื• ื›ื™ ืจืฉืช ื”ื‘ื™ื˜ื—ื•ืŸ ืคืจื•ืฉื” ื‘ื’ื•ื‘ื” ื”ืฉื•ื•ื” ืœื’ื•ื‘ื” ื ื™ืชื•ืง ื”ืคืขืœื•ืœืŸ ืžื”ืงืคื™ืฅโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_011.htmlโ€ฌโ€ฌ
โ€ซืคื•ื˜ื ืฆื™ืืœ ืžืจื›ื–ื™ ื•ื™ื•ื ื™ื–ืฆื™ื”โ€ฌ
โ€ซโ€ช ,โ€ฌื›ืืฉืจ โ€ช rโ€ฌื”ื•ื ืžืจื—ืง ื”ื’ื•ืฃ ืžืžืจื›ื–โ€ฌ
โ€ซื—ืœืงื™ืง ืงืฉื•ืจ ืœื ืงื•ื“ื” ืžืกื•ื™ื™ืžืช )ืžืจื›ื– ื”ื›ื•ื—(โ€ช .โ€ฌื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช ืฉืœ ื”ื—ืœืงื™ืง ื”ื™ืโ€ฌ
โ€ซื”ื›ื•ื—โ€ช A .โ€ฌื•โ€ช B -โ€ฌืงื‘ื•ืขื™ื ื—ื™ื•ื‘ื™ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืžืจื—ืง ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ ื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืฉโ€ช.โ€ฌืžโ€ช .โ€ฌื”ื•ื ื™ืฆื™ื‘โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืื ืจื’ื™ื™ืช ื”ื™ื•ื ื™ืฆื™ื” โ€ช .โ€ฌื›ืœื•ืžืจโ€ช ,โ€ฌื”ืขื‘ื•ื“ื” ืฉื™ืฉ ืœื”ืฉืงื™ืข ื›ื“ื™ ืœื”ืจื—ื™ืง ืืช ื”ื—ืœืงื™ืง ืžืžืจื—ืงโ€ฌ
โ€ซื‘ืขื–ืจืช ื•โ€ฌ
โ€ซโ€ช .3โ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซโ€ช .4โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ืฉืžื‘ืฆืข ื”ื›ื•ื— ื”ืžืจื›ื–ื™ ื”ื ื’ื–ืจ ืžโ€ช-โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช ? xyโ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ื‘ืขื–ืจืชโ€ฌ
โ€ซโ€ช .5โ€ฌืื ื ืชื•ืŸ ืฉื”ืื ืจื’ื™ื” ื”ื›ื•ืœืœืช ืฉืœ ื”ื—ืœืงื™ืง ื”ื™ืโ€ฌ
โ€ซื›ืฉื”ื—ืœืงื™ืง ืขื•ื‘ืจ ืžื ืงื•ื“ื” โ€ช1โ€ฌโ€ฌ
โ€ซืœืื™ื ืกื•ืฃโ€ช.โ€ฌโ€ฌ
โ€ซืœื ืงื•ื“ื” โ€ช2โ€ฌโ€ฌ
โ€ซื‘ืžื™ืฉื•ืจโ€ฌ
โ€ซโ€ช ,โ€ฌื•ื›ืŸ ื ืชื•ืŸ ืฉืชื ื•ืขืชื• ืจื“ื™ืืœื™ืช ื‘ืœื‘ื“โ€ช ,โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ื”ื ืงื•ื“ื•ืช ื‘ื”ืŸ ื”ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืชืืคืกืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_012.htmlโ€ฌโ€ฌ
โ€ซืฉืืœืช ื”ื•ืงโ€ฌ
โ€ซืื—ื“ ื”ื’ื•ืจืžื™ื ืฉืขื•ืจืจื• ืืช ืขื‘ื•ื“ืชื• ืฉืœ ื ื™ื•ื˜ื•ืŸ ืขืœ ื”ืžื›ื ื™ืงื” ื”ื™ืชื” ืฉืืœื” ืฉืฉืœื— ืœื• ื”ื•ืงโ€ช:โ€ฌโ€ฌ
โ€ซืื ืงื•ื“ื—ื™ื ื‘ื•ืจ ื‘ืื“ืžื” ืขื“ ืœืฆื™ื“ื• ื”ืฉื ื™ ืฉืœ ื›ื“ื•ืจ ื”ืืจืฅโ€ช ,โ€ฌื•ืžืคื™ืœื™ื ืœืชื•ื›ื• ื›ื“ื•ืจโ€ช ,โ€ฌืžื” ืชื”ื™ื” ืชื ื•ืขืชื•?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_013.htmlโ€ฌโ€ฌ
โ€ซืฉืจืฉืจืช ืขืœ ืฉื•ืœื—ืŸโ€ฌ
โ€ซื ืชื•ื ื” ืฉืจืฉืจืช ื‘ืขืœืช ืžืกื” โ€ช mโ€ฌื•ืื•ืจืš โ€ช Lโ€ฌื”ืžื•ื ื—ืช ืขืœ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช ,โ€ฌื›ืืฉืจ ืจื‘ืข ืžืื•ืจื›ื” ื ืฉืืจ ืชืœื•ื™ ื‘ืื•ื•ื™ืจ ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช .โ€ฌื›ืžื”โ€ฌ
โ€ซืขื‘ื•ื“ื” ื™ืฉ ืœื”ืฉืงื™ืข ื‘ื›ื“ื™ ืœืžืฉื•ืš ืืช ื”ืฉืจืฉืจืช ื‘ืžืœื•ืื” ื—ื–ืจื” ืœืฉื•ืœื—ืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_014.htmlโ€ฌโ€ฌ
โ€ซื—ื˜ื™ืคืช ื›ื“ื•ืจื™ืโ€ฌ
โ€ซื‘ืกืจื˜ื™ื ืจื•ืื™ื ื‘ื“"ื› ื›ื™ ื”ืื“ื )ืž"ื”ืจืขื™ื" ื›ืžื•ื‘ืŸ( ื”ื—ื•ื˜ืฃ ื›ื“ื•ืจ ืขืฃ ืœืื—ื•ืจ ื‘ืขื•ืฆืžื” ืจื‘ื”โ€ช .โ€ฌื”ืื ืกืฆื™ื ื•ืช ืืœื• ืžืฆื™ืื•ืชื™ื•ืช?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืžืฉืงืœ ืื“ื ืžืžื•ืฆืข โ€“ โ€ช.kg 80โ€ฌืขื‘ื•ืจ ื›ื“ื•ืจ ืืงื“ื— ื’ื“ื•ืœ )ืงืœื™ื‘ืจ โ€ช :(inch 0.45โ€ฌืžืกื” โ€ช, grain 230 -โ€ฌืžื”ื™ืจื•ืช โ€“ โ€ช.ft./sec 830โ€ฌืขื‘ื•ืจ ื›ื“ื•ืจ โ€ช7.62โ€ฌโ€ฌ
โ€ซืž"ืž ืฉืœ ืจื•ื‘ื” โ€ช :M-24โ€ฌืžืกื” โ€“ โ€ช , grain 115โ€ฌืžื”ื™ืจื•ืช โ€“ โ€ช.mi/hr 2180โ€ฌโ€ฌ
โ€ซื”ืขืจื”โ€ช grain :โ€ฌื”ื™ื ืžื™ื“ืช ืžืฉืงืœ ืขื‘ื•ืจ ืงืœื™ืขื™ืโ€ช grain .โ€ฌืื—ื“ ืฉื•ื•ื” ืœืžืฉืงืœ ื’ืจื’ืจ ื—ื™ื˜ื” ื‘ื•ื“ื“ ืžื”ืฉื™ื‘ื•ืœืช ื”ืžืจื›ื–ื™ืชโ€ชgrain=0.0648 gram 1 .โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_015.htmlโ€ฌโ€ฌ
โ€ซืฉืœื™ื—ืช ื—ืœืœื™ืช ืœืฆื“ืงโ€ฌ
โ€ซื›ืืฉืจ โ€ช NASAโ€ฌืžืฉื’ืจืช ื—ืœืœื™ืช ืœื›ื™ื•ื•ืŸ ืฆื“ืงโ€ช ,โ€ฌื”ื™ื ืฉื•ืœื—ืช ืื•ืชื” ืœื”ืงื™ืฃ ืงื•ื“ื ืืช ื ื•ื’ื”โ€ช .โ€ฌื”ืื ื™ืฉ ื”ื™ื’ื™ื•ืŸ ื‘ืฉื™ื’ื•ืจ ื›ื–ื”? ื”ืื ื–ื”ื• ืขื•ื“ ืžื—ื“ืœ ืฉืœโ€ฌ
โ€ซโ€ช?NASAโ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืช ื”ื—ืœืœื™ืช โ€ช 12 -โ€ฌืงืžโ€ช/โ€ฌืฉื ' โ€ช ,โ€ฌืžื”ื™ืจื•ืช ื ื•ื’ื” โ€ช 35 -โ€ฌืงืžโ€ช/โ€ฌืฉื 'โ€ช.โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌื ื™ืชืŸ ืœื”ืชื™ื™ื—ืก ืœื”ืงืคืช ื ื•ื’ื” ื›ืืœ ื”ืชื ื’ืฉื•ืช ืืœืกื˜ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_016.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืขืœ ืงืคื™ืฅ ืžืชืคืจืงโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื’ื•ืคื™ื ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช .โ€ฌื‘ืจื’ืข ืžืกื•ื™ื ืžืชืจื—ืฉ ืคื™ืฆื•ืฅ ื‘ื™ืŸ ืฉื ื™ื”ื ื•ื’ื•ืฃ โ€ช m1โ€ฌื ื™ืชื– ืœื›ื™ื•ื•ืŸ ื™ืžื™ืŸ ื‘ืžื”ื™ืจื•ืช โ€ช.v1โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžื” ื”ื™ื ืžื”ื™ืจื•ืชื• ื”ืžืจื‘ื™ืช ืฉืœ ื’ื•ืฃ โ€ช ?m2โ€ฌื‘ืื™ื–ื• ืžื™ืงื•ื?โ€ฌ
โ€ซโ€ช .2โ€ฌืžื” ื”ื™ื ืžืฉืจืขืช ื”ืชื ื•ืขื”?โ€ฌ
โ€ซโ€ช .3โ€ฌืžื” ื”ื•ื ื”ื”ืขืชืง ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_017.htmlโ€ฌโ€ฌ
โ€ซื”ืชืคื•ืฆืฆื•ืช ืคื’ื– ื‘ืขืช ืชื ื•ืขืชื•โ€ฌ
โ€ซืคื’ื– ื ื•ืจื” ื‘ื–ื•ื•ื™ืช โ€ช 60โ€ฌืžืขืœื•ืช ืžืขืœ ืœืื•ืคืง ื‘ืžื”ื™ืจื•ืช ืœื•ืข ืฉืœ โ€ช .m/s 360โ€ฌื‘ืคืกื’ืช ืžืกืœื•ืœื• ื”ื•ื ืžืชืคื•ืฆืฅ ืœืฉื ื™ ืจืกื™ืกื™ื ืฉื•ื•ื™ ืžืกื”โ€ช .โ€ฌืจืกื™ืก ืื—ื“โ€ฌ
โ€ซื ื•ืคืœ ืื ื›ื™ืช ื›ืœืคื™ ืžื˜ื” ื‘ื ืงื•ื“ืช ื”ืคื™ืฆื•ืฅโ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืžื ืงื•ื“ืช ื”ื™ืจื™ ืคื•ื’ืข ื”ืจืกื™ืก ื”ืฉื ื™ ื‘ืงืจืงืข?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_018.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขืช ืงืจื•ื ื•ืช ื‘ื’ืฉืโ€ฌ
โ€ซื‘ื™ื•ื ื’ืฉื ืžืฉืœื—ื™ื ืฉื ื™ ืงืจื•ื ื•ืช ื–ื”ื™ื ื‘ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช ืฉื•ื•ื”โ€ช .โ€ฌื‘ืงืจื•ืŸ ื”ืื—ื“ ืžืฆื˜ื‘ืจื™ื ืžื™ ื”ื’ืฉื ื‘ืงืฆื‘ ืงื‘ื•ืขโ€ช ,โ€ฌื•ื‘ืงืจื•ืŸ ื”ืฉื ื™ ื™ืฉื ื• ื—ื•ืจโ€ฌ
โ€ซื”ืžืืคืฉืจ ืœืžื™ ื”ื’ืฉื ืœื–ืจื•ื ื”ื—ื•ืฆื” ื‘ื ื™ืฆื‘ ืœื›ื™ื•ื•ืŸ ื”ืชื ื•ืขื”โ€ช .โ€ฌื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ืงืจื•ืŸ ื•ื”ืžืกื™ืœื” ื–ื ื™ื—โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืจืฉื•ื ื‘ื™ื˜ื•ื™ ืœืžื”ื™ืจื•ืช ื›ืœ ืงืจื•ืŸ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌืื™ื–ื” ืงืจื•ืŸ ื™ื’ื™ืข ืœืžืจื—ืง ื’ื“ื•ืœ ื™ื•ืชืจ ืขื“ ืขืฆื™ืจืชื• ื”ืกื•ืคื™ืช?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_019.htmlโ€ฌโ€ฌ
โ€ซื‘ื™ืงื•ืขโ€ฌ
โ€ซื‘ื™ืงื•ืข ื’ืจืขื™ื ื™โ€ช ,โ€ฌืชื”ืœื™ืš ื”ืžืกืคืง ืื ืจื’ื™ื” ื‘ืชื—ื ื•ืช ื›ื•ื— ื’ืจืขื™ื ื™ื•ืชโ€ช ,โ€ฌืžืชืจื—ืฉ ื›ืืฉืจ ื’ืจืขื™ืŸ ื›ื‘ื“ ืžืชื‘ืงืข ืœืฉื ื™ ื’ืจืขื™ื ื™ื ืงื˜ื ื™ื ื™ื•ืชืจโ€ช .โ€ฌืจื™ืืงืฆื™ื” ืื—ืชโ€ฌ
โ€ซื›ื–ื• ืžืชืจื—ืฉืช ื›ืืฉืจ ื ื•ื™ื˜ืจื•ืŸ ืžืชื ื’ืฉ ื‘ื’ืจืขื™ืŸ ืฉืœ โ€ช U235โ€ฌื•ืžืคืฆืœ ืื•ืชื• ืœื’ืจืขื™ืŸ โ€ช Ba141โ€ฌื•ืœื’ืจืขื™ืŸ โ€ช .Kr92โ€ฌื‘ืจื™ืืงืฆื™ื” ื–ื• ืžืชืคืฆืœื™ื ื•ื ืชื–ื™ื ื’ืโ€ฌ
โ€ซืฉื ื™ ื ื•ื™ื˜ืจื•ื ื™ื ื ื•ืกืคื™ื ืžื’ืจืขื™ืŸ ื”โ€ช U235 -โ€ฌื”ืžืงื•ืจื™โ€ช .โ€ฌืื™ื•ืจ ื' ืžืชืืจ ืืช ื”ืžืฆื‘ ืœืคื ื™ ื”ื”ืชื ื’ืฉื•ืชโ€ช .โ€ฌืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช ื ืข ื’ืจืขื™ืŸ ื”โ€ชBa141 -โ€ฌโ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ โ€ช ,+zโ€ฌื•ื’ืจืขื™ืŸ โ€ช-โ€ฌื”โ€ช Kr92โ€ฌื‘ื›ื™ื•ื•ืŸ โ€ช .-zโ€ฌืฉืœื•ืฉืช ื”ื ื•ื™ื˜ืจื•ื ื™ื ื ืขื™ื ื‘ืžื™ืฉื•ืจ โ€ช xyโ€ฌื›ืžืชื•ืืจ ื‘ืื™ื•ืจ ื‘'โ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ื ื•ื™ื˜ืจื•ืŸโ€ฌ
โ€ซื”ื™ื โ€ช 4โ€ฌืžื™ืœื™ื•ืŸ ืžโ€ช/โ€ฌืฉื 'โ€ช ,โ€ฌื•ืžื”ื™ืจื•ืชื• ื”ืกื•ืคื™ืช โ€ช 2โ€ฌืžื™ืœื™ื•ืŸ ืžโ€ช/โ€ฌืฉื 'โ€ช .โ€ฌืžื” ื”ืŸ ืžื”ื™ืจื•ื™ื•ืช ืฉื ื™ ื”ื ื•ื™ื˜ืจื•ื ื™ื ื”ืื—ืจื™ื? ื•ืžื” ื ื™ืชืŸ ืœื•ืžืจ ืขืœ ืžื”ื™ืจื•ื™ื•ืช ื’ืจืขื™ื ื™ ื”โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช Ba141โ€ฌื•ื”โ€ช?Kr92 -โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_031.htmlโ€ฌโ€ฌ
โ€ซืชื ืขโ€ฌ
โ€ซื”ืžื•ื ื—ืช ืขืœ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌื‘ื™ืŸ ื”ืžืกื•ืช ืงื™ื™ื ื—ื™ื›ื•ืš ื‘ืขืœ ืžืงื“ื ื—ื™ื›ื•ืš ืงื™ื ื˜ื™โ€ฌ
โ€ซืžื•ื ื—ืช ืขืœ ื’ื‘ื™ ืžืกื”โ€ฌ
โ€ซืžืกื”โ€ฌ
โ€ซืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช โ€ช .โ€ฌืžื” ื™ื”ื™ื• ืžื”ื™ืจื•ื™ื•ืช ื”ืžืกื•ืช ืœืื—ืจ ื–ืžืŸ ืจื‘?โ€ฌ
โ€ซืžืขื™ื ืงื™ื ืœืžืกื”โ€ฌ
โ€ซืโ€ช .โ€ฌืคืชื•ืจโ€ช/โ€ฌื™ ืขืœ ื™ื“ื™ ืฉื™ืงื•ืœื™ ืชื ืขโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืคืชื•ืจโ€ช/โ€ฌื™ ืขืœ ื™ื“ื™ ืฉื™ืงื•ืœื™ ืื ืจื’ื™ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_032.htmlโ€ฌโ€ฌ
โ€ซืชื ืขโ€ฌ
โ€ซื—ื‘ืœ ื’ืžื™ืฉ ื•ืื—ื™ื“ ื‘ืขืœ ืื•ืจืš ื•ืžืกื” ืžื•ื ื— ืขืœ ื”ืงืจืงืขโ€ช .โ€ฌืžื”ื• ื”ื›ื•ื— ืฉื™ืฉ ืœื”ืคืขื™ืœ ืขืœ ืžื ืช ืœื”ืจื™ื ืื ื›ื™ืชโ€ฌ
โ€ซืืช ืื—ื“ ืžืงืฆื•ืช ื”ื—ื‘ืœ ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ? ืžืฆืโ€ช/โ€ฌื™ ื‘ื™ื˜ื•ื™ ืœื›ื•ื— ื›ืคื•ื ืงืฆื™ื” ืฉืœ ืื•ืจืš ื”ื—ื‘ืœ ื”ืžื•ืจื โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_033.htmlโ€ฌโ€ฌ
โ€ซืชื ืขโ€ฌ
โ€ซื ื•ืกืข ืขืœ ืžืกื™ืœื” ื—ืกืจืช ื—ื™ื›ื•ืš ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืงืจื•ืŸ ื‘ืขืœื™ ืžืกื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื›ืš ืฉืงืฆื‘ ื’ื™ื“ื•ืœ ื”ืžืกื” ืฉืœื•โ€ฌ
โ€ซโ€ช .โ€ฌืœืคืชืข ืžืชื—ื™ืœ ืœืจื“ืช ื’ืฉื ื•ืžื™ื ื ืื’ืจื™ื ื‘ืงืจื•ืŸโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืงืจื•ืŸ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ื™ืชื” ืžื”ื™ืจื•ืช ื”ืงืจื•ืŸ ืื ื‘ืงืจื•ืŸ ื”ื™ื” ื—ื•ืจ ื›ืš ืฉื”ืžื™ืโ€ฌ
โ€ซื”ื™ื• ื–ื•ืจืžื™ื ื”ื—ื•ืฆื” ื‘ื ื™ืฆื‘ ืœื›ื™ื•ื•ืŸ ื”ื ืกื™ืขื”? )ื”ื ื—โ€ช/โ€ฌื™ ื›ื™ ื”ืžื™ื ืœื ืžืฆื˜ื‘ืจื™ื ื‘ืงืจื•ืŸ(โ€ช.โ€ฌโ€ฌ
โ€ซื•โ€ช. -โ€ฌโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซื‘ื˜ืโ€ช/โ€ฌื™ ืชืฉื•ื‘ื•ืชื™ืš ื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_034.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืฉืจืฉืจืช ื‘ืžืกื” โ€ช mโ€ฌื•ื‘ืื•ืจืš โ€ช dโ€ฌืžื•ื ื—ืช ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™โ€ช ,โ€ฌื›ืš ืฉืจื‘ืข ืžืžื ื” ืชืœื•ื™ ืžืขื‘ืจ ืœืงืฆื” ื”ืฉื•ืœื—ืŸโ€ช .โ€ฌื‘ื™ืŸ ื”ืฉืจืฉืจืช ื•ื”ืฉื•ืœื—ืŸ ืžืงื“ื ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช.0.2โ€ฌโ€ฌ
โ€ซื”ืฉืจืฉืจืช ืžืฉื•ื—ืจืจืช ืžืžืฆื‘ ืกื˜ื˜ื™ ื•ืžื—ืœื™ืงื” ืžืŸ ื”ืฉื•ืœื—ืŸโ€ช .โ€ฌื—ืฉื‘ื• ืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืช ื”ืฉืจืฉืจืช ื‘ืจื’ืข ืฉืชืกื™ื™ื ืœื”ื—ืœื™ืง ืขืœ ื”ืฉื•ืœื—ืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_2_035.htmlโ€ฌโ€ฌ
โ€ซืคื•ื˜ื ืฆื™ืืœ ืžืจื›ื–ื™โ€ฌ
โ€ซื”ื•ืงื˜ื•ืจ ืจื•ื ื’ื” ืœื ืฅ ืžื•ื’ื“ืจ ืข"ื™โ€ช:โ€ฌโ€ฌ
โ€ซื”ืจืื• ืฉืขื‘ื•ืจ ืคื•ื˜ื ืฆื™ืืœ ื”ื›ื‘ื™ื“ื”โ€ช:โ€ฌโ€ฌ
โ€ซื•ืงื˜ื•ืจ ืจื•ื ื’ื” ืœื ืฅ ื”ื•ื ืงื‘ื•ืข ืชื ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_100.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืžื—ืœื™ืง ืขืœ ืคื ื™ ืžืฉื˜ื— ืขืงื•ื )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช .โ€ฌื”ื’ื•ืฃ ืžืฉื•ื—ืจืจ ืžืžื ื•ื—ื” ื‘ื ืงื•ื“ื” ื•ื ืขืฆืจ ื‘ื ืงื•ื“ื” ื”ื ืžืฆืืชโ€ฌ
โ€ซื‘ืžืจื—ืง ืื•ืคืงื™ ืžืžื ื”โ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืฃ ืœื‘ื™ืŸ ื”ืžืฉื˜ื— ื”ื•ื โ€ช .โ€ฌืžื”ื• ื”ืคืจืฉ ื”ื’ื‘ื”ื™ื ื‘ื™ืŸ ื”ื ืงื•ื“ื•ืช ื•โ€ช? -โ€ฌโ€ฌ
โ€ซื”ื ื—โ€ช/โ€ฌื™ ืฉืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื”ื™ื ืงื˜ื ื” ื›ืš ืฉื ื™ืชืŸ ืœื”ื–ื ื™ื— ืืช ื”ืชื•ืกืคืช ืœื›ื•ื— ื”ื ื•ืจืžืœื™ ื”ื ื•ืฆืจืช ื‘ืฉืœ ื”ืชื ื•ืขื” ื‘ืžืกืœื•ืœ ืขืงื•ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_101.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื ืข ื‘ืžืขืœื” ืžื™ืฉื•ืจ ืžืฉื•ืคืขโ€ช .โ€ฌื–ื•ื™ืช ื”ื ื˜ื™ื” ืฉืœ ื”ืžื™ืฉื•ืจ ื‘ื™ื—ืก ืœืื•ืคืง ื”ื™ืโ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืฉืžืกืชื•โ€ฌ
โ€ซื‘ื”ืฉืคืขืช ื”ื›ื•ื—ื•ืช ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ ื”ืื•ืคืงื™โ€ช.โ€ฌโ€ฌ
โ€ซโ€ฌโ€ซืฉื›ื™ื•ื•ื ื• ื‘ืžืขืœื” ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืขโ€ช.โ€ฌโ€ฌ
โ€ซโ€ฌโ€ซื›ื•ื— ื—ื™ื›ื•ืš ืฉื’ื•ื“ืœื• ืงื‘ื•ืขโ€ช.โ€ฌโ€ฌ
โ€ซโ€ฌโ€ซโ€ช -โ€ฌื›ื•ื— ื”ื›ื•ื‘ื“โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื”ื’ื•ืฃ ื ืขโ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืขื‘ื•ื“ื” ืฉื ืขืฉืชื” ืขืœ ื™ื“ื™ ื”ื›ื•ื— ื”ืฉืงื•ืœโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ืขื‘ื•ื“ื” ืฉื ืขืฉืชื” ืขืœ ื™ื“ื™ ื›ืœ ืื—ื“ ืžื”ื›ื•ื—ื•ืช ื•ื”ืจืื” ืฉืกื›ื•ืžื ืฉื•ื•ื” ืœืขื‘ื•ื“ืช ื”ื›ื•ื— ื”ืฉืงื•ืœโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_102.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืžื•ื ื— ื‘ืชื—ืชื™ืช ืžื™ืฉื•ืจ ืžืฉื•ืคืข ืฉื–ื•ื™ืช ื ื˜ื™ื™ืชื• ื‘ื™ื—ืก ืœืื•ืคืง โ€ช .โ€ฌืžืขื ื™ืงื™ื ืœื’ื•ืฃ ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืชโ€ฌ
โ€ซื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืขโ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื•ื”ืงื™ื ื˜ื™ ื‘ื™ืŸ ื”ื’ื•ืฃ ืœื‘ื™ืŸ ื”ืžื™ืฉื•ืจ ื”ื•ื โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื” ื”ืžืจื—ืง ืฉื™ืขื‘ื•ืจ ื”ื’ื•ืฃ ื‘ืžืขืœื” ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข ืขื“ ืฉื™ื™ืขืฆืจ? ื‘ื˜ืโ€ช/โ€ฌื™ ืชืฉื•ื‘ืชืš ื‘ืืžืฆืขื•ืช โ€ช , ,โ€ฌื•โ€ช. -โ€ฌโ€ฌ
โ€ซื‘ืืžืฆืขื•ืช ื•โ€ช. -โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื‘ื”ื’ื™ืขื• ืœืชื—ืชื™ืช ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข โ€ช .โ€ฌื”ื‘ืขโ€ช/โ€ฌื™ ืืช ื”ื™ื—ืกโ€ฌ
โ€ซ? ื”ืกื‘ืจโ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืงื•ืจื” ืœื™ื—ืก ืฉื—ื•ืฉื‘ ื‘ืกืขื™ืฃ ื”ืงื•ื“ื ื›ืืฉืจโ€ฌ
โ€ซื‘ืžืขืœื”โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_103.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืžื‘ืขืœ ืžืกื” ื ืข ื‘ืžื™ืฉื•ืจโ€ช .โ€ฌืžื™ืงื•ืžื• ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ ื ืชื•ืŸ ืขืœ ื™ื“ื™ ื”ื•ืงื˜ื•ืจโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช , ,โ€ฌื•โ€ช -โ€ฌืงื‘ื•ืขื™ื ื•ื—ื™ื•ื‘ื™ื™ืโ€ช .โ€ฌื›ืžื• ื›ืŸโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืžืกืœื•ืœ ื”ื—ืœืงื™ืง ื™ื•ืฆืจ ืืœื™ืคืกื” ืฉืžืจื›ื–ื” ื‘ืจืืฉื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ื”ื›ื•ื— ื”ืคื•ืขืœ ืขืœ ื”ื—ืœืงื™ืง ืžื•ืคื ื” ืชืžื™ื“ ืœื›ื™ื•ื•ืŸ ื”ืจืืฉื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ืขื‘ื•ื“ื” ืฉื ืขืฉื™ืช ืขืœ ื”ื—ืœืงื™ืง ื›ืืฉืจ ื”ื•ื ื ืข ืžื”ื ืงื•ื“ื”โ€ฌ
โ€ซื“โ€ช .โ€ฌืื™ื–ื” ืขืจืš ืžืงื‘ืœ ื”ื‘ื™ื˜ื•ื™ ืฉื—ื•ืฉื‘ ื‘ืกืขื™ืฃ ื”ืงื•ื“ื ื›ืืฉืจโ€ฌ
โ€ซ?โ€ฌ
โ€ซืœื ืงื•ื“ื”โ€ฌ
โ€ซ? ื”ืกื‘ืจโ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซืœื”ื ืืชื›ื ืžืฉื•ื•ืืช ืืœื™ืคืกื”โ€ช:โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_104.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื ืžืฆื ื‘ืžืขืœื” ืฉื™ืคื•ืข ืฉืœโ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซืžื›ื•ื•ืฆื• ื‘โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื‘ืชื—ืชื™ืช ื”ืฉื™ืคื•ืข ื ืžืฆื ืงืคื™ืฅโ€ช ,โ€ฌืืฉืจ ื›ื•ื— ืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื”ืงืคื™ืฅ ืžื›ื•ื•ืฅ ื‘โ€ช-โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื‘ื”ื’ื™ืขื• ืœืงืคื™ืฅ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ืžืจื—ืง ืฉื”ื’ื•ืฃ ืขื‘ืจ ื‘ืžื•ืจื“ ื”ืฉื™ืคื•ืข ืขื“ ืœืขืฆื™ืจืชื•?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ืœืื•ืจืš ื”ืฉื™ืคื•ืข ืื•ืชื• ื™ืขืœื” ื”ื’ื•ืฃ ื—ื–ืจื”?โ€ฌ
โ€ซ)ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜( ื‘ืขืœ ืžืงื“ื ื—ื™ื›ื•ืš ืงื™ื ื˜ื™โ€ฌ
โ€ซื›ืš ืฉื”ื•ื ื ืขืฆืจ ื›ืืฉืจโ€ฌ
โ€ซโ€ช .โ€ฌื”ื’ื•ืฃ ืžืฉื•ื—ืจืจ ืžื’ื•ื‘ื”โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_105.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ื‘ืขืœ ืžืกื” ืžื—ืœื™ืง ืœืœื ื—ื™ื›ื•ืš ืขืœ ืžืกื™ืœื” ื›ืžืชื•ืืจ ื‘ืฉืจื˜ื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ื’ื•ืฃ ืžืฉื•ื—ืจืจ ืžืžื ื•ื—ื” ื‘ื ืงื•ื“ื” โ€ช .โ€ฌืžื”ื• ื”ื›ื•ื— ื”ืคื•ืขืœ ืขืœ ื”ื’ื•ืฃ ื‘ื ืงื•ื“ื” ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื”ื’ื•ื‘ื” ื”ืžื™ื ื™ืžืœื™ ืžืขืœ ื”ื ืงื•ื“ื” ืžืžื ื• ื™ืฉ ืœืฉื—ืจืจ ืืช ื”ืžืกื” ืขืœ ืžื ืช ืฉืชืฉืœื™ื ืกื™ื‘ื•ื‘ ืžืœื ื‘ืœื•ืœืื”?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_106.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืžื—ืœื™ืง ืขืœ ืžืกื™ืœื” ื›ืžืชื•ืืจ ื‘ืฉืจื˜ื•ื˜โ€ช .โ€ฌื”ืคืจืฉ ื”ื’ื‘ื”ื™ื ื‘ื™ืŸ ื ืงื•ื“ื” ืœื‘ื™ืŸ ื ืงื•ื“ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื”ื’ื•ืฃ ืžืชื ื’ืฉ ื‘ืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืข ืงืคื™ืฅ โ€ช.โ€ฌโ€ฌ
โ€ซืงื• ื™ืฉืจ ื•ืื•ืคืงื™โ€ช .โ€ฌื”ื’ื•ืฃ ืžืฉื•ื—ืจืจ ืžืžื ื•ื—ื” ื‘ื ืงื•ื“ื” โ€ช .โ€ฌื‘ื ืงื•ื“ื”โ€ฌ
โ€ซื”ื•ืโ€ฌ
โ€ซโ€ช .โ€ฌืงื˜ืขโ€ฌ
โ€ซื”ื™ื ื•โ€ฌ
โ€ซืžื”ื• ื”ื›ื™ื•ื•ืฅ ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ืงืคื™ืฅ ื›ืืฉืจโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ืžืกื™ืœื” ื—ืกืจืช ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซืฉืื•ืจื›ื• ืงื™ื™ื ื—ื™ื›ื•ืš ื‘ืขืœ ืžืงื“ื ื—ื™ื›ื•ืš ืงื™ื ื˜ื™โ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ืงื˜ืขโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_107.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื ื•ื—ืช ืขืœ ืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืขโ€ฌ
โ€ซื•ื’ื•ืจืžืช ืœื• ื›ื™ื•ื•ืฅ ืžืงืกื™ืžืœื™ ืฉืœโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื›ืžื” ืขื‘ื•ื“ื” ื ืขืฉืชื” ืขืœ ื™ื“ื™ ื›ื•ื— ื”ื›ื•ื‘ื“ ื‘ืžื”ืœืš ื›ื™ื•ื•ืฅ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื›ืžื” ืขื‘ื•ื“ื” ื ืขืฉืชื” ืขืœ ื™ื“ื™ ื”ืงืคื™ืฅ ื‘ืžื”ืœืš ื›ื™ื•ื•ืฅ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ื™ืชื” ืžื”ื™ืจื•ืช ื”ืžืกื” ื”ืจื’ืข ืคื’ื™ืขืชื” ื‘ืงืคื™ืฅ?โ€ฌ
โ€ซื“โ€ช .โ€ฌืœื• ืžื”ื™ืจื•ืช ื”ืžืกื” ื‘ืจื’ืข ืคื’ื™ืขืชื” ื‘ืงืคื™ืฅ ื”ื™ืชื” ื›ืคื•ืœื”โ€ช ,โ€ฌืžื” ื”ื™ื” ื›ื™ื•ื•ืฅ ื”ืงืคื™ืฅ ื”ืžืงืกื™ืžืœื™? ื›ืžื” ืคืชืจื•ื ื•ืช ืœืžืฉื•ื•ืื”โ€ฌ
โ€ซืฉืงื‘ืœืช ื•ืžื” ืžื™ื™ืฆื’ ื›ืœ ืคืชืจื•ืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_108.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืžืฉื•ื—ืจืจ ืžืžื ื•ื—ื” ื‘ืคืกื’ืช ืžืฉื˜ื— ื›ื“ื•ืจื™ )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื”ื–ื ื—ืช ื”ื—ื™ื›ื•ืšโ€ช ,โ€ฌื‘ืื™ื–ื• ื–ื•ื™ืช ื™ืชื ืชืง ื”ื’ื•ืฃ ืžื”ืžืฉื˜ื—?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื—ื•ื–ืจื™ื ืขืœ ื”ืชื”ืœื™ืš ื‘ื™ืจื—โ€ช .โ€ฌื‘ืื™ื–ื• ื–ื•ื™ืช ื™ืชื ืชืง ื”ื’ื•ืฃ ืžื”ืžืฉื˜ื—?โ€ฌ
โ€ซื’โ€ช .โ€ฌื‘ื—ื–ืจื” ื‘ื›ื“ื•ืจ ื”ืืจืฅโ€ช ,โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืœืœื ื”ื–ื ื—ืช ื”ื—ื™ื›ื•ืš ื”ื–ื•ื™ืช ื‘ื” ืžืชื ืชืง ื”ื’ื•ืฃ ืžื”ืžืฉื˜ื— ื’ื“ื•ืœื” ื™ื•ืชืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_109.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื›ืš ืฉืžื”ื™ืจื•ืชื” ืงื‘ื•ืขื” ื›ืœ ื”ื–ืžืŸโ€ช .โ€ฌื‘ื–ืžืŸ ืžืกื•ื™ื™ื ืžื ื™ื—ื™ื ืขืœ ื”ืขื’ืœื” ืชื™ื‘ื” ื‘ืขืœืชโ€ฌ
โ€ซืื™ืฉ ื—ืจื•ืฅ ื’ื•ืจืจ ืขื’ืœื” ื‘ืขืœืช ืžืกื”โ€ฌ
โ€ซโ€ช .โ€ฌื‘ืจื’ืข ื”ื ื—ืช ื”ืชื™ื‘ื”โ€ช ,โ€ฌืžื”ื™ืจื•ืชื” ืฉืœ ื”ืชื™ื‘ื” โ€ช .โ€ฌื‘ื™ืŸ ื”ืขื’ืœื” ืœื‘ื™ืŸ ื”ืชื™ื‘ื” ื—ื™ื›ื•ืš โ€ช.โ€ฌโ€ฌ
โ€ซืžืกื”โ€ฌ
โ€ซืโ€ช .โ€ฌืฉืจื˜ื˜โ€ช/โ€ฌื™ ืชืจืฉื™ื ื›ื•ื—ื•ืช ืขื‘ื•ืจ ื”ืขื’ืœื” ื•ืขื‘ื•ืจ ื”ืชื™ื‘ื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ืฉื‘ื•ืฆืขื” ืขืœ ื”ืชื™ื‘ื” ืžืจื’ืข ื”ื ื—ืชื” ืขืœ ื”ืขื’ืœื” ื•ืขื“ ืฉื”ื’ื™ืข ืœืžื”ื™ืจื•ืช ? ืžื™ ื‘ื™ืฆืข ืขื‘ื•ื“ื” ื–ื•?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ืฉืžื‘ืฆืข ื”ืื™ืฉ ื”ื—ืจื•ืฅ ืžืจื’ืข ื”ื ื—ืช ื”ืชื™ื‘ื” ืขืœ ื”ืขื’ืœื” ื•ืขื“ ืฉื”ื’ื™ืขื” ืœืžื”ื™ืจื•ืช ?โ€ฌ
โ€ซื“โ€ช .โ€ฌื”ืกื‘ืจโ€ช/โ€ฌื™ ืžื“ื•ืข ื”ืชืฉื•ื‘ื•ืช ืœืกืขื™ืคื™ื ื‘' ื•โ€ช-โ€ฌื’' ืฉื•ื ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_201.htmlโ€ฌโ€ฌ
โ€ซืžืชืงืฃ ื•ืชื ืขโ€ฌ
โ€ซืžื•ื ื— ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ื•ืžื—ื•ื‘ืจ ืœืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืข ืงืคื™ืฅโ€ฌ
โ€ซื‘ื•ืœ ืขืฅ ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื•ื ื ืขืฅ ื‘ื‘ื•ืœโ€ช .โ€ฌืžื”ื™ ื”ื”ืชื›ื•ื•ืฆื•ืช ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซืโ€ช .โ€ฌื›ืืฉืจ ื‘ื™ืŸ ื”ื‘ื•ืœ ืœื‘ื™ืŸ ื”ืžืฉื˜ื— ืื™ืŸ ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื›ืืฉืจ ื‘ื™ืŸ ื”ื‘ื•ืœ ืœื‘ื™ืŸ ื”ืžืฉื˜ื— ื—ื™ื›ื•ืš ืงื™ื ื˜ื™ โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืงืœื™ืข ื‘ืขืœ ืžืกื”โ€ฌ
โ€ซื ื•ืจื” ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_202.htmlโ€ฌโ€ฌ
โ€ซืžืชืงืฃ ื•ืชื ืขโ€ฌ
โ€ซืžื—ืœื™ืง ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ื—ืœืง ื‘ืžื”ื™ืจื•ืช ืœืื•ืจืš ืฆื™ืจ )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช .โ€ฌื”ื—ืœืงื™ืง ืžืชื ื’ืฉโ€ฌ
โ€ซื—ืœืงื™ืง ื ืงื•ื“ืชื™ ืžืกื”โ€ฌ
โ€ซื‘ื›ื™ื•ื•ืŸ ื”ื™ื•ืฆืจ ื–ื•ื™ืชโ€ฌ
โ€ซื‘ื—ืœืงื™ืง ืื—ืจ ื‘ืขืœ ืžืกื” ื”ื ืžืฆื ื‘ืžื ื•ื—ื”โ€ช .โ€ฌืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช ื ืข ื”ื—ืœืงื™ืง ืฉืžืกืชื• ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืขื ืฆื™ืจ โ€ช.โ€ฌโ€ฌ
โ€ซื‘ืชโ€ฌ
โ€ซืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช?โ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช )ื’ื•ื“ืœ ื•ื›ื™ื•ื•ืŸ( ืฉืœ ื”ื—ืœืงื™ืง ืฉืžืกืชื•โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืื ื”ื”ืชื ื’ืฉื•ืช ืืœืกื˜ื™ืช ืœื—ืœื•ื˜ื™ืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_3_203.htmlโ€ฌโ€ฌ
โ€ซืžืชืงืฃ ื•ืชื ืขโ€ฌ
โ€ซืžื•ื ื—ืช ืขืœ ืขื’ืœื” ืฉืžืกืชื”โ€ฌ
โ€ซื”ื™ื›ื•ืœื” ืœื ื•ืข ืœืœื ื—ื™ื›ื•ืš ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช .โ€ฌืžืขื ื™ืงื™ืโ€ฌ
โ€ซืชื™ื‘ื” ืฉืžืกืชื”โ€ฌ
โ€ซืœืชื™ื‘ื” ืžื”ื™ืจื•ืช โ€ช.โ€ฌโ€ฌ
โ€ซืฉืœ ื”ืขื’ืœื” ืฉื™ืžื ืข ืืช ืžืขื‘ืจ ื”ืชื™ื‘ื” ืœืฆื™ื“ื” ื”ืฉื ื™?โ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื”ื’ื•ื‘ื” ื”ืžื™ื ื™ืžืœื™โ€ฌ
โ€ซืฉื—ื•ืฉื‘ ื‘ืกืขื™ืฃ ื'โ€ช ,โ€ฌืžื”ืŸ ืžื”ื™ืจื•ื™ื•ืช ื”ืขื’ืœื” ื•ื”ืชื™ื‘ื” ื›ืืฉืจ ื”ืชื™ื‘ื” ื—ื•ื–ืจืช ืœืชื—ืชื™ืช ื”ืขื’ืœื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืื ื’ื•ื‘ื” ื”ืขื’ืœื” ื’ื“ื•ืœ ืžโ€ช-โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_071.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื‘ืกืจื˜ื™ื ืจื•ืื™ื ื‘ื“" ื› ื›ื™ ื”ืื“ื )ืž" ื”ืจืขื™ื" ื›ืžื•ื‘ืŸ( ื”ื—ื•ื˜ืฃ ื›ื“ื•ืจ ืขืฃ ืœืื—ื•ืจ ื‘ืขื•ืฆืžื” ืจื‘ื”โ€ช .โ€ฌื”ืื ืกืฆื™ื ื•ืช ืืœื• ืžืฆื™ืื•ืชื™ื•ืช?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืžืฉืงืœ ืื“ื ืžืžื•ืฆืข โ€“ โ€ช.kg 80โ€ฌโ€ฌ
โ€ซืขื‘ื•ืจ ื›ื“ื•ืจ ืืงื“ื— ื’ื“ื•ืœ )ืงืœื™ื‘ืจ โ€ช :(inch 0.45โ€ฌืžืกื” โ€ช, grain 230 -โ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืช โ€“ โ€ช.ft./sec 830โ€ฌโ€ฌ
โ€ซืขื‘ื•ืจ ื›ื“ื•ืจ โ€ช 7.62โ€ฌืž" ืž ืฉืœ ืจื•ื‘ื” โ€ช :M-24โ€ฌืžืกื” โ€“ โ€ช, grain 115โ€ฌโ€ฌ
โ€ซืžื”ื™ืจื•ืช โ€“ โ€ช.mi/hr 2180โ€ฌโ€ฌ
โ€ซื”ืขืจื”โ€ช grain :โ€ฌื”ื™ื ืžื™ื“ืช ืžืฉืงืœ ืขื‘ื•ืจ ืงืœื™ืขื™ืโ€ช grain .โ€ฌืื—ื“ ืฉื•ื•ื” ืœืžืฉืงืœ ื’ืจื’ืจ ื—ื™ื˜ื” ื‘ื•ื“ื“ ืžื”ืฉื™ื‘ื•ืœืช ื”ืžืจื›ื–ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_072.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืงืœื™ืข ื‘ืžืกื” โ€ช 3.5โ€ฌืง" ื’ ื ื•ืจื” ืื•ืคืงื™ืช ืœืขื‘ืจ ืฉืชื™ ืงื•ื‘ื™ื•ืช ื”ื ืžืฆืื•ืช ื‘ืžื ื•ื—ื” ืขืœโ€ช-โ€ฌื’ื‘ื™ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌืžืกืช ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” โ€ช 1.20โ€ฌืง" ื’โ€ฌ
โ€ซื•ืžืกืช ื”ืงื•ื‘ื™ื” ื”ืฉื ื™ื” โ€ช 1.80โ€ฌืง" ื’โ€ช .โ€ฌื”ืงืœื™ืข ืขื•ื‘ืจ ื“ืจืš ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” ื•ื ืชืงืข ื‘ืฉื ื™ื”โ€ช .โ€ฌื‘ืขื•ื‘ืจื• ื“ืจืš ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” ื”ื•ื ืžืขื ื™ืง ืœืงื•ื‘ื™ื”โ€ฌ
โ€ซืžื”ื™ืจื•ืช ืฉืœ โ€ช 0.63โ€ฌืžื˜ืจ ืœืฉื ื™ื”โ€ช ,โ€ฌื•ืœืื—ืจ ืฉื”ื•ื ื ื ืขืฅ ื‘ืงื•ื‘ื™ื” ื”ืฉื ื™ื”โ€ช ,โ€ฌืžื”ื™ืจื•ืชื” โ€ช 1.4โ€ฌืžื˜ืจ ืœืฉื ื™ื”โ€ช.โ€ฌโ€ฌ
โ€ซ·ืžื”ื™ ืžื”ื™ืจื•ืช ื”ืงืœื™ืข ืœืื—ืจ ืฉื—ื“ืจ ื“ืจืš ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื”?โ€ฌ
โ€ซ·ืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ืงืœื™ืข?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_073.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืขื’ืœื” ืขืžื•ืกืช ื—ื•ืœ ื ื’ืจืจืช ืขืœ ื™ื“ื™ ื›ื•ื— ืงื‘ื•ืขโ€ฌ
โ€ซื”ื ื™ื—ื• ื›ื™ ื‘ื–ืžืŸโ€ฌ
โ€ซโ€ช .โ€ฌืขืงื‘ ื—ื•ืจ ื‘ืงืจืงืขื™ืช ื ืฉืคืš ื—ื•ืœ ื‘ืงืฆื‘ ืื—ื™ื“โ€ฌ
โ€ซื”ืขื’ืœื” ื‘ืžื ื•ื—ื”โ€ช ,โ€ฌื•ืžืกืชื” ืื– ื”ื™ืโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซื( ืžืฆืื• ืืช ื”ืžืกื” ื›ืชืœื•ืช ื‘ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืžืฆืื• ืืช ืžื”ื™ืจื•ืช ื”ืขื’ืœื” ื›ืชืœื•ืช ื‘ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_074.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื—ื•ื“ืจ ืœืชื•ืšโ€ฌ
โ€ซื•ืžื”ื™ืจื•ืชื•โ€ฌ
โ€ซืžื•ื ื— ืขืœ ื’ื‘ื™ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌืงืœื™ืข ืฉืžืกืชื•โ€ฌ
โ€ซื•ื‘ืชื•ื›ื• ื“ื—ื•ืก ื’ื– ืฉืžืกืชื•โ€ฌ
โ€ซื›ื“ื•ืจ ืฉืžืกืชื•โ€ฌ
โ€ซื”ื›ื“ื•ืจ ื•ื ืœื›ื“ ื‘ืชื•ื›ื• ื‘ืจื’ืข โ€ช) .t=0โ€ฌื ื™ืชืŸ ืœื”ื–ื ื™ื— ืืช ืคืœื™ื˜ืช ื”ื’ื– ื‘ืžื”ืœืš ื”ื”ืชื ื’ืฉื•ืช(โ€ช .โ€ฌื”ื’ื– ื”ื“ื—ื•ืก ื‘ื›ื“ื•ืจ ื ืคืœื˜ ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ฌ
โ€ซื‘ื™ื—ืก ืœื›ื“ื•ืจ ื•ื‘ืงืฆื‘ ืงื‘ื•ืขโ€ช,โ€ฌโ€ฌ
โ€ซ)โ€ช -aโ€ฌืคืจืžื˜ืจ ื—ื™ื•ื‘ื™(โ€ช.โ€ฌโ€ฌ
โ€ซ·ืžื”ื™ ืžื”ื™ืจื•ืชื• ืฉืœ ื”ื›ื“ื•ืจ ืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช ืขื ื”ืงืœื™ืขโ€ช.โ€ฌโ€ฌ
โ€ซ·ืžืฆื ืืช ืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ )ื”ื’ื– ืžืชื—ื™ืœ ืœื”ื™ืคืœื˜ ื‘โ€ช.(t=0-โ€ฌโ€ฌ
โ€ซ·ืžื”ื™ ืžื”ื™ืจื•ืชื• ื”ืกื•ืคื™ืช ืฉืœ ื”ื›ื“ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_075.htmlโ€ฌโ€ฌ
โ€ซืื ืจื’ื™ื” ืžื”ื™ืจื•ืช ื•ื–ืžืŸโ€ฌ
โ€ซืฉื ื™ ื’ื•ืคื™ื ื–ื”ื™ื ืžืชื—ื™ืœื™ื ืœื ื•ืข ื™ืžื™ื ื”โ€ช ,โ€ฌื›ืคื™ ืฉืžืจืื” ื”ืื™ื•ืจโ€ช ,โ€ฌื‘ืžื”ื™ืจื•ื™ื•ืช ื–ื”ื•ืชโ€ช ,โ€ฌืžื ืงื•ื“ื”โ€ฌ
โ€ซโ€ช .โ€ฌืื™ื–ื” ื’ื•ืฃ ื™ื’ื™ืข ืจืืฉื•ืŸ ืœื ืงื•ื“ื”โ€ฌ
โ€ซ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_076.htmlโ€ฌโ€ฌ
โ€ซืฉื™ืžื•ืจ ืื ืจื’ื™ื”โ€ฌ
โ€ซื( ื”ื•ื›ื—โ€ช/โ€ฌื™ ืžื™ ืžื‘ื™ืŸ ื”ื›ื•ื—ื•ืช ื”ื‘ืื™ื ื”ื•ื ื›ื•ื— ืžืฉืžืจโ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช Bโ€ฌื”ื•ื ื•ืงื˜ื•ืจ ื›ืœืฉื”ื• ื•โ€ช V-โ€ฌื”ื™ื ืžื”ื™ืจื•ืช )ืจืžื–โ€ช :โ€ฌืœืฉื™ื ืœื‘ ืœืžื›ืคืœื” ื”ื•ืงื˜ื•ืจื™ืช ื•ืœื ืœืฉื›ื•ื— ืฉโ€ช-โ€ฌโ€ฌ
โ€ซ(โ€ฌ
โ€ซื‘( ื”ื™ื—ื™ื“ื•ืช ืฉืœ ืžื”ื™ืจื•ืช ื”ืื•ืจ โ€ช cโ€ฌื”ืŸโ€ฌ
โ€ซโ€ช ,โ€ฌืฉืœ ืงื‘ื•ืข ื”ื›ื‘ื™ื“ื” โ€ช Gโ€ฌื”ืŸโ€ฌ
โ€ซื•ืฉืœ ืงื‘ื•ืข ืคืœื ืง โ€ช hโ€ฌื”ืŸโ€ฌ
โ€ซโ€ช.โ€ฌืขืœ ื™ื“ื™ ืฉื™ืžื•ืฉ ื‘ื’ื“ืœื™ืโ€ฌ
โ€ซืืœื• ืžืฆืโ€ช/โ€ฌื™ ื’ื•ื“ืœ ื‘ืขืœ ื™ื—ื™ื“ื•ืช ืฉืœโ€ช (I:โ€ฌื–ืžืŸโ€ช (II ,โ€ฌืžืจื—ืงโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_077.htmlโ€ฌโ€ฌ
โ€ซื”ืกืคืงโ€ฌ
โ€ซื( ืœื‘ ื”ืื“ื ื”ื•ื ืžืฉืื‘ื” ืจื‘ืช ื›ื•ื— ื•ืืžื™ื ื” ื‘ื™ื•ืชืจโ€ช .โ€ฌื‘ื›ืœ ื™ืžืžื” ื”ื•ื ืฉื•ืื‘ ื•ืคื•ืœื˜ ื›โ€ช 7500-โ€ฌืœื™ื˜ืจื™ื ืฉืœ ื“ืโ€ช .โ€ฌืื ื”ืขื‘ื•ื“ื” ื”ื ืขืฉื™ืช ืขืœ ื™ื“ื™โ€ฌ
โ€ซื”ืœื‘ ืฉื•ื•ื” ืœืขื‘ื•ื“ื” ื”ื“ืจื•ืฉื” ืœื”ืขืœื•ืช ื›ืžื•ืช ื›ื–ื• ืฉืœ ื“ื ืœื’ื•ื‘ื” ืžืžื•ืฆืข ืฉืœ ืื™ืฉื” โ€“ โ€ช ,1.63mโ€ฌื•ืื ืฆืคื™ืคื•ืช ื”ื“ื ืฉื•ื•ื” ืœืฆืคื™ืคื•ืช ื”ืžื™ืโ€ฌ
โ€ซโ€ช (Iโ€ฌื›ืžื” ืขื‘ื•ื“ื” ืขื•ืฉื” ื”ืœื‘ ื‘ื™ืžืžื”?โ€ฌ
โ€ซโ€ช (IIโ€ฌืžื” ืชืคื•ืงืช ื”ื”ืกืคืง ืฉืœื• ื‘ื•ืื˜ื™ื?โ€ฌ
โ€ซื‘( ื ืชื•ืŸ ืจื›ื‘ ื‘ืžืฉืงืœ โ€ช1.2โ€ฌื˜ื•ืŸโ€ช ,โ€ฌื•ื”ืกืคืง ืžื ื•ืข ืฉืœ โ€ช .125hpโ€ฌื‘ื ืกื™ืขื” ืžื™ืฉื•ืจื™ืช ืžืคืชื— ื”ืจื›ื‘ ืžื”ื™ืจื•ืช ืžื›ืกื™ืžืœื™ืช ืฉืœ โ€ช 165โ€ฌืงืž"ืฉโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช (Iโ€ฌืžื” ื”ื›ื•ื— ืฉืžืคืชื— ื”ืžื ื•ืข ื‘ืžื”ื™ืจื•ืช ื–ื•?โ€ฌ
โ€ซโ€ช(IIโ€ฌื›ืขืช ืขื•ืœื” ื”ืจื›ื‘ ื‘ืžืขืœื” ืžื“ืจื•ืŸ ื‘ืฉื™ืคื•ืข ืฉืœ โ€ช .5ºโ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืื™ื–ื• ืžื”ื™ืจื•ืช ืžื›ืกื™ืžืืœื™ืช ื™ืคืชื— ื”ืจื›ื‘โ€ช.โ€ฌโ€ฌ
โ€ซ(? ืžื” ืขืจื›ื” ื‘โ€ช? SI-โ€ฌโ€ฌ
โ€ซื’( ืžื” ืžืฆื™ื™ื ืช ื”ื™ื—ื™ื“ื” ืงื™ืœื•ื•ืื˜โ€ช-โ€ฌืฉืขื” )โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_078.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืืžืช ื‘ืคืจืกื•ืโ€ฌ
โ€ซื‘ื”ื™ื•ืชืš ืžื”ื ื“ืกโ€ช/โ€ฌืช ืžื—ืœืงืช ื—ื’ื•ืจื•ืช ื‘ื˜ื™ื—ื•ืช ืฉืœ ื™ืฆืจื ื™ืช ืจื›ื‘ ื™ื“ื•ืขื” ื ื™ื’ืฉ ืืœื™ืš ืžื ื”ืœ ื”ืžื—ืœืงื” ื•ืื•ืžืจ ืœืš ืฉื”ื•ื ืฉืžืข ื‘ืจื“ื™ื• ืฉื‘ืขืชโ€ฌ
โ€ซื”ืชื ื’ืฉื•ืช ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 60โ€ฌืงืž"ืฉโ€ช ,โ€ฌืชื™ื ื•ืง ื‘ืžืฉืงืœ โ€ช 2.5โ€ฌืง"ื’ ืžืคืขื™ืœ ืขืœ ืืžื• ืฉืžื—ื–ื™ืงื” ืื•ืชื• ื›ื•ื— ื”ืฉื•ื•ื” ืœืžืฉืงืœ ืฉืœ ื˜ื•ืŸ ืื—ื“!โ€ฌ
โ€ซื”ืื ื–ื” ื ื›ื•ืŸ? )ื”ืชื ื’ืฉื•ืช ื ืžืฉื›ืช โ€ช msec 5โ€ฌื‘ืžืžื•ืฆืข(โ€ฌ
โ€ซื( ืžื” ืคืชืื•ื! ื–ื” ืฉื•ื‘ ื”ืจื“ื™ื• ืขื ื”ืฉื˜ื•ื™ื•ืช ืฉืœื•!โ€ฌ
โ€ซื‘( ื›ืŸโ€ช ,โ€ฌื–ื” ืžื” ืฉืงื•ืจื” ื›ืฉื”ื™ืœื“ ืœื ื‘ื›ื™ืกื ื‘ื˜ื™ื—ื•ืช!โ€ฌ
โ€ซื’( ืžื”???โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_079.htmlโ€ฌโ€ฌ
โ€ซืžื–ื—ืœืช โ€ช -โ€ฌืžืกื” ืžืฉืชื ื”โ€ฌ
โ€ซืžื–ื—ืœืช ืžืœืื” ื‘ื—ื•ืœ ืžื—ืœื™ืงื” ืœืœื ื—ื™ื›ื•ืš ื‘ืžื•ืจื“ ืžื™ืฉื•ืจ ืฉืฉื™ืคื•ืขื• โ€ช .30°โ€ฌื ืชื•ืŸ ื›ื™ ื”ื—ื•ืœ ื“ื•ืœืฃ ืžืชื—ืชื™ืช ื”ืžื–ื—ืœืช ื‘ืงืฆื‘ ืฉืœ โ€ช .kg/sec 2โ€ฌื”ืžื–ื—ืœืชโ€ฌ
โ€ซืžืชื—ื™ืœื” ืžืžื ื•ื—ื” ื•ืžืกืชื” โ€ช.kg 40โ€ฌโ€ฌ
โ€ซื( ืžื” ืชืื•ืฆืช ื”ืžื–ื—ืœืช ื•ืžื” ื”ื”ื‘ื“ืœ ื‘ื™ืŸ ื”ื—ื•ืœ ืœื‘ื™ืŸ ื”ืžื™ื ื‘ืฉืืœื” ืžืกืคืจ โ€ช?1โ€ฌโ€ฌ
โ€ซื‘( ื›ืžื” ื–ืžืŸ ื™ื™ืงื— ืœื” ืœืขื‘ื•ืจ โ€ช m 120โ€ฌืœืื•ืจืš ื”ืžื•ืจื“?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_080.htmlโ€ฌโ€ฌ
โ€ซืฆืœื—ืช ื—ืจืก ื•ืฉื™ืžื•ืจ ืชื ืขโ€ฌ
โ€ซืฆืœื—ืช ื—ืจืก ืฉืžืกืชื”โ€ฌ
โ€ซื ื•ืจื™ืช ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซื”ื ืข ื›ืœืคื™ ืžืขืœื” ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌ
โ€ซื( ืœืื™ื–ื” ื’ื•ื‘ื” ื ื•ืกืฃ ืชื’ื™ืข ื”ืฆืœื—ืช?โ€ฌ
โ€ซื‘( ืœืื™ื–ื” ืชื•ืกืคืช ืžืจื—ืง ืชื’ื™ืข ื”ืฆืœื—ืช?โ€ฌ
โ€ซื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช .30°โ€ฌื‘ืฉื™ื ื”ื’ื•ื‘ื” ื”ื™ื ื ืคื’ืขืช ืžืœืžื˜ื” ืขืœ ื™ื“ื™ ืงืœื™ืข ืฉืžืกืชื• โ€ช15gโ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื”ืงืœื™ืข ื ืชืงืข ื‘ืฆืœื—ืช ื”ื—ืจืกโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_081.htmlโ€ฌโ€ฌ
โ€ซืฉื™ืžื•ืจ ืชื ืขโ€ฌ
โ€ซื ืชื•ืŸ ื’ื•ืฃ โ€ช m1โ€ฌื”ืžืฉื•ื—ืจืจ ืžืžื ื•ื—ื” ืžืจืืฉื• ืฉืœ ืžื“ืจื•ืŸ ืžืฉื•ืคืข ื‘ื–ื•ื•ื™ืช โ€ชฮฑโ€ฌโ€ฌ
โ€ซื•ื’ื•ื‘ื” โ€ช .hโ€ฌืžืกืชื• ืฉืœ ื”ืžื“ืจื•ืŸ ื”ืžืฉื•ืคืข ื”ื•ื โ€ช ,m2โ€ฌื›ืžื•"ื› ื ืชื•ืŸ ื›ื™ ืื™ืŸ ื—ื™ื›ื•ืš ื‘ื™ืŸ โ€ช m1โ€ฌืœโ€ช m2 -โ€ฌื•ืœืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซื( ืžืฆืโ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืชื ืฉืœ โ€ช m1 , m2โ€ฌื›ืืฉืจ โ€ช m1โ€ฌืžื’ื™ืข ืœืงืจืงืขโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืžื” ื”ื•ื ื”ืžืจื—ืง ื”ืื•ืคืงื™ ืฉืขื‘ืจ โ€ชm1โ€ฌืขื“ ืืฉืจ ื”ื’ื™ืข ืœืงืจืงืขโ€ฌ
โ€ซ)ืจืžื–โ€ช :โ€ฌื–ื›ื•ืจ ืืช ืืฉืจ ืืžืจ ืœืš ืžืจื›ื– ื”ืžืกื”!(?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_082.htmlโ€ฌโ€ฌ
โ€ซืจืงื˜ื”โ€ฌ
โ€ซื ืชื•ื ื” ืจืงื˜ื” ื‘ืขืœืช ืžืกื” ื”ืชื—ืœืชื™ืช โ€ช . m0โ€ฌื”ืžื ื•ืข ืฉืœ ื”ืจืงื˜ื” ืฉื•ืจืฃ ื“ืœืง ื‘ืงืฆื‘โ€ฌ
โ€ซโ€ช ,โ€ฌื•ื’ื–ื™ ื”ืคืœื™ื˜ื” ื™ื•ืฆืื™ื ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช Vrโ€ฌื™ื—ืกื™ืช ืœืจืงื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซื( ืžืฆืโ€ช/โ€ฌื™ ืืช ื”ื›ื•ื— )ื”ื“ื—ืฃ( ืฉืžืคื™ืง ื”ืžื ื•ืขโ€ช ,โ€ฌื•ืืช ืชืื•ืฆืช ื”ืจืงื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซื”ื ื“ืœืงโ€ช .โ€ฌืงืฆื‘ ื”ื‘ืขื™ืจื” ื”ื•ื โ€ช , kg/sec 480โ€ฌืžื”ื™ืจื•ืช ื’ื–ื™ ื”ืคืœื™ื˜ื” ื”ื™ื โ€ช3.27โ€ฌโ€ฌ
โ€ซโ€ช ,โ€ฌืžืชื•ื›ืโ€ฌ
โ€ซื‘(โ€ฌ
โ€ซโ€ช .km/secโ€ฌื”ืื ื”ืจืงื˜ื” ืชืฆืœื™ื— ืœื”ืžืจื™ื ืžื›ื“ื•"ื”?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_083.htmlโ€ฌโ€ฌ
โ€ซืขื’ืœื” ืขืœ ืžืกื™ืœื”โ€ฌ
โ€ซืขื’ืœื” ืขื ืžืกื” โ€ช 67โ€ฌืง"ื’โ€ช ,โ€ฌื ืขื” ืขืœ ื’ื‘ื™ ืžืกื™ืœื” ืฉืžืชื•ืืจืช ืข"ื™ ื”ืžืฉื•ื•ืื”โ€ช:โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื›ืืฉืจโ€ฌ
โ€ซืื™ืŸ ื—ื™ื›ื•ืš ืขืœ ื”ืžืกื™ืœื” )ื ื™ืชืŸ ืœื”ื ื™ื— ืฉื™ืžื•ืจ ืื ืจื’ื™ื”(โ€ฌ
โ€ซืœื—ืฉื‘ ืืช ื”ืขื‘ื•ื“ื” ืฉื ืขืฉื™ืช ืข"ื™ ื›ืœ ื”ื›ื•ื—ื•ืช ื›ืืฉืจ ื”ืขื’ืœื” ื ืขื” ื‘ื™ืŸโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_084.htmlโ€ฌโ€ฌ
โ€ซืฉื™ืžื•ืจ ืชื ืขโ€ฌ
โ€ซื”ืจืื” ื“ืจืš ืžืจื›ื– ื”ืžืกื” ืฉืื ืื™ืŸ ื›ื•ื—ื•ืช ื—ื™ืฆื•ื ื™ื™ื ื™ืฉ ืฉื™ืžื•ืจ ืชื ืข?โ€ฌ
โ€ซื”ืื ื–ื” ืื•ืžืจ ืฉื™ืฉ ืฉื™ืžื•ืจ ืื ืจื’ื™ื”?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_085.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื” ืฉืœ ืชื™ืœโ€ฌ
โ€ซื ืชื•ืŸ ืชื™ืœ ื‘ืื•ืจืš ื•ืžืกื” โ€ช .โ€ฌืฆืคื™ืคื•ืช ื”ืžืกื” ื”ื™ื โ€ช:โ€ฌโ€ฌ
โ€ซืงืฆื”ื• ื”ืฉืžืืœื™ ืฉืœ ื”ืชื™ืœ ื ืžืฆื ื‘ืจืืฉื™ืช ื”ืฆื™ืจื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื‘ื˜ื ืืช ื‘ืืžืฆืขื•ืชโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ื™ื›ืŸ ืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืชื™ืœ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_086.htmlโ€ฌโ€ฌ
โ€ซืื‘ ื‘ืŸ ื•ืกื™ืจื”โ€ฌ
โ€ซืื‘ ื•ื‘ื ื• ื ืžืฆืื™ื ื‘ืงืฆื”ื• ื”ืฉืžืืœื™ ืฉืœ ืกื™ืจื” ื”ื ืžืฆืืช ื‘ืžื ื•ื—ื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘ืจื’ืข ืžืกื•ื™ื ื”ื‘ืŸ ืฆื•ืขื“ ืœืžืจื›ื– ื”ืกื™ืจื” ื•ื”ืื‘ ืœืงืฆื” ื”ื™ืžื ื™ ืฉืœ ื”ืกื™ืจื”โ€ช.โ€ฌโ€ฌ
โ€ซืžืกืช ื”ืกื™ืจื” โ€ช-โ€ฌโ€ฌ
โ€ซืื•ืจืš ื”ืกื™ืจื” โ€ช-โ€ฌโ€ฌ
โ€ซืžืกืช ื”ืื‘ โ€ช-โ€ฌโ€ฌ
โ€ซืžืกืช ื”ื‘ืŸ โ€ช-โ€ฌโ€ฌ
โ€ซื‘ื›ืžื” ื”ืกื™ืจื” ื–ื–ื”?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_087.htmlโ€ฌโ€ฌ
โ€ซื”ืชื ื’ืฉื•ืช ืื ืฉื™ื ื‘ืžื™ืฉื•ืจ ืžืฉื•ืคืขโ€ฌ
โ€ซืื‘ืจื”ื ื ืžืฆื ื‘ืงืฆื” ืžื™ืฉื•ืจ ืžืฉื•ืคืขโ€ช .โ€ฌื”ื•ื ืžื—ืœื™ืง ืœืžื˜ื” )ื‘ืœื™ ื—ื™ื›ื•ืš(โ€ช.โ€ฌโ€ฌ
โ€ซื”ื•ื ืžืชื ื’ืฉ ื‘ื‘ื ื™ ื•ื”ื ืžืžืฉื™ื›ื™ื ืœื”ื—ืœื™ืง ื‘ื™ื—ื“โ€ช.โ€ฌโ€ฌ
โ€ซืœืื—ืจ ืฉืขื‘ืจโ€ฌ
โ€ซืขื“ ืื™ื–ื” ื’ื•ื‘ื” ื”ื ื™ื’ื™ืขื•? )ื‘ื™ื—ืก ืœื ืงื•ื“ืช ื”ืžืคื’ืฉ(โ€ฌ
โ€ซื ืชื•ืŸ ืฉื‘ื ื™ ืฉื•ืงืœ ืฉื ื™ ืฉืœื™ืฉ ืžืื‘ืจื”ืโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_088.htmlโ€ฌโ€ฌ
โ€ซืขื’ืœื” ื•ื’ื‘ืขื”โ€ฌ
โ€ซืขื•ืœื” ืขืœ ื’ื‘ืขื” ืฉื™ื›ื•ืœื” ืœื ื•ืข ืขืœ ืžืฉื˜ื— ื—ืกืจ ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซืขื’ืœื” ืฉืžืกืชื”โ€ฌ
โ€ซื”ื’ื‘ืขื” ื ืžืฆืืช ื‘ืžื ื•ื—ื” ืœืคื ื™ ืขืœื™ื™ืช ื”ืขื’ืœื”โ€ช .โ€ฌื”ืžื”ื™ืจื•ืช ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ืขื’ืœื” ื”ื™ืโ€ฌ
โ€ซืโ€ช .โ€ฌืขื“ ืœืื™ื–ื” ื’ื•ื‘ื” ืชืขืœื” ืขื’ืœื”?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืช ื”ืขื’ืœื” ื•ืžื”ื™ืจื•ืช ื”ื’ื‘ืขื” ืœืื—ืจ ืฉื”ืขื’ืœื” ืชืจื“ ืžืžื ื”?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_089.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืงืคื™ืฅโ€ฌ
โ€ซืžืกื” ืžื•ืคืœืช ืขืœ ืงืคื™ืฅ ืื ื›ื™ ื‘ืžื ื•ื—ื”โ€ช .โ€ฌืงื‘ื•ืข ื”ืงืคื™ืฅ ื”ื•ื โ€ช.โ€ฌโ€ฌ
โ€ซื”ืžืกื” ื ืฆืžื“ืช ืœืงืคื™ืฅ ื•ืžื›ื•ื•ืฆืช ืื•ืชื•โ€ช .โ€ฌืื•ืจืš ื”ื›ื™ื•ื•ืฅ ื”ืžืงืกื™ืžืœื™ ื”ื•ื โ€ช.โ€ฌโ€ฌ
โ€ซื‘ื–ืžืŸ ืฉื”ืงืคื™ืฅ ืžืชื›ื•ื•ืฅ ืื™ื–ื” ืขื‘ื•ื“ื” ื ืขืฉื™ืชโ€ฌ
โ€ซืโ€ช .โ€ฌืขืœ ื™ื“ื™ ื”ื’ืจื•ื™ื˜ืฆื™ื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืขืœ ื™ื“ื™ ื”ืงืคื™ืฅโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ื”ื™ื™ืชื” ืžื”ื™ืจื•ืช ื”ืžืกื” ื‘ืจื’ืข ื”ืคื’ื™ืขื”?โ€ฌ
โ€ซื“โ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ื”ืžืกื” ื ืคืœื” ืžืžื ื•ื—ื”โ€ช ,โ€ฌืžืื™ื–ื” ื’ื•ื‘ื” ืžืขืœ ื”ืงืคื™ืฅ ื”ื™ื ื ืคืœื”?โ€ฌ
โ€ซื”โ€ช .โ€ฌืื ื’ื•ื‘ื” ื”ื ืคื™ืœื” ื™ื•ื›ืคืœ ืžื” ื™ื”ื™ื” ื”ื›ื™ื•ื•ืฅ ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ืงืคื™ืฅโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_090.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื’ื•ืฃ ืขื ื—ื™ื›ื•ืšโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช .โ€ฌืžื—ืœื™ืง ื‘ืžื•ืจื“ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ืœืžืจื—ืง โ€ช .โ€ฌื”ืžื™ืฉื•ืจ ื ื˜ื•ื™ ื‘ื–ื•ื™ืชโ€ฌ
โ€ซโ€ช .โ€ฌื—ืฉื‘โ€ฌ
โ€ซโ€ช:โ€ฌื”ื’ื•ืฃ ืœืžืฉื˜ื— ื”ื•ืโ€ฌ
โ€ซืžืชื—ืช ืœืื•ืคืงโ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸโ€ฌ
โ€ซโ€ช.โ€ฌื( ืืช ื”ืขื‘ื•ื“ื” ื”ื ืขืฉื™ืช ืขืœ ื™ื“ื™ ื”ื—ื™ื›ื•ืš ืขืœ ื”ืžืกื”โ€ฌ
โ€ซโ€ช.โ€ฌื‘( ืืช ืขื‘ื•ื“ืช ื›ื— ื”ื›ื•ื‘ื“โ€ฌ
โ€ซโ€ช.โ€ฌื’( ืืช ืขื‘ื•ื“ืช ื”ื›ื— ื”ื ื•ืจืžืœื™โ€ฌ
โ€ซโ€ช.โ€ฌื“( ืืช ืกืš ื”ืขื‘ื•ื“ื” ื”ื ืขืฉื™ืช ืขืœ ื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช.โ€ฌื”( ืืช ืžื”ื™ืจื•ืชื• ืฉืœ ื”ื’ื•ืฃ ืื ื”ื•ื ื”ื—ืœ ืืช ืชื ื•ืขืชื• ืžืžื ื•ื—ื”โ€ช ,โ€ฌื”ื ื— ื›ื™ ื”ื—ื™ื›ื•ืš ืžืกืคื™ืง ื—ืœืฉ ื›ืš ืฉื”ื’ื•ืฃ ืœื ื ืฉืืจ ื‘ืžื ื•ื—ื”โ€ฌ
โ€ซโ€ช.โ€ฌื•( ื—ื–ื•ืจ ืขืœ ื”ืกืขื™ืฃ ื”ืื—ืจื•ืŸ ืชื•ืš ืฉื™ืžื•ืฉ ื‘ืžืฉื•ื•ืื•ืช ื”ืงื™ื ืžื˜ื™ืงื”โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_091.htmlโ€ฌโ€ฌ
โ€ซืื ืจื’ื™ื” ืืœืกื˜ื™ืชโ€ฌ
โ€ซืžื›ื•ื•ืฆืช ืงืคื™ืฅ ืฉืงื‘ื•ืข ื”ื›ื— ืฉืœื• ื”ื•ื โ€ช ,โ€ฌืื•ืจืš ื”ื›ื™ื•ื•ืฅ ื”ื•ืโ€ฌ
โ€ซโ€ช .โ€ฌืžืกื”โ€ฌ
โ€ซืขื“ ื”ื’ื™ืขื• ืœืžื ื•ื—ื”โ€ฌ
โ€ซโ€ช.โ€ฌืœืื—ืจ ืฉื—ืจื•ืจื•โ€ช ,โ€ฌื ืขื” ื”ืžืกื” ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ืœืžืจื—ืง ืฉืœโ€ฌ
โ€ซืžื” ืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ืžืกื” ืœืžืฉื˜ื—โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_092.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ื•ืงืคื™ืฅโ€ฌ
โ€ซืžืกื” ืžื•ื ื—ืช ืขืœ ื’ื‘ื™ ืžืฉื˜ื— ืžืื•ื ืš ื‘ื–ื•ื™ืชโ€ฌ
โ€ซืขืœ ื”ืžืฉื˜ื— ืžื•ื ื— ืงืคื™ืฅ ืฉืงื‘ื•ืข ื”ืงืคื™ืฅ ื”ื•ื โ€ช.โ€ฌโ€ฌ
โ€ซื”ืžืกื” ืžืฉื•ื—ืจืจืช ืžืžื ื•ื—ื”โ€ช ,โ€ฌื”ื›ื™ื•ื•ืฅ ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ืงืคื™ืฅ ื”ื•ืโ€ฌ
โ€ซืโ€ช .โ€ฌืื™ื–ื” ืžืจื—ืง ื ืขื” ื”ืžืกื” ืขื“ ืœื›ื™ื™ื•ืฅ ื”ืžืงืกื™ืžืœื™ ืฉืœ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ืžื”ื™ืจื•ืช ื”ืžืกื” ื‘ืจื’ืข ื”ืžื’ืข ืขื ื”ืงืคื™ืฅ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืื™ื–ื• ืขื‘ื•ื“ื” ื ืขืฉื™ืช ืขืœ ื™ื“ื™ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_4_093.htmlโ€ฌโ€ฌ
โ€ซืงืจื•ืŸ ื”ืชืคื•ื–ื™ืโ€ฌ
โ€ซืื“ื ืขื•ืžื“ ื‘ืงืจื•ืŸ ืขื ืชืคื•ื–ื™ืโ€ช ,โ€ฌื‘ื™ื—ื“ ื™ืฉ ืœื”ื ืžืกื”โ€ฌ
โ€ซื”ืื“ื ื–ื•ืจืง ืชืคื•ื–ื™ื ืื—ื•ืจื” ื‘ืงืฆื‘ ืฉืœ ืชืคื•ื– ืœืฉื ื™ื” ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื‘ื™ื—ืก ืœืงืจื•ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ ืฉืžื”ื™ืจื•ืช ื”ืงืจื•ืŸ ื‘ื”ืชื—ืœื”โ€ฌ
โ€ซืžื” ื”ืฉื™ื ื•ื™ ื‘ืžื”ื™ืจื•ืช ืœืื—ืจ ื–ืงื™ืจืช ืชืคื•ื– ื™ื—ื™ื“?โ€ฌ
โ€ซืžื” ืชื”ื™ื” ืชืื•ืฆืช ื”ืงืจื•ืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_051.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืžื”ื™ ืžื”ื™ืจื•ืช ื”ืžืกื•ืช ืœืื—ืจ ืฉื”ืžืขืจื›ืช ื”ืžื•ืจืื™ืช ื‘ืฆื™ื•ืจ ื ืขื” โ€ช 1โ€ฌืžื˜ืจ )ื”ืžืขืจื›ืช ืžืชื—ื™ืœื” ืžืžื ื•ื—ื”( ?โ€ฌ
โ€ซืโ€ช.โ€ฌืื™ืŸ ื—ื™ื›ื•ืšโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌื™ืฉ ื—ื™ื›ื•ืš ื•ื™ื“ื•ืข ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ืฉื•ื•ื” ืœโ€ช.0.2 -โ€ฌโ€ฌ
โ€ซื ืชื•ืŸ โ€ชm1=4 kg, m2=6 kg :โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_052.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื›ื•ื— ืฉืœ โ€ช 30โ€ฌื ื™ื•ื˜ื•ืŸ ืžื•ืฉืš ืื•ืคืงื™ืช ื’ื•ืฃ ืฉืžืกืชื• โ€ช 30โ€ฌืง" ื’โ€ช .โ€ฌื”ื’ื•ืฃ ื ืข ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื” ืฉืœ โ€ช 5โ€ฌืž\ืฉ )ื”ืžืฉื˜ื— ืœื ื—ืœืง(โ€ช .โ€ฌื”ื›ื•ื— ืคื•ืขืœ ื‘ืžืฉืš โ€ช10โ€ฌโ€ฌ
โ€ซืฉื ื™ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ืขื‘ื•ื“ืช ื”ื›ื•ื— ื”ืžื•ืฉืš ? ืžื”ื• ื”ืกืคืงื• ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ืขื‘ื•ื“ืช ื›ื•ื— ื”ื—ื™ื›ื•ืš ? ืžื”ื• ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืงื™ื ื˜ื™ ?โ€ฌ
โ€ซื’โ€ช.โ€ฌืื™ื–ื” ืžืจื—ืง ื™ืขื‘ื•ืจ ื”ื’ื•ืฃ ืขื“ ืœืขืฆื™ืจื” ืžืœืื”โ€ช ,โ€ฌืžืจื’ืข ืฉื”ื•ืคืกืงื” ืคืขื•ืœืช ื”ื›ื•ื— ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_053.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื›ื“ื•ืจ ืฉืžืกืชื• โ€ช 1โ€ฌืง" ื’ ืชืœื•ื™ ืขืœ ื—ื•ื˜ ืฉืื•ืจื›ื• โ€ช 1.5โ€ฌืžื˜ืจื™ืโ€ช .โ€ฌื”ื›ื“ื•ืจ ืžื•ืกื˜ ื‘ื–ื•ื•ื™ืช โ€ช) 300โ€ฌืจืื• ืฆื™ื•ืจ ืžืฆื“ ืฉืžืืœ ืฉืœ ื”ืงื• ื”ืžืงื•ื•ืงื•( ื•ืžืฉื•ื—ืจืจโ€ช.โ€ฌโ€ฌ
โ€ซืžืกืœื•ืœ ื”ื›ื“ื•ืจ ืžื•ืคืจืข ืข" ื™ ืžืกืžืจ ื”ืชืงื•ืข โ€ช 1โ€ฌืžื˜ืจ ืžืชื—ืช ืœื ืงื•ื“ืช ื”ื—ื™ื‘ื•ืจ ืฉืœ ื”ื—ื•ื˜ )ืจืื• ืฆื™ื•ืจ ืžืฆื“ ื™ืžื™ืŸ ืฉืœ ื”ืงื• ื”ืžืงื•ื•ืงื•(โ€ช .โ€ฌืžื”ื™ ื”ื–ื•ื•ื™ืชโ€ฌ
โ€ซื”ืžื›ืกื™ืžืœื™ืช ืืœื™ื” ื™ื›ื•ืœ ืœื”ื’ื™ืข ื”ื›ื“ื•ืจ ?โ€ฌ
โ€ซื‘ื•ื ื•ืก โ€ช:โ€ฌืžื” ืชื”ื™ื” ื”ื–ื•ื•ื™ืช ื”ืžื›ืกื™ืžืœื™ืช ืื ื™ื“ื•ืข ืฉืขื“ ืœื”ืชื ื’ืฉื•ืช ืคืขืœ ืขืœ ื”ื›ื“ื•ืจ ื›ื•ื— ืื•ืคืงื™ ืงื‘ื•ืข ืœื›ื™ื•ื•ืŸ ื™ืžื™ืŸ ืฉื’ื•ื“ืœื• โ€ช 10โ€ฌื ื™ื•ื˜ื•ืŸ ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_054.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ืžื—ืœื™ืง ืขืœ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื—ืœืงโ€ช .โ€ฌื”ืžืฆื‘ ื”ื”ืชื—ืœืชื™ ืžื•ืจืื” ื‘ืฆื™ื•ืจ )ื”ื’ื•ืฃ ืžืชื—ื™ืœ ืžืžื ื•ื—ื”(โ€ช .โ€ฌื‘ื›ืžื” ื™ื™ืชื›ื•ื•ืฅ ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซื ืชื•ืŸ โ€ช :โ€ฌืžืกืช ื”ื’ื•ืฃ โ€ช 4โ€ฌืง" ื’โ€ช ,โ€ฌืงื‘ื•ืข ื”ืงืคื™ืฅ โ€ช 100โ€ฌื ื™ื•ื˜ื•ืŸ ืœืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_055.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืฉืชื™ ืžืกื•ืช โ€ช m1, m2โ€ฌื ืžืฆืื•ืช ืขืœ ืžืฉื˜ื— ื—ืœืงโ€ช .โ€ฌืœืžืกื” ื”ื™ืžื ื™ืช ื”ื ืžืฆืืช ื‘ืžื ื•ื—ื” ืžื—ื•ื‘ืจ ืงืคื™ืฅ ืจืคื•ื™ ื‘ืขืœ ืงื‘ื•ืข ืงืคื™ืฅ โ€ช .kโ€ฌื”ืžืกื” ื”ืฉืžืืœื™ืช ื ืขื”โ€ฌ
โ€ซืœืขื‘ืจ ื”ื™ืžื ื™ืช )ื•ื”ืงืคื™ืฅ( ื‘ืžื”ื™ืจื•ืช โ€ช .vโ€ฌื”ืงืคื™ืฅ ืจืคื•ื™ ื•ื—ืกืจ ืžืกื”โ€ช .โ€ฌืžื”ื™ ื”ื”ืชื›ื•ื•ืฆื•ืช ื”ืžื›ืกื™ืžืœื™ืช ืฉืœ ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_056.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื’ื•ืฃ ื‘ืขืœ ืžืกื” โ€ช 0.2โ€ฌืง" ื’ ื ื•ืคืœ ืžืžื ื•ื—ื” ืžื’ื•ื‘ื” โ€ช 2โ€ฌืžื˜ืจื™ื ืขืœ ืงืคื™ืฅ ืื ื›ื™ ื‘ืขืœ ืงื‘ื•ืข ืงืคื™ืฅ โ€ช 16โ€ฌื ื™ื•ื˜ื•ืŸ\ืžื˜ืจโ€ช .โ€ฌืงืคื™ืฅ ื—ืกืจ ืžืกื” ื•ืจืคื•ื™โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืžืกื” ืžืžืฉ ืœืคื ื™ ื”ืคื’ื™ืขื” ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ื”ืชื›ื•ื•ืฆื•ืช ื”ืžืงืกื™ืžืœื™ืช ืฉืœ ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_061.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืžื˜ื•ืก ื˜ืก ื‘ืžืขื’ืœ ืื•ืคืงื™ ื‘ืžื”ื™ืจื•ืช ืžืฉื™ืงื™ืช ืฉืœ โ€ช 480โ€ฌืงืž"ืฉโ€ช .โ€ฌืื ื”ื›ื ืคื™ื™ื ืžื•ื˜ื•ืช ื‘ื–ื•ื™ืช ืฉืœ โ€ช 40โ€ฌืžืขืœื•ืช ืœืื•ืคืงโ€ช ,โ€ฌืžื”ื• ืจื“ื™ื•ืก ื”ืžืขื’ืœ ืฉื”ืžื˜ื•ืกโ€ฌ
โ€ซืžื‘ืฆืข? )ื”ื ื—โ€ช/โ€ฌื”ื ื™ื—ื™ ื›ื™ ื”ื›ื•ื—ื•ืช ื”ืคื•ืขืœื™ื ืขืœ ื”ืžื˜ื•ืก ื”ื ื›ื•ื— ื”ื›ื•ื‘ื“ ื•ื›ื•ื— ื”ืขื™ืœื•ื™ ื”ืคื•ืขืœ ืขืœ ื”ื›ื ืคื™ื™ื ื‘ืžืื•ื ืš ืœื”ืโ€ช(.โ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_062.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื‘โ€ช 1901โ€ฌื“ื™ืื•ื•ืœื• ืจื›ื‘ ืขื ืื•ืคื ื™ื™ื ืขืœ ืžืกืœื•ืœ ืžืขื’ืœื™ ื›ืคื™ ืฉืžืจืื” ื”ื”ื•ื“ืขื”โ€ช .โ€ฌื”ื ื— ื›ื™ ืจื“ื™ื•ืก ื”ืžืกืœื•ืœ ื”ื•ื โ€ช .2.7mโ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืžื™ื ื™ืžืœื™ืชโ€ฌ
โ€ซืฉืฆืจื™ื›ื” ืœื”ื™ื•ืช ืœื“ื™ืื•ื•ืœื• ื‘ืงืฆื” ื”ืขืœื™ื•ืŸ ืฉืœ ื”ืžืกืœื•ืœ ื‘ื›ื“ื™ ืฉืœื ื™ื™ืคื•ืœ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_063.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซื›ื“ื•ืจ ืžืชื›ืช ื‘ืขืœ ืžืกื” ืฉืœ โ€ช 500โ€ฌื’ืจื ืงืฉื•ืจ ืœื—ื‘ืœ )ื—ืกืจ ืžืกื”( ื‘ืื•ืจืš โ€ช 70โ€ฌืก" ืž ื”ืงืฉื•ืจ ื‘ืงืฆื”ื•โ€ช .โ€ฌื”ื›ื“ื•ืจ ืžืฉื•ื—ืจืจ ืžืžืฆื‘ ืื•ืคืงื™โ€ช .โ€ฌื‘ืชื—ืชื™ืชโ€ฌ
โ€ซื”ืžืกืœื•ืœ ืฉืœื• ื”ื•ื ืคื•ื’ืข ื‘ืงื•ื‘ื™ืช ื‘ืจื–ืœ ื‘ืžืกื” ืฉืœ โ€ช 2.5โ€ฌืง"ื’ ื”ื ื—ื” ืขืœ ืžืฉื˜ื— ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌื”ื”ืชื ื’ืฉื•ืช ืืœืกื˜ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื›ื“ื•ืจ ืœืคื ื™ ื•ืื—ืจื™ ื”ื”ืชื ื’ืฉื•ืช?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืงื•ืคืกื ืœืคื ื™ ื•ืื—ืจื™ ื”ื”ืชื ื’ืฉื•ืช?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_064.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ื”ืชื ื’ืฉื•ื™ื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌืงืœื™ืข ืฉืžืกืชื• โ€ช 3.5โ€ฌืงื™ืœื•ื’ืจืžื™ื ื ื•ืจื” ืื•ืคืงื™ืช ืœืขื‘ืจ ืฉืชื™ ืงื•ื‘ื™ื•ืช ื”ื ืžืฆืื•ืช ื‘ืžื ื•ื—ื” ืขืœ ื’ื‘ื™ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ฌ
โ€ซโ€ช.โ€ฌืžืกืช ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” โ€ช 1.2โ€ฌืงื™ืœื•ื’ืจืžื™ื ื•ืžืกืช ื”ืงื•ื‘ื™ื” ื”ืฉื ื™ื” โ€ช 1.8โ€ฌืงื™ืœื•ื’ืจืžื™ืโ€ฌ
โ€ซื”ืงืœื™ืข ืขื•ื‘ืจ ื“ืจืš ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” ื•ื ืชืงืข ื‘ืฉื ื™ื”โ€ช .โ€ฌื‘ืขื•ื‘ืจื• ื‘ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื” ื”ื•ื ืžืขื ื™ืง ืœื” ืžื”ื™ืจื•ืช ืฉืœ โ€ช 0.63โ€ฌืžื˜ืจ ืœืฉื ื™ื” ื•ืœืื—ืจ ืฉื”ื•ืโ€ฌ
โ€ซโ€ช.โ€ฌื ื ืขืฅ ื‘ืงื•ื‘ื™ื” ื”ืฉื ื™ื” ืžื”ื™ืจื•ืชื” โ€ช 1.4โ€ฌืžื˜ืจ ืœืฉื ื™ื”โ€ฌ
โ€ซ?ืโ€ช .โ€ฌืžื” ืžื”ื™ืจื•ืช ื”ืงืœื™ืข ืœืื—ืจ ืฉื—ื“ืจ ื“ืจืš ื”ืงื•ื‘ื™ื” ื”ืจืืฉื•ื ื”โ€ฌ
โ€ซ?ื‘โ€ช .โ€ฌืžื” ืžื”ื™ืจื•ืชื• ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ืงืœื™ืขโ€ฌ
โ€ซืจืื• ืชืจืฉื™ืโ€ฌ
13_Energy_momentum/e_13_5_065.html
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
v0 โ€ซ ืขื•ืžื“ ืขืœ ื”ืงืจื•ื ื™ืช ื”ื ืขื” ื‘ืžื”ื™ืจื•ืชโ€ฌw โ€ซืื“ื ื‘ืžืฉืงืœโ€ฌ-โ€ซ ื‘ืŸโ€ฌ,โ€ซ ื‘ื”ืชื—ืœื”โ€ฌ.โ€ซ ื™ื›ื•ืœื” ืœื ื•ืข ืœืœื ื—ื™ื›ื•ืš ืขืœ ื’ื‘ื™ ืžืกื™ืœื” ืื•ืคืงื™ืชโ€ฌW โ€ซืงืจื•ื ื™ืช ื‘ืžืฉืงืœโ€ฌ
.โ€ซื‘ื™ื—ืก ืœืžืกื™ืœื”โ€ฌ
vrel โ€ซื‘ื›ืžื” ืžืฉืชื ื” ืžื”ื™ืจื•ืช ื”ืงืจื•ื ื™ืช ืื ื”ืื“ื ืจืฅ ื‘ื›ื™ื•ื•ืŸ ื”ืคื•ืšืœื›ื™ื•ื•ืŸ ืชื ื•ืขืช ื”ืงืจื•ื ื™ืช ื›ืš ืฉืžื”ื™ืจื•ืชื• ื‘ื™ื—ืก ืœืงืจื•ื ื™ืช ื”ื™ืโ€ฌ
13_Energy_momentum/e_13_5_066.html
Momentum
A body is thrown vertically up with initial speed of
with the influence of gravity. After
momentum changes by
.
1. What is the bodies mass?
2. What is the force acting on the body (use only the given)?
3. What is the velocity after
?
its
13_Energy_momentum/e_13_5_067.html
Momentum
A bullet with mass
and speed
is shot towards a ballistic pendulum with mass
will the pendulum rise after the collision? what is the percentage of the lost energy?
. how high
13_Energy_momentum/e_13_5_068.html
Momentum
A disk with mass
slides on a smooth horizontal surface with velocity
and collides with a second disk
with mass
which is at rest. After the collision the second disk moves with velocity
in a direction of
relative to the direction of the first disks initial motion.
1. What is the velocity of the first disk after the collision?
2. Was the collision totally elastic?
13_Energy_momentum/e_13_5_069.html
โ€ซโ€ชMomentumโ€ฌโ€ฌ
โ€ซโ€ชa ball with massโ€ฌโ€ฌ
โ€ซโ€ชis thrown vertically up and blow up to three pieces at the top of its path. The first piece,โ€ฌโ€ฌ
โ€ซโ€ชwith massโ€ฌโ€ฌ
โ€ซโ€ช, moves up with velocityโ€ฌโ€ฌ
โ€ซโ€ชand the second piece, whose mass isโ€ฌโ€ฌ
โ€ซโ€ช, moves right withโ€ฌโ€ฌ
โ€ซโ€ชa velocity ofโ€ฌโ€ฌ
โ€ซ?โ€ช. What is the velocity of the third pieceโ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_5_080.htmlโ€ฌโ€ฌ
โ€ซืขื‘ื•ื“ื” ืฉืœ ื›ื•ื— ืžืฉืชื ื”โ€ฌ
โ€ซื›ื•ื— ืžืฉืชื ื” ืคื•ืขืœ ืขืœ ืžืกื” ืฉืœ โ€ช 5โ€ฌืง"ื’ ื‘ืขืœืช ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช โ€ช 4โ€ฌืžื˜ืจ ืœืฉื ื™ื”โ€ช .โ€ฌื”ื›ื•ื— ืชืœื•ื™ ื‘ื”ืขืชืง โ€ช xโ€ฌื•ืžืชื ื”ื’ ืื—ืจืช ื‘ืฉืœื•ืฉื” ืื™ื–ื•ืจื™ ืคืขื•ืœื”โ€ฌ
โ€ซืฉื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืขื‘ื•ื“ื” ื‘ื›ืœ ืื—ื“ ืžืื™ื–ื•ืจื™ ื”ืคืขื•ืœื” ื”ืฉื•ื ื™ื?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ืฉื™ื ื•ื™ ื‘ืื ืจื’ื™ื” ืœืื—ืจ โ€ช 20โ€ฌืžื˜ืจ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ืœืื—ืจ โ€ช 20โ€ฌืžื˜ืจ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_7_001.htmlโ€ฌโ€ฌ
โ€ซื”ืชื ื’ืฉื•ืช ื‘ื“ื• ืžื™ืžื“โ€ฌ
โ€ซื“ื™ืกืงื” ืฉืžืกืชื” โ€ช 2mโ€ฌืžื—ืœื™ืงื” ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ื—ืœืง ื‘ืžื”ื™ืจื•ืช โ€ช v0โ€ฌืœืื•ืจืš ืฆื™ืจ ืก ื•ืžืชื ื’ืฉืช ื‘ื“ื™ืกืงื” ืฉื ื™ื” ืฉืžืกืชื” โ€ช mโ€ฌื”ื ืžืฆืืช ื‘ืžื ื•ื—ื”โ€ช.โ€ฌโ€ฌ
โ€ซืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช ื ืขื” ื”ื“ื™ืกืงื” ืฉืžืกืชื” โ€ช mโ€ฌื‘ืžื”ื™ืจื•ืช โ€ช 0.5v0โ€ฌื‘ื›ื™ื•ื•ืŸ ื”ื™ื•ืฆืจ ื–ื•ื•ื™ืช ื‘ืช โ€ช 30โ€ฌืžืขืœื•ืช ืขื ื›ื™ื•ื•ืŸ ืชื ื•ืขืชื” ืฉืœ ื”ื“ื™ืกืงื” ื”ืคื•ื’ืขืชโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื• ื•ืงื˜ื•ืจ ื”ืžื”ื™ืจื•ืช ืฉืœ ื”ื“ื™ืกืงื” โ€ช 2mโ€ฌืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืื ื”ื”ืชื ื’ืฉื•ืช ื”ื™ื™ืชื” ืืœืกื˜ื™ืช ืœื—ืœื•ื˜ื™ืŸ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_7_002.htmlโ€ฌโ€ฌ
โ€ซืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซื ืชื•ืŸ ืžื•ื˜ ืฉืžืฉืงืœื• โ€ช 25โ€ฌืง"ื’ ืฉื ืžืฆื ื‘ืฉื™ื•ื•ื™ ืžืฉืงืœ )ืจืื” ืฆื™ื•ืจ(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆื ืืช ื”ืžืชื™ื—ื•ืช ื‘ื—ื•ื˜ ืฉืžื—ื‘ืจ ืืช ื”ืžื•ื˜ ืœืงื™ืจโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžืฆื ืืช ื”ื›ื•ื— ืฉืžืคืขื™ืœ ื”ืงื™ืจ ืขืœ ื”ืžื•ื˜โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_7_003.htmlโ€ฌโ€ฌ
โ€ซืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆื ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื’ื•ืฃ ื‘ืฆื•ืจืช ื—' ื›ืืฉืจ ืื•ืจืš ื›ืœ ืฆืœืข โ€ช lโ€ฌื•ืžืกืช ื›ืœ ืฆืœืข โ€ช. mโ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžืฆื ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ืžืฉื•ืœืฉ ืฉื•ื•ื” ืฆืœืขื•ืช ื›ืืฉืจ ืื•ืจืš ื›ืœ ืฆืœืข โ€ช lโ€ฌื•ืžืกืช ื›ืœ ืฆืœืข โ€ชmโ€ฌโ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_7_004.htmlโ€ฌโ€ฌ
โ€ซืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆื ืืช ืกื”"ื› ื”ืžื•ืžื ื˜ ืกื‘ื™ื‘ ืฆื™ืจ ื”ืกื™ื‘ื•ื‘ )ืžืกื•ืžืŸ ื‘ืืคื•ืจ(โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_8_084.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืจ'โ€ช ,โ€ฌืฉื—ืงืŸ ื‘ื™ื™ืกื‘ื•ืœ ืžืฆื˜ื™ื™ืŸโ€ช ,โ€ฌื”ืœืš ืœืฉื—ืง ืขื ื—ื‘ืจื™ื•โ€ช .โ€ฌื›ื“ื•ืจ ื”ื‘ื™ื™ืกื‘ื•ืœ ืฉืžืกืชื• โ€ช [g] 100โ€ฌื ืข ื‘ืžื”ื™ืจื•ืช ืื•ืคืงื™ืช ืฉืœ โ€ช [m/sec] 40โ€ฌืœืขื‘ืจ ื”ืžื—ื‘ื˜โ€ฌ
โ€ซืฉืœ ืจ'โ€ช .โ€ฌืœืื—ืจ ืฉืกื•ืคื’ ื—ื‘ื˜ื” ืขืฃ ื”ื›ื“ื•ืจ ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช [m/sec] 50โ€ฌื•ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 37°โ€ฌืžืขืœ ืœืื•ืคืง )ืœื›ื™ื•ื•ืŸ ื”ื ื’ื“ื™โ€ช ,โ€ฌืจืื”โ€ช/โ€ฌื™ ืฆื™ื•ืจ(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ืžืชืงืฃ ืฉื”ื•ืคืขืœ ืขืœ ื”ื›ื“ื•ืจ ื‘ื–ืžืŸ ื”ื—ื‘ื˜ื” ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌื‘ื”ื ื—ื” ืฉื”ื—ื‘ื˜ื” ื ืžืฉื›ื” โ€ช ,[msec] 1.5โ€ฌืžื”ื• ื”ื›ื•ื— ื”ืžืžื•ืฆืข ืฉืคืขืœ ืขืœ ื”ื›ื“ื•ืจ ื‘ื–ืžืŸ ื–ื” ?โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื• ื”ืžืชืงืฃ ืฉืคืขืœ ืขืœ ื”ืžื—ื‘ื˜ ?โ€ฌ
โ€ซื“โ€ช.โ€ฌืœืื™ื–ื” ืžืจื—ืง ืขืฃ ื”ื›ื“ื•ืจ ืื ื”ื•ื ืขื•ื–ื‘ ืืช ื”ืžื—ื‘ื˜ ื‘ื’ื•ื‘ื” ืฉืœ โ€ช [m] 1โ€ฌืžืขืœ ื”ืงืจืงืข ?โ€ฌ
โ€ซโ€ช13_Energy_momentum/e_13_8_085.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
โ€ซืžืกื” ืฉืœ โ€ช [kg] 10โ€ฌื”ื ืžืฆืืช ื‘ืžื ื•ื—ื” ืžืชื—ืœืงืช ืœืฉืœื•ืฉื” ื—ืœืงื™ืโ€ช ,โ€ฌื”ื ืขื™ื ืœืื—ืจ ื”ื—ืœื•ืงื” ื‘ืžื™ืฉื•ืจ โ€ช .x-yโ€ฌื—ืœืง ืื—ื“ ืžืกืชื• โ€ช m1 = 2 kgโ€ฌื ืขโ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช u1 = 4 m/secโ€ฌื‘ื›ื™ื•ื•ืŸ ืฆื™ืจ โ€ช xโ€ฌื•ื—ืœืง ืฉื ื™ ืžืกืชื• โ€ช ,m2 = 3 kgโ€ฌื ืข ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช u2 = 3 m/secโ€ฌื•ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30°โ€ฌื‘ื™ื—ืก ืœืฆื™ืจ โ€ชxโ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืชื• ืฉืœ ื”ื—ืœืง ื”ืฉืœื™ืฉื™โ€ช.โ€ฌโ€ฌ
13_Energy_momentum/e_13_8_086.html
โ€ซืชื ืข ื•ืื ืจื’ื™ื”โ€ฌ
.m/sec 3 โ€ซ ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌkg 4 โ€ซ ืœืคื ื™ื” ื ืขื” ื‘ืื•ืชื• ื›ื™ื•ื•ืŸ ืžืกื” ืฉืœโ€ฌ.m/sec 10 โ€ซ ืžื—ืœื™ืงื” ืขืœ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืš ื‘ืžื”ื™ืจื•ืช ืฉืœโ€ฌkg 2 โ€ซืžืกื” ืฉืœโ€ฌ
โ€ซ ืžื” ืชื”ื™ื” ื”ื”ืชื›ื•ื•ืฆื•ืช ื”ืžื›ืกื™ืžืœื™ืช ืฉืœ ื”ืงืคื™ืฅ ? ืžื” ื™ื”ื™ื• ื”ืžื”ื™ืจื•ื™ื•ืช ืฉืœ ืฉืชื™โ€ฌ.N/m 11 โ€ซืœืžืกื” ื–ื• ืžื—ื•ื‘ืจ ืงืคื™ืฅ ื—ืกืจ ืžืกื” ื‘ืขืœ ืงื‘ื•ืข ืฉืœโ€ฌ
? โ€ซื”ืžืกื•ืช ืœืื—ืจ ืฉื”ืงืคื™ืฅ ื™ื—ื–ื•ืจ ืœืžืฆื‘ ืจืคื•ื™ ื•ื”ืžืกื•ืช ื™ื™ืคืจื“ื•โ€ฌ
M =4 kg
m =2 kg
โ€ซืงืคื™ืฅ ื—ืกืจ ืžืกื”โ€ฌ
13_Energy_momentum/e_13_8_087.html
To push a 52-kg crate across a floor, a worker applies a force of 190 N, directed 22° below the horizontal.
As the crate moves 3.3 m, how much work is done on the crate by
(a) the worker,
(b) the force of gravity, and
(c) the normal force of the floor on the crate?
13_Energy_momentum/e_13_8_088.html
A l06-kg object is initially moving in a straight line with a speed of 51.3 m/s.
(a) If it is brought to a stop with a dece1eration of 1.97 m/s2, what force is required, what distance does the
object travel, and how much work is done by the force?
(b) Answer the same questions if the object's deceleration is 4.82 m/s2.
13_Energy_momentum/e_13_8_089.html
To push a 25-kg crate up a 27° incline, a worker exerts a force of 120 N, parallel to the incline.
As the crate slides 3.6 m, how much work is done on the crate by
(a) the worker,
(b) the force of gravity, and
(c) the normal force of the incline?
13_Energy_momentum/e_13_8_090.html
A 52.3-kg trunk is pushed 5.95 m at constant speed up a 28.0° incline by a constant horizontal force.
The coefficient of kinetic friction between the trunk and the incline is 0.19. Calculate the work done by
(a) the applied force and
(b) the force of gravity.
13_Energy_momentum/e_13_8_091.html
A 47.2-kg block of ice slides down an incline 1.62 m long and 0.902 m high. A worker pushes up on the ice
parallel to the incline so that it slides down at constant speed.
The coefficient of kinetic friction between the ice and the incline is 0.110. Find
(a) the force exerted by the worker,
(b) the work done by the worker on the block of ice, and
(c) the work done by gravity on the ice.
13_Energy_momentum/e_13_8_092.html
In a 100-person ski lift, a machine raises passengers averaging 667 N in weight a height of 152 m in 55.0 s, at
constant speed.
Find the power output of the motor, assuming no frictional losses.
13_Energy_momentum/e_13_8_093.html
What power is developed by a grinding machine whose wheel has a radius of 20.7 cm and runs at 2.53 rev/s
when the tool to be
sharpened is held against the wheel with a force of 180 N? The coefficient of friction between the tool and the
wheel is 0.32.
13_Energy_momentum/e_13_8_094.html
A fully loaded freight elevator has a total mass of 1220 kg. It is required to travel downward 54.5 m in 43.0 s.
The counter weight has a mass of 1380 kg. Find the power output, in hp, of the elevator motor.
Ignore the work required to start and stop the elevator; that is, assume that it travels at constant speed.
13_Energy_momentum/e_13_8_095.html
The figure below shows a spring with a pointer attached, hanging next to a scale graduated in millimeters.
Three different weights are hung from the spring, in turn, as shown.
(a) If all weight is removed from the spring, which mark on the scale will the pointer indicate?
(b) Find the weight W.
13_Energy_momentum/e_13_8_096.html
A spring has a force constant of 15.0 N/cm.
(a) How much work is required to extend the spring 7.60 mm from its relaxed position?
(b) How much work is needed to extend the spring an additional 7.60 mm?
13_Energy_momentum/e_13_8_097.html
An object of mass 0.675 kg on a frictionless table is attached to a string that passes through a hole in the table at
the center of the horizontal circle in which the object moves with constant speed.
(a) If the radius of the circle is 0.500 m and the speed is 10.0 m/s, compute the tension in the string.
(b) It is found that drawing an additional 0.200 m of the string down through the hole, thereby reducing the
radius of the circle to 0.300 m, has the effect of multiplying the original tension in the string by 4.63. Compute
the total work done by the string on the revolving object during the reduction of the radius.
13_Energy_momentum/e_13_8_098.html
A force acts on a 2.80-kg particle in such a way that the position of the particle as a function of time is given by
x = (3.0 m/s)t - (4.0 m/s2)t2 + (1.0 m/s3)t3.
(a) Find the work done by the force during the first 4.0 s.
(b) At what instantaneous rate is the force doing work on the particle at the instant t = 3.0 s?
13_Energy_momentum/e_13_8_099.html
A 3700-lb automobile (m = 1600 kg) starts from rest on a level road and gains a speed of 45 mi/h (= 72 km/h)
in 33 s.
(a) What is the kinetic energy of the auto at the end of the 33 s?
(b) What is the average net power delivered to the car during the 33-s interval?
(c) What is the instantaneous power at the end of the 33-s interval assuming that the acceleration was constant?
13_Energy_momentum/e_13_8_100.html
The figure below shows an arrangement of pulleys designed to facilitate the lifting of a heavy load L. Assume
that friction can be ignored everywhere and that the pulleys
to which the load is attached weigh a total of 20.0 lb. An 840-lb load is to be raised 12.0 ft.
(a) What is the minimum applied force F that can lift the load?
(b) How much work must be done against gravity in lifting the 840-lb load 12.0 ft?
(c) Through what distance must the applied force be exerted to lift the load 12.0 ft?
(d) How much work must be done by the applied force F to accomplish this task?
13_Energy_momentum/e_13_8_101.html
A 1380-kg block of granite is dragged up an incline at a con¬stant speed of 1.34 m/s by a steam winch (see
figure below).
The coefficient of kinetic friction between the block and the incline is 0.41. How much power must be supplied
by the winch?
13_Energy_momentum/e_13_8_102.html
An escalator joins one floor with another one 8.20 m above. The escalator is 13.3 m long and moves along its
length at 62.0 cm/s.
(a) What power must its motor deliver if it js required to carry a maximum of 100 persons per minute, of
average mass 75.0 kg?
(b) An 83.5-kg man walks up the escalator in 9.50 s. How much work does the motor do on him?
(c) If this man turned around at the middle and walked down the escalator so as to stay at the same level in
space, would the mo¬tor do work on him? If so, what power does it deliver for this purpose?
(d) Is there any (other?) way the man could walk along the escalator without consuming power from the motor?
13_Energy_momentum/e_13_8_103.html
(a) Estimate the work done by the force shown on the graph below in displacing a particle from x = 1 m to x =3
m.
Refine your method to see how close you can come to the exact answer of 6 J.
(b) The curve is given analytically by F = A/x2, where A = 9 N. m2. Show how to calculate the work by the
rules of integration.
13_Energy_momentum/e_13_8_104.html
A 0.550-kg projectile is launched from the edge of a cliff with an initial kinetic energy of 1550 J and at its
highest point. is 140 m above the launch point.
(a) What is the horizontal component of its velocity?
(b) What was the vertical component of its velocity just after launch?
(c) At one instant during its flight the vertical component of its velocity is found to be 65.0 m/s. At that time,
how far is it above or below the launch point?
13_Energy_momentum/e_13_8_105.html
A 263-g block is dropped onto a vertical spring with force constant k = 2.52 N/cm (see the figure below).
The block sticks to the spring, and the spring compresses 11.8 cm before coming momentarily to rest.
While the spring is being compressed, how much work is done
(a) by the force of gravity and
(b) by the spring?
(c) What was the speed of the block just before it hit the spring?
(d) If this initial speed of the block is doubled, what is the maximum compression of the spring? Ignore friction.
13_Energy_momentum/e_13_8_106.html
An object of mass m accelerates uniformly from rest to a speed vf in time tf.
(a) Show that the work done W on the object as a function of time t is W = m(vf/ tf)2 t2/2.
(b) As a function of time t, what is the instantaneous power delivered to the object?
13_Energy_momentum/e_13_8_107.html
A 220-lb man jumps out a window into a fire net 36 ft below. The net stretches 4.4 ft before bringing him to
rest and tossing him back into the air. What is the potential energy of the stretched net?
13_Energy_momentum/e_13_8_108.html
A frictionless roller-coaster car starts at point A in the figure below with speed vo. What will be the speed of
the car
(a) at point B,
(b) at point C, and
(c) at point D? Assume that the car can be considered a particle and that it always remains on the track.
13_Energy_momentum/e_13_8_109.html
The figure below shows a 7.94-kg stone resting on a spring. The spring is compressed 10.2 cm by the stone.
(a) Calculate the force constant of the spring.
(b) The stone is pushed down an additional 28.6 cm and released. How much potential energy is stored in the
spring just before the stone is released?
(c) How high above this new (lowest) position will the stone rise?
13_Energy_momentum/e_13_8_110.html
A 1.93-kg block is placed against a compressed spring on a frictionless 27.0° incline (see the figure below).
The spring, whose force constant is 20.8 N/cm, is compressed 18.7 cm, after which the block is released. How
far up the incline will the block go before coming to rest?
Measure the final position of the block with respect to its position just before being released.
13_Energy_momentum/e_13_8_111.html
A 2.14-kg block is dropped from a height of 43.6 cm onto a spring of force constant k = 18.6 N/cm, as shown in
the figure below.
Find the maximum distance the spring will be compressed.
13_Energy_momentum/e_13_8_112.html
Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a
spring-loaded gun that is mounted on a table.
The target box is 2.20 m horizontally from the edge of the table; see the figure below. Bobby compresses the
spring 1.10 cm, but the marble falls 27.0 cm short.
How far should Rhoda compress the spring to score a hit?
13_Energy_momentum/e_13_8_113.html
A particle moves along the x axis through a region in which its potential energy U(x) varies as shown in the
figure below.
(a) Make a quantitative plot of the force F(x) that acts on the particle, us¬ing the same x axis scale as the figure.
(b) The particle has a (constant) mechanical energy E of 4.0 J. Sketch a plot of its kinetic energy K(x) directly
on this figure.
13_Energy_momentum/e_13_8_114.html
An ideal massless spring can be compressed 2.33 cm by a force of 268 N. A block whose mass is m = 3.18 kg
is released from rest at the top of the incline as shown in the figure below,
the angle of the incline being 32.0°. The block comes to rest momentarily after it has compressed this spring by
5.48 cm.
(a) How far has the block moved down the incline at this moment?
(b) What is the speed of the block just as it touches the spring?
13_Energy_momentum/e_13_8_115.html
A small block of mass m slides along the frictionless loop-the-loop track shown in the figure below.
(a) The block is released from rest at point P. What is the net force acting on it at point Q?
(b) At what height above the bottom of the loop should the block be released so that it is on the verge of losing
contact with the track at the top of the loop?
13_Energy_momentum/e_13_8_116.html
A block of mass m at the end of a string swings in a vertical circle of radius R under the influence of gravity
only.
Find the difference between the magnitudes of the tension in the string at the top of the loop and at the bottom
of the loop assuming the
block is always moving fast enough so that the string never goes slack.
13_Energy_momentum/e_13_8_117.html
A boy is seated on the top of a hemispherical mound of ice (see figure below). He is given a very small push
and starts sliding down the ice.
Show that he leaves the ice at a point whose height is 2R/3 if the ice is frictionless. (Hint: The normal force
vanishes as he leaves the ice.)
13_Energy_momentum/e_13_8_118.html
A particle is projected horizontally along the interior of a frictionless hemispherical bowl of radius r, which is
kept at rest (see figure below).
We wish to find the initial speed vo required for the particle to just reach the top of the bowl. Find vo as a
function of ฮธo, the initial angular position of the particle.
13_Energy_momentum/e_13_8_119.html
3. A 4.88-kg object with a speed of 31.4 m/s strikes a steel plate at an angle of 42.0º and rebounds at the same
speed and angle.
What is the change (magnitude and direction) of the linear momentum of the object?
13_Energy_momentum/e_13_8_120.html
5. A ball of mass m and speed v strikes a wall perpendicularly and rebounds with undiminished speed.
(a) If the time of collision is t, what is the average force exerted by the ball on the wall?
(b) Evaluate this average force numerically for a rubber ball with mass 140 g moving at 7.8 rn/s; the duration of
the collision is 3.9 ms.
13_Energy_momentum/e_13_8_121.html
6. A golfer hits a golf ball, imparting to it an initial velocity of magnitude 52.2 m/s directed 30º above the
horizontal.
Assuming that the mass of the ball is 46.0 g and the club and ball are in contact for 1.20 ms, find
(a) the impulse imparted to the ball,
(b) the impulse imparted to the club, and
(c) the average force exerted on the ball by the club.
13_Energy_momentum/e_13_8_122.html
9. The figure below shows an approximate representation of force versus time during the collision of a 58-g
tennis ball with a wall.
The initial velocity of the ball is 32 m/s perpendicular to the wall; it rebounds with the same speed, also
perpendicular to the wall.
What is the value of Fmax, the maximum contact force during the collision?
13_Energy_momentum/e_13_8_123.html
11. A croquet ball with a mass 0.50 kg is struck by a mallet, receiving the impulse shown in the graph.
What is the ball's velocity just after the force has become zero?
13_Energy_momentum/e_13_8_124.html
14. A pellet gun fires ten 2.14-g pellets per second with a speed of 483 m/s. The pellets are stopped by a rigid
wall.
(a) Find the momentum of each pellet.
(b) Calculate the average force exerted by the stream of pellets on the wall.
(c) If each pellet is in contact with the wall for 1.25 ms, what is the average force exerted on the wall by each
pellet while in contact? Why is this so different from (b)?
13_Energy_momentum/e_13_8_125.html
18. A railroad flatcar of weight W can roll without friction along a straight horizontal track. Initially, a man of
weight w is standing on the car,
which is moving to the right with speed Vo. What is the change in velocity of the car if the man runs to the left
(Figure) so that his speed
relative to the car is vrel just before he jumps off at the left end?
13_Energy_momentum/e_13_8_126.html
20. The blocks in the figure below slide without friction. What is the velocity v of the l.6-kg block after the
collision?
13_Energy_momentum/e_13_8_127.html
22. Meteor Crater in Arizona (see Fig. below) is thought to have been formed by the impact of a meteorite with
the Earth some 20,000 years ago.
The mass of the meteorite is estimated to be 5 X 1010 kg and its speed to have been 7.2 km/s. What speed
would such a meteorite impart to the Earth in a head-on collision?
13_Energy_momentum/e_13_8_128.html
23. A 5.18-g bullet moving at 672 m/s strikes a 715-g wood~n block at rest on a frictionless surface.
The bullet emerges with its speed reduced to 428 m/s. Find the resulting speed of the block.
13_Energy_momentum/e_13_8_129.html
25. Two objects, A and B, collide. A has mass 2.0 kg, and B has mass 3.0 kg. The velocities before the collision
are viA = (15 m/s)i + (30 m/s)j
and viB = (-10 m/s)i + (5.0 m/s)j. After the collision, VfA = (- 6.0 m/s)i + (30 m/s)j. What is the final velocity
of B?
13_Energy_momentum/e_13_8_130.html
27. A barge with mass 1.50 X 105 kg is proceeding downriver at 6.20 m/s in heavy fog when it collides
broadside with a barge heading directly across the river; see figure below.
The second barge has mass 2.78 X 105 kg and was moving at 4.30 m/s. Immediately after impact, the second
barge finds its course deflected by 18.0° in the downriver direction and its speed increased to 5.10 m/s. The
river current was practically zero at the time of the accident.
What is the speed and direction of motion of the first barge immediately after the collision?
13_Energy_momentum/e_13_8_131.html
29. Two titanium spheres approach each other head-on with the same speed and collide elastically.
After the collision, one of the spheres, whose mass is 300 g, remains at rest. What is the mass of the other
sphere?
13_Energy_momentum/e_13_8_132.html
30. A cart with mass 342 g moving on a frictionless linear air¬track at an initial speed of 1.24 m/s strikes a
second cart of unknown mass at rest.
The collision between the carts is elastic. After the collision, the first cart continues in its original direction at
0.636 m/s.
(a) What is the mass of the second cart?
(b) What is its speed after impact?
13_Energy_momentum/e_13_8_133.html
31. An object of 2.0-kg mass makes an elastic collision with another object at rest and continues to move in the
original direction but with one-fourth of its original speed.
What is the mass of the struck object?
13_Energy_momentum/e_13_8_134.html
32. A railroad freight car weighing 31.8 tons and traveling at 5.20 ft/s overtakes one weighing 24.2 tons and
traveling at 2.90 ft/s in the same direction.
(a) Find the speeds of the cars after collision if the cars couple together.
(b) If instead, as is very unlikely, the collision is elastic, find the speeds of the cars after collision.
13_Energy_momentum/e_13_8_135.html
3. A 325-g ball with a speed v of 6.22 m/s strikes a wall at an angle ฮธ of 33.0° and then rebounds with the same
speed and angle (Figure).
It is in contact with the wall for 10.4 ms.
(a) What impulse was experienced by the ball?
(b) What was the average force exerted by the ball on the wall?
13_Energy_momentum/e_13_8_136.html
4. It is well known that bullets and other missiles fired at Super¬man simply bounce off his chest as in figure.
Suppose that a gangster sprays Superman's chest with 3.0-g bullets at the rate of 100 bullets/min, the speed of
each bullet being 500 m/s.
Suppose too that the bullets rebound straight back with no loss in speed.
Find the average force exerted by the stream of bullets on Superman's chest.
13_Energy_momentum/e_13_8_137.html
5. During a violent thunderstorm, hail the size of marbles (diam¬eter = 1.0 cm) falls at a speed of 25 m/s. There
are estimated to be 120 hailstones per cubic meter of air.
Ignore the bounce of the hail on impact.
(a) What is the mass of each hailstone?
(b) What force is exerted by hail on a 10 m X 20 m flat roof during the storm? Assume that, as for ice, 1.0 cm3
of hail has a mass of 0.92 g.
13_Energy_momentum/e_13_8_138.html
2. Show that the ratio of the distances xl and x2 of two particles from their center of mass is the inverse ratio of
their masses; that is, xl/x2 = m2/m1.
13_Energy_momentum/e_13_8_139.html
3. A Plymouth car with a mass of 2210 kg is moving along a straight stretch of road at 105 km/h. It is followed
by a Ford with mass
2080 kg moving at 43.5 km/h.
How fast is the center of mass of the two cars moving?
13_Energy_momentum/e_13_8_140.html
4. Two skaters, one with mass 65 kg and the other with mass 42 kg, stand on an ice rink holding a pole with a
length of 9.7 m and a mass that is negligible.
Starting from the ends of the pole, the skaters pull themselves along the pole until they meet.
How far will the 42-kg skater move?
13_Energy_momentum/e_13_8_141.html
5. Two particles P and Q are initially at rest 1.64 m apart. P has a mass of 1.43 kg and Q a mass of 4.29 kg. P
and Q attract each other with a constant force
of 1.79 X 10-2 N. No external forces act on the system.
(a) Describe the motion of the center of mass.
(b) At what distance from P's original position do the particles collide?
13_Energy_momentum/e_13_8_142.html
6. A shell is fired from a gun with a muzzle velocity of 466 m/s, at an angle of 57.4º with the horizontal.
At the top of the trajectory, the shell explodes into two fragments of equal mass. One fragment, whose speed
immediately after the explosion is zero,
falls vertically. How far from the gun does the other fragment land, assuming level terrain?
13_Energy_momentum/e_13_8_143.html
10. Where is the center of mass of the three particles shown in the figure below?
13_Energy_momentum/e_13_8_144.html
13. Three thin rods each of length L are arranged in an inverted U, as shown in the figure below. The two rods
on the arms of the U each have mass M; the third rod has mass 3M. Where is the center of mass of the
assembly?
13_Energy_momentum/e_13_8_145.html
14. The figure below shows a composite slab with dimensions 22.0 cm X 13.0 cm X 2.80 cm. Half of the slab is
made of aluminum
(density = 2.70 g/cm3) and half of iron (density = 7.85 g/cm3), as shown. Where is the center of mass of the
slab?
13_Energy_momentum/e_13_8_146.html
16. A vessel at rest explodes, breaking into three pieces. Two pieces, one with twice the mass of the other, fly
off perpendicular to one another with the same speed of 31.4 m/ s.
The third piece has three times the mass of the lightest piece.
Find the magnitude and direction of its ve1ocity immediately after the explosion.
(Specify the direction by giving the angle from the line of travel of the least massive piece.)
13_Energy_momentum/e_13_8_147.html
18. A railway flat car is rushing along a level frictionless track at a speed of 45 m/s.
Mounted on the car and aimed forward is a cannon that fires 65-kg cannon balls with a muzzle speed of 625
m/s.
The total mass of the car, the cannon, and the large supply of cannon balls on the car is 3500 kg.
How many cannon balls must be fired to bring the car as close to rest as possible?
13_Energy_momentum/e_13_8_148.html
20. A rocket at rest in space, where there is virtual1y no gravity, has a mass of 2.55 X 105 kg, of which 1.81 X
105 kg is fuel.
The engine consumes fuel at the rate of 480 kg/s, and the exhaust speed is 3.27 km/s. The engine is fired for
250 s.
(a) Find the thrust of the rocket engine.
(b) What is the mass of the rocket after the engine burn?
(c) What is the final speed attained?
13_Energy_momentum/e_13_8_149.html
21. Consider a rocket at rest in empty space. What must be its mass ratio (ratio of initial to final mass) in order
that, after firing its engine, the rocket's speed is
(a) equal to the exhaust speed and
(b) equal to twice the exhaust speed?
13_Energy_momentum/e_13_8_150.html
23. A rocket of total mass 1.11 X 105 kg, of which 8.70 X 104 kg is fuel, is to be launched vertically.
The fuel will be burned at the constant rate of 820 kg/s.
Relative to the rocket, what is the minimum exhaust speed that allows liftoff at launch?
13_Energy_momentum/e_13_8_151.html
3. A uniform flexible chain of length L, with weight per unit length , passes over a small, frictionless peg; see
figure below.
It is released from a rest position with a length of chain x hanging from one side and a length L -x from the
other side.
Find the acceleration a as a function of x.
13_Energy_momentum/e_13_8_152.html
7. A 1400-kg cannon, which fires a 70.0-kg shell with a muzzle speed of 556 m/s, is set at an elevation angle of
39.0° above the horizontal.
The cannon is mounted on frictionless rails, so that it recoils freely.
(a) What is the speed of the shell with respect to the Earth?
(b) At what angle with the ground is the shell projected?
(Hint: The horizontal component of the momentum of the system remains unchanged as the gun is fired.)
13_Energy_momentum/e_13_8_153.html
10. A 5860-kg rocket is set for vertical firing. The exhaust speed is 1.17 km/s.
How much gas must be ejected each second to supply the thrust needed
(a) to overcome the weight of the rocket and
(b) to give the rocket an initial upward acceleration of 18.3 m/s2?
Note that gravity is present here as an external force.
13_Energy_momentum/e_13_8_154.html
9. A 2.9-ton weight falling through a distance of 6.5 ft drives a 0.50-ton pile 1.5 inches into the ground.
(a) Assuming that the weight - pile collision is completely inelastic, find the average force of resistance exerted
by the ground.
(b) Assuming the force of resistance by the ground remains constant at the value found in (a), how far into the
ground would the pile be driven if the collision were elastic?
(c) Which is more effective in this case, elastic or inelastic collisions?
13_Energy_momentum/e_13_8_155.html
10. Two 22.7-kg ice sleds are placed a short distance apart, one directly behind the other, as shown in Fig. 6-30.
A 3.63-kg cat, standing on one sled, jumps across to the other and im¬mediately back to the first.
Both jumps are made at a speed of 3.05 m/s relative to the sled the cat is standing on when the jump is made.
Find the final speeds of the two sleds.
13_Energy_momentum/e_13_8_156.html
11. Two vehicles A and B are traveling west and south, respectively, toward the same intersection where they
collide and lock together.
Before the collision, A (weight 2720 lb) is moving with a speed of 38.5 mi/h and B (weight 3640 lb) has a
speed of 58.0 mi/h.
Find the magnitude and direction of the velocity of the (interlocked) vehicles immediately after the collision.
13_Energy_momentum/e_13_8_157.html
12. Two balls A and B, having different but unknown masses, collide. A is initially at rest and B has a speed v.
After collision, B has a speed v/2 and moves at right angles to its original motion.
(a) Find the direction in which ball A moves after the collision.
(b) Can you determine the speed of A from the information given? Explain.
13_Energy_momentum/e_13_8_158.html
13. In a game of pool, the cue ball strikes another ball initially at rest. After the collision, the cue ball moves at
3.50 m/s along a line making an angle of 65.0°
with its original direction of motion. The second ball acquires a speed of 6.75 m/s.
Using momentum conservation, find
(a) the angle between the direction of motion of the second ball and the original direction of motion of the cue
ball and
(b) the original speed of the cue ball.
13_Energy_momentum/e_13_8_159.html
19. A 3.54-g bullet is fired horizontally at two blocks resting on a frictionless tabletop, as shown in Fig. 6-34a.
The bullet passes through the first block, with mass 1.22 kg, and embeds itself in the second, with mass 1.78
kg. Speeds of 0.630 m/s and 1.48 m/s,
respectively, are thereby imparted to the blocks, as shown in Fig. 6-34b. Neglecting the mass removed from the
first block by the bullet, find
(a) the speed of the bullet immediately after emerging from the first block and
(b) the original speed of the bullet.
13_Energy_momentum/e_13_8_160.html
20. A 2.0-kg block is released from rest at the top of a 22° frictionless inclined plane of height 0.65 m (Fig. 635).
At the bottom of the plane it collides with and sticks to a block of mass 3.5 kg. The two blocks together slide a
distance of 0.57 m
across a horizontal plane before coming to rest.
What is the coefficient of friction of the horizontal surface?
13_Energy_momentum/e_13_8_161.html
21. Two cars A and B slide on an icy road as they attempt to stop at a traffic light. The mass of A is 1100 kg
and the mass of B is 1400 kg.
The coefficient of kinetic friction between the locked wheels of both cars and the road is 0.130. Car A succeeds
in coming to rest at the light,
but car B cannot stop and rear-ends (collides with) car A. After the collision, A comes to rest 8.20 m ahead of
the impact point and B 6.10 m ahead: see Fig. 6-36.
Both drivers had their brakes locked throughout the incident.
(a) From the distances each car moved after the collision, find the speed of each car immediately after impact.
(b) Use conservation of momentum to find the speed at which car B struck car A.
On what grounds can the use of momentum conservation be criticized here?
13_Energy_momentum/e_13_8_162.html
2. Show that 1 rev/min = 0.105 rad/s.
13_Energy_momentum/e_13_8_163.html
3. The angle turned through by the flywheel of a generator during a time interval t is given by ฯ† = at + bt3 - ct4,
where a, b, and c are constants.
What is the expression for its
(a) angular velocity and
(b) angular acceleration?
13_Energy_momentum/e_13_8_164.html
4. Our Sun is 2.3 X 104 ly (light-years) from the center of our Milky Way galaxy and is moving in a circle
around this center at a speed of 250 km/s.
(One light year is the distance that light travels in one year.)
(a) How long does it take the Sun to make one revolution about the galactic center?
(b) How many revolutions has the Sun completed since it was formed about 4.5 X 109 years ago?
13_Energy_momentum/e_13_8_165.html
5. A wheel rotates with an angular acceleration z given by z = 4at3 - 3bt2, where t is the time and a and b are
constants.
If the wheel has an initial angular velocity Wo, write the equations for
(a) the angular velocity and
(b) the angle turned through as functions of time.
13_Energy_momentum/e_13_8_166.html
6. What is the angular speed of
(a) the second hand,
(b) the minute hand, and
(c) the hour hand of a watch?
13_Energy_momentum/e_13_8_167.html
7. A good baseball pitcher can throw a baseball toward home plate at 85 mi/h with a spin of 1800 rev/min.
How many revolutions does the baseball make on its way to home plate?
For simplicity, assume that the 60-ft trajectory is a straight line.
13_Energy_momentum/e_13_8_168.html
While waiting to board a helicopter, you notice that the rotor's motion changed from 315 rev/min to 225
rev/min in 1.00 min.
(a) Find the average angular acceleration during the interval.
(b) Assuming that this acceleration remains constant, calculate how long it will take for the rotor to stop.
(c) How many revolutions will the rotor make after your second observation?
13_Energy_momentum/e_13_8_169.html
17. A certain wheel turns through 90 rev in 15 s, its angular speed at the end of the period being 10 rev/s.
(a) What was the angular speed of the wheel at the beginning of the 15-s interval, assuming constant angular
acceleration?
(b) How much time had elapsed between the time the wheel was at rest and the beginning of the 15-s interval?
13_Energy_momentum/e_13_8_170.html
24. A threaded rod with 12.0 turns/cm and diameter 1.18 cm is mounted horizontally.
A bar with a threaded hole to match the rod is screwed onto the rod; see Fig. 8-17.
The bar spins at 237 rev/min. How long will it take for the bar to move 1.50 cm along the rod?
13_Energy_momentum/e_13_8_171.html
25. (a) What is the angular speed about the polar axis of a point on the Earth's surface at a latitude of 40° N?
(b) What is the linear speed?
(c) What are the values for a point at the equator?
13_Energy_momentum/e_13_8_172.html
26. A gyroscope flywheel of radius 2.83 cm is accelerated from rest at 14.2 rad/s2 until its angular speed is
2760 rev/min.
(a) What is the tangential acceleration of a point on the rim of the flywheel?
(b) What is the radial acceleration of this point when the flywheel is spinning at full speed?
(c) Through what distance does a point on the rim move during the acceleration?
13_Energy_momentum/e_13_8_173.html
17. Fig. 9-43 shows a uniform block of mass, M and edge lengths a, b, and c.
Calculate its rotational inertia (moment of inertia) about an axis through one corner and perpendicular to the
large face of the block.
13_Energy_momentum/e_13_8_174.html
18. Calculate the rotational inertia of a meter stick, with mass 0.56 kg, about an axis perpendicular to the stick
and located at the 20-cm mark.
13_Energy_momentum/e_13_8_175.html
19. Two particles, each with mass m, are fastened to each other and to a rotation axis by two rods, each with
length L and mass M, as shown in Fig. 9-44.
The combination rotates around the rotation axis with angular velocity . Obtain an algebraic expression for
the rotational inertia of the combination about this axis.
13_Energy_momentum/e_13_8_176.html
12. Nine square holes have been cut in a fiat square plate, as shown in Fig. 9-62. The plate has edge length L,
and the holes have edge length a.
The holes are located at the centers of the small squares formed by dividing each side of the square into three
equal sections.
Find the rotational inertia for rotations about an axis perpendicular to the plate passing through its center.
13_Energy_momentum/e_13_8_177.html
The figure below shows a spring with a pointer attached, hanging next to a scale graduated in millimeters.
Three different weights are hung from the spring, in turn, as shown.
(a) If all weight is removed from the spring, which mark on the scale will the pointer indicate?
(b) Find the weight W.
13_Energy_momentum/e_13_8_178.html
The figure below shows an arrangement of pulleys designed to facilitate the lifting of a heavy load L. Assume
that friction can be ignored everywhere and that the pulleys
to which the load is attached weigh a total of 20.0 lb. An 840-lb load is to be raised 12.0 ft.
(a) What is the minimum applied force F that can lift the load?
(b) How much work must be done against gravity in lifting the 840-lb load 12.0 ft?
(c) Through what distance must the applied force be exerted to lift the load 12.0 ft?
(d) How much work must be done by the applied force F to accomplish this task?
13_Energy_momentum/e_13_8_179.html
A 1380-kg block of granite is dragged up an incline at a con¬stant speed of 1.34 m/s by a steam winch (see
figure below).
The coefficient of kinetic friction between the block and the incline is 0.41. How much power must be supplied
by the winch?
13_Energy_momentum/e_13_8_180.html
(a) Estimate the work done by the force shown on the graph below in displacing a particle from x = 1 m to x =3
m.
Refine your method to see how close you can come to the exact answer of 6 J.
(b) The curve is given analytically by F = A/x2, where A = 9 N. m2. Show how to calculate the work by the
rules of integration.
13_Energy_momentum/e_13_8_181.html
A 263-g block is dropped onto a vertical spring with force constant k = 2.52 N/cm (see the figure below).
The block sticks to the spring, and the spring compresses 11.8 cm before coming momentarily to rest.
While the spring is being compressed, how much work is done
(a) by the force of gravity and
(b) by the spring?
(c) What was the speed of the block just before it hit the spring?
(d) If this initial speed of the block is doubled, what is the maximum compression of the spring? Ignore friction.
13_Energy_momentum/e_13_8_182.html
A frictionless roller-coaster car starts at point A in the figure below with speed vo. What will be the speed of
the car
(a) at point B,
(b) at point C, and
(c) at point D? Assume that the car can be considered a particle and that it always remains on the track.
13_Energy_momentum/e_13_8_183.html
The figure below shows a 7.94-kg stone resting on a spring. The spring is compressed 10.2 cm by the stone.
(a) Calculate the force constant of the spring.
(b) The stone is pushed down an additional 28.6 cm and released. How much potential energy is stored in the
spring just before the stone is released?
(c) How high above this new (lowest) position will the stone rise?
13_Energy_momentum/e_13_8_184.html
A 1.93-kg block is placed against a compressed spring on a frictionless 27.0° incline (see the figure below).
The spring, whose force constant is 20.8 N/cm, is compressed 18.7 cm, after which the block is released. How
far up the incline will the block go before coming to rest?
Measure the final position of the block with respect to its position just before being released.
13_Energy_momentum/e_13_8_185.html
A 2.14-kg block is dropped from a height of 43.6 cm onto a spring of force constant k = 18.6 N/cm, as shown in
the figure below.
Find the maximum distance the spring will be compressed.
13_Energy_momentum/e_13_8_186.html
Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a
spring-loaded gun that is mounted on a table.
The target box is 2.20 m horizontally from the edge of the table; see the figure below. Bobby compresses the
spring 1.10 cm, but the marble falls 27.0 cm short.
How far should Rhoda compress the spring to score a hit?
13_Energy_momentum/e_13_8_187.html
A particle moves along the x axis through a region in which its potential energy U(x) varies as shown in the
figure below.
(a) Make a quantitative plot of the force F(x) that acts on the particle, us¬ing the same x axis scale as the figure.
(b) The particle has a (constant) mechanical energy E of 4.0 J. Sketch a plot of its kinetic energy K(x) directly
on this figure.
13_Energy_momentum/e_13_8_188.html
An ideal massless spring can be compressed 2.33 cm by a force of 268 N. A block whose mass is m = 3.18 kg
is released from rest at the top of the incline as shown in the figure below,
the angle of the incline being 32.0°. The block comes to rest momentarily after it has compressed this spring by
5.48 cm.
(a) How far has the block moved down the incline at this moment?
(b) What is the speed of the block just as it touches the spring?
13_Energy_momentum/e_13_8_189.html
A small block of mass m slides along the frictionless loop-the-loop track shown in the figure below.
(a) The block is released from rest at point P. What is the net force acting on it at point Q?
(b) At what height above the bottom of the loop should the block be released so that it is on the verge of losing
contact with the track at the top of the loop?
13_Energy_momentum/e_13_8_190.html
A boy is seated on the top of a hemispherical mound of ice (see figure below). He is given a very small push
and starts sliding down the ice.
Show that he leaves the ice at a point whose height is 2R/3 if the ice is frictionless. (Hint: The normal force
vanishes as he leaves the ice.)
13_Energy_momentum/e_13_8_191.html
A particle is projected horizontally along the interior of a frictionless hemispherical bowl of radius r, which is
kept at rest (see figure below).
We wish to find the initial speed vo required for the particle to just reach the top of the bowl. Find vo as a
function of ฮธo, the initial angular position of the particle.
19_Vibrations/e_19_1_011.html
Oscillations
Find the frequency of small oscillations of a particle (mass
.
) near the equilibrium in the potential
19_Vibrations/e_19_1_012.html
Oscillations
A particle is in the stable equilibrium in the potential energy
. Suddenly it gets a
small addition of energy
. Assuming that the oscillations are small find the frequency and amplitude.
19_Vibrations/e_19_1_013.html
Oscillations
A particle moves in a well of the shape
without friction (potential energy
motion can be described as a harmonic oscillation and find the frequency.
). Show that the
19_Vibrations/e_19_1_014.html
Oscillations
A body with the mass
is attached to a spring (spring constant ). The other end of the spring is brought into
the motion according to the law
. The friction acting on the body is
. Show that
the body can oscillate with a constant amplitude and find this amplitude.
19_Vibrations/e_19_1_015.html
Oscillations
Find the motion of an oscillator with the natural frequency
,
.
and mass
under the force
19_Vibrations/e_19_1_016.html
Oscillations
Find the frequency of small radial oscillations of a particle with a mass
central potential
.
19_Vibrations/e_19_1_017.html
near a circular orbit
in a
Oscillations
Find the frequency of small radial oscillations of a particle with a mass
central potential
.
near a circular orbit
in a
19_Vibrations/e_19_1_018.html
Oscillations
Find the average power of the external force
for the oscillator:
.
19_Vibrations/e_19_1_019.html
Oscillations
A bead of the mass
can move on a straight wire along
axis without friction. The bead is connected to two
springs (spring constant , length ). The springs are connected to the points
and
,
,
respectively. Initially the bead starts moving from
with the velocity . Assuming that the the oscillations
are small, find their frequency and amplitude.
19_Vibrations/e_19_1_020.html
Oscillations
A particle is moving in the magnetic field
. Find
.
,
, and electric field
,
19_Vibrations/e_19_1_021.html
Oscillations
A particle with the mass
is moving in
plane with the potential energy
the particle is in the position
and its velocity is
. Find the trajectory.
. Initially
19_Vibrations/e_19_1_022.html
Oscillations
A satellite (mass ) is moving along a strait line
between two stars of mass
.
The distance from each star to the line is (at the nearest point).
1. What is the condition for having harmonic oscillations?
2. Find the frequency of small oscillations.
19_Vibrations/e_19_1_023.html
Oscillations
A particle is moving in the magnetic field
. Find
,
, and electric field
,
.
19_Vibrations/e_19_1_024.html
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โ€ซื—ื•ืคืฉื™ ืœื ื•ืข ืื•ืคืงื™ืช ืœืœื ื—ื™ื›ื•ืš ืขืœ ืขื’ืœื” ืฉืžืกืชื”โ€ฌ
โ€ซื’ื•ืฃ ื ืงื•ื“ืชื™ ืฉืžืกืชื•โ€ฌ
โ€ซืงื‘ื•ืข โ€ช .โ€ฌื”ืขื’ืœื” ืžื•ื ื—ืช ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌืžื•ืชื—ื™ื ืืช ื”ืงืคื™ืฅ ื•ืžืฉื—ืจืจื™ื ืืช ื”ืžืขืจื›ืช ืžืžื ื•ื—ื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ื“ื•ืช ืฉื™ื‘ืฆืขื• ื”ื’ื•ืฃ ื”ื ืงื•ื“ืชื™ ื•ื”ืขื’ืœื”?โ€ฌ
โ€ซ?โ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืชื“ื™ืจื•ืช ื‘ื’ื‘ื•ืœื•ืชโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_3_103.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื”ืžื—ื•ื‘ืจ ื‘ืงืฆื”ื• )ืจืื”โ€ช/โ€ฌื™ ืฉืจื˜ื•ื˜(โ€ช .โ€ฌื‘ื–ืžืŸโ€ฌ
โ€ซื•ื’ื•ืฃ ื ืงื•ื“ืชื™ ืฉืžืกืชื•โ€ฌ
โ€ซืžื˜ื•ื˜ืœืช ืขืฉื•ื™ื” ืžื•ื˜ ืื—ื™ื“ ื‘ืขืœ ืื•ืจืš ื•ืžืกื”โ€ฌ
โ€ซโ€ช .โ€ฌื ื™ืชืŸ ืœื”ื ื™ื— ื›ื™ ื”ื–ื•ื™ืช ืงื˜ื ื”โ€ช.โ€ฌโ€ฌ
โ€ซื”ืžื˜ื•ื˜ืœืช ื ืžืฆืืช ื‘ื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœ ื•ืžืขื ื™ืงื™ื ืœื” ืžื”ื™ืจื•ืช ื–ื•ื™ืชื™ืชโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ื“ื•ืช ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ื–ื•ื™ืชโ€ช ,โ€ฌื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ื•ื”ืชืื•ืฆื” ื”ื–ื•ื™ืชื™ืช ื›ืชืœื•ืช ื‘ื–ืžืŸ )โ€ฌ
โ€ซ(?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_3_104.htmlโ€ฌโ€ฌ
โ€ซืชื“ื™ืจื•ืช ืชื ื•ื“ื•ืช ืงื˜ื ื•ืช ืฉืœ ืžื™ืžืŸ ื‘ืจื•ืžื™โ€ฌ
โ€ซื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช ืฉืœ ืžื•ืœืงื•ืœืช ืžื™ืžืŸ ื‘ืจื•ืžื™ )โ€ช (HBrโ€ฌื ื™ืชื ืช ืข"ื™ ื”ื ื•ืกื—ืโ€ช:โ€ฌโ€ฌ
โ€ซื›ืืฉืจ โ€ช,โ€ฌโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซืงื‘ื•ืขื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžื”ื• ื”ื›ื— ื”ืคื•ืขืœ ื‘ื™ืŸ ื”ืื˜ื•ืžื™ื?โ€ฌ
โ€ซโ€ช .2โ€ฌืžื”ื• ืžืจื—ืง ื”ืฉื™ื•ื•ื™ ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช .3โ€ฌืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ื“ื•ืช ื”ืงื˜ื ื•ืช ืฉืœ ื”ืžื•ืœืงื•ืœื” ืกื‘ื™ื‘ ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_001.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืชโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื’ื•ืคื™ื ืื—ื“ ืžืขืœ ืœืฉื ื™ ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช .โ€ฌืžืกืช ื”ื’ื•ืฃ ื”ืขืœื™ื•ืŸ ื”ื™ื โ€ช 1.22โ€ฌืง"ื’โ€ช ,โ€ฌื•ืžืกืช ื”ื’ื•ืฃ ื”ืฉื ื™ ื”ื™ืโ€ฌ
โ€ซื”ืืžืคืœื™ื˜ื•ื“ื” ื”ืžื›ืกื™ืžืœื™ืช ื‘ื” ื™ื ื•ืข โ€ช M2โ€ฌื›ืš ืฉื”ื’ื•ืฃ ื”ืขืœื™ื•ืŸ ืœื ื™ื—ืœื™ืงโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_002.htmlโ€ฌโ€ฌ
โ€ซื‘ื•ืœื ื–ืขื–ื•ืขื™ืโ€ฌ
โ€ซื‘ื•ืœื ื–ืขื–ื•ืขื™ื ื‘ืจื›ื‘ ืขืฉื•ื™ ืžืงืคื™ืฅ ื”ื ืžืฆื ื‘ืชื•ืš ื‘ื•ื›ื ื” ื”ืžืœืื” ื‘ืฉืžืŸ ื”ื™ื“ืจืื•ืœื™โ€ช .โ€ฌื›ื•ื— ื”ื—ื™ื›ื•ืš ื ืชื•ืŸ ืข"ื™โ€ฌ
โ€ซืžืงื“ื ื”ืฆืžื™ื’ื•ืช ืฉืœ ื”ืฉืžืŸ ื”ื•ืโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช ,โ€ฌืžืกืช ื”ืจื›ื‘ ื”ื™ื โ€ช 0.823โ€ฌื˜ื•ืŸโ€ช ,โ€ฌืžืงื“ื ื”ืงืคื™ืฅ ื”ื•ื โ€ช .k=35000 N/mโ€ฌื”ืจื›ื‘ ื ื›ื ืกโ€ฌ
โ€ซืœืžื”ืžื•ืจื” ื‘ืขื•ืžืง โ€ช.10cmโ€ฌโ€ฌ
โ€ซื( ืžื” ืชื”ื™ื” ื”ื”ืชื›ื•ื•ืฆื•ืช ื”ืžื›ืกื™ืžืœื™ืช ืฉืœ ื”ืงืคื™ืฅ?โ€ฌ
โ€ซื‘( ืชื•ืš ื›ืžื” ื–ืžืŸ ื™ืคืกื™ืง ื”ืจื›ื‘ ืœื”ืชื ื“ื ื“ )ืžื‘ื—ื™ื ื” ืžืขืฉื™ืช(?โ€ฌ
โ€ซื’( ืžื” ืชื“ืจ ื”ืชื ื•ื“ื•ืช? ื›ืžื” ืชื ื•ื“ื•ืช ื™ืขืฉื” ื”ืจื›ื‘ ืขื“ ืœืขืฆื™ืจืชื•?โ€ฌ
โ€ซื“( ืžืฆืโ€ช/โ€ฌื™ ืืช ืžื”ื™ืจื•ืช ื”ืชื ื•ื“ื•ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_003.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืช ื”ืจืžื•ื ื™ื•ืชโ€ฌ
โ€ซื( ื ืชื•ืŸ ื›ื“ื•ืจ ื”ืžืงืคืฅ ืขืœ ืคื ื™ ื”ืงืจืงืข ื‘ืงืคื™ืฆื•ืช ืืœืกื˜ื™ื•ืช ืœื—ืœื•ื˜ื™ืŸโ€ช ,โ€ฌื›ืœื•ืžืจ ืœืœื ืื™ื‘ื•ื“ ืื ืจื’ื™ื” )ื”ื•ื ื—ื•ื–ืจ ืœืื•ืชื• ื’ื•ื‘ื” ื›ืœ ืคืขื(โ€ช .โ€ฌื”ืืโ€ฌ
โ€ซื”ื›ื“ื•ืจ ืžื‘ืฆืข ืชื ื•ืขื” ื”ืจืžื•ื ื™ืช? ื”ื•ื›ื—โ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื ืชื•ื ื” ืžื˜ื•ื˜ืœืช ื”ืžืชื ื“ื ื“ืช ื‘ื–ืžืŸ ืžื—ื–ื•ืจ ืฉืœ โ€ช .T=0.5 sโ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ืื•ืจืš ื”ืžื˜ื•ื˜ืœืช ื›ืš ืฉื’ื ืขืœ ืคื ื™ ื”ื™ืจื— ื™ื”ื™ื” ืœื” ืืช ืื•ืชื• ื–ืžืŸโ€ฌ
โ€ซืžื—ื–ื•ืจ? โ€ช.gm=1.67 m/s2โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_004.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืช ื”ืจืžื•ื ื™ื•ืช ื•ืฉื›ืจ ื”ื•ืœืโ€ฌ
โ€ซืžืคืขื™ืœืช ืžืขืœื™ืช ืจื•ืฉืžืช ืืช ื–ืžืŸ ืขื‘ื•ื“ืชื” ื‘ืืžืฆืขื•ืช ื“ืคื™ืงืช ื›ืจื˜ื™ืก ื‘ืฉืขื•ืŸ ื–ืžืŸโ€ช .โ€ฌื–ื•ื”ื™ ืžืขืœื™ืช ื™ืฉื ื” ืžืื•ื“โ€ช ,โ€ฌื•ืฉืขื•ืŸ ื”ื–ืžืŸ ืฉื ืžืฆื ื‘ื” ื”ื•ื ืฉืขื•ืŸโ€ฌ
โ€ซืกื‘ื ืขื ืžื˜ื•ื˜ืœืชโ€ช .โ€ฌื”ื”ืืฆื” ื•ื”ื”ืื˜ื” ืฉืœ ื”ืžืขืœื™ืช ื—ืœืงื•ืช ืœื—ืœื•ื˜ื™ืŸ ื•ืงื‘ื•ืขื•ืชโ€ช ,โ€ฌื•ื‘ืขืœื•ืช ืื•ืชื• ืขืจืš ืžื•ื—ืœื˜ )ืงื˜ืŸ ื™ื•ืชืจ ืžโ€ช (g-โ€ฌื‘ื™ืŸ ืื ื”ืžืขืœื™ืชโ€ฌ
โ€ซืขื•ืœื” ืื• ื™ื•ืจื“ืชโ€ช .โ€ฌืื ืžืฉืœืžื™ื ืœืžืคืขื™ืœืช ื”ืžืขืœื™ืช ืœืคื™ ืฉืขื”โ€ช ,โ€ฌื”ืื ืžืฉืœืžื™ื ืœื” ื™ื•ืชืจ ืžื“ื™ ืื• ืคื—ื•ืช ืžื“ื™?โ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌื”ื™ื ืขื•ื‘ื“ืช ื—ื‘ืจืช ื›ื•ื— ืื“ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_005.htmlโ€ฌโ€ฌ
โ€ซืžื•ืœืงื•ืœื” ื“ื•โ€ช-โ€ฌืื˜ื•ืžื™ืชโ€ฌ
โ€ซื ืชื•ื ื” ืžื•ืœืงื•ืœื” ื“ื• ืื˜ื•ืžื™ืช ื”ืžื•ืจื›ื‘ืช ืžืฉื ื™ ืื˜ื•ืžื™ื ื‘ืขืœื™ ืžืกื” ืฉืœ โ€ช.mโ€ฌโ€ฌ
โ€ซื”ืงืฉืจ ื”ื‘ื™ื ืื˜ื•ืžื™ ื”ืชื•ืŸ ืขืœ ื™ื“ื™ ื”ืคื•ื˜ื ืฆื™ืืœโ€ช:โ€ฌโ€ฌ
โ€ซื( ืžื”ื™ ื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ ืฉืœ ื”ืžื•ืœืงื•ืœื”?โ€ฌ
โ€ซื‘( ืžื”ื™ ื”ืžืกื” ื”ืžืฆื•ืžืฆืžืช ื‘ื‘ืขื™ื”? )ืจืื• ืข"ืž โ€ช 332โ€ฌืคืจืง โ€ช 15โ€ฌื‘ืจื–ื ื™ืง(โ€ฌ
โ€ซื’( ื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืขื‘ื•ืจ ืชื ื•ื“ื•ืช ืงื˜ื ื•ืช ืกื‘ื™ื‘ ื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœโ€ช ,โ€ฌื ื™ืชืŸ ืœืคืชื— ืืช ื”ื›ื•ื— ืœื˜ื•ืจ ื˜ื™ื™ืœื•ืจ ื‘ืกื“ืจ โ€ช ,Iโ€ฌื›ืš ืฉื”ืžื•ืœืงื•ืœื” ืชื ื“ื ื“ ื‘ืชื ื•ืขื”โ€ฌ
โ€ซื”ืจืžื•ื ื™ืช ืคืฉื•ื˜ื”โ€ช.โ€ฌโ€ฌ
โ€ซื“( ืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœ ื”ืชื ื•ื“ื”?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_4_006.htmlโ€ฌโ€ฌ
โ€ซืงืคื™ืฆื™ืโ€ฌ
โ€ซืงืคื™ืฅ "ืงืฉื”" ืžื•ื’ื“ืจ ื›ืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืข โ€ช kโ€ฌื’ื“ื•ืœโ€ช ,โ€ฌื•ืงืคื™ืฅ "ืจืš" ืžื•ื’ื“ืจ ื›ืงืคื™ืฅ ื‘ืขืœ ืงื‘ื•ืข โ€ช kโ€ฌืงื˜ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื‘ื•ืœืžื™ ื–ืขื–ื•ืขื™ื ืฉืœ ืจื›ื‘ )ืœืคื—ื•ืช ื‘ื“ื’ืžื™ื ื™ืฉื ื™ื( ืขืฉื•ื™ื™ื ืงืคื™ืฆื™ืโ€ช .โ€ฌืžื” ื”ื ื”ื™ืชืจื•ื ื•ืช ื•ื”ื—ืกืจื•ื ื•ืช ืฉืœ ื‘ื•ืœืžื™ ื–ืขื–ื•ืขื™ื ืจื›ื™ื ื•ืงืฉื™ื? ื”ื›ื•ื•ื ื”โ€ฌ
โ€ซื”ื™ื ืœืชื—ื•ืฉืช ืื™ ื”ื ื•ื—ื•ืช ืฉืœ ื”ื ื•ืกืข โ€“ ืžืฉืจืขืช ื”ืชื ื•ื“ื”โ€ช ,โ€ฌื–ืžืŸ ืžื—ื–ื•ืจ ื”ืชื ื•ื“ื” ื•ื›ืžื•ืช ื”ื˜ืœื˜ื•ืœื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_001.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซืžื’ืฉ ืฉืžืกืชื• โ€ช 4Kgโ€ฌืžื•ื ื— ืขืœ ืงืคื™ืฅ ืื ื›ื™โ€ช ,โ€ฌืฉืงื‘ื•ืข ื”ืงืคื™ืฅ ืฉืœื• โ€ช.100N/mโ€ฌโ€ฌ
โ€ซืžื’ื•ื‘ื” โ€ช 1mโ€ฌื ื•ืคืœ ืขืœื™ื• ื›ื“ื•ืจ ืฉืžืกืชื• โ€ช 1Kgโ€ฌื•ืžืชื ื’ืฉ ื‘ื• ื”ืชื ื’ืฉื•ืช ืคืœืกื˜ื™ืช โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซโ€ช.5โ€ฌโ€ฌ
โ€ซืžื”ื™ ืžืฉืจืขืช ื”ืชื ื•ื“ื•ืช?โ€ฌ
โ€ซืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœ ื”ืชื ื•ื“ื•ืช?โ€ฌ
โ€ซืžื”ื• ื”ื‘ื™ื˜ื•ื™ ืœื”ืขืชืง ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซืžื”ื• ื”ื‘ื™ื˜ื•ื™ ืœืžื”ื™ืจื•ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ?โ€ฌ
โ€ซืžื”ื™ ื”ืื ืจื’ื™ื” ื”ื›ืœืœื™ืช ืฉืœ ื”ืชื ื•ืขื” ื‘ื™ื—ืก ืœื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_002.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื‘ื•ืœ ืฉืžืกืชื• โ€ช M=0.5kgโ€ฌื ืžืฆื ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื—ืœืง ื•ืžื—ื•ื‘ืจ ืืœ ืงืฆื”ื• ื”ื™ืžื ื™ ืฉืœ ืงื‘ื™ืฅ ืฉืงื‘ื•ืขื• โ€ช .k=40N/mโ€ฌืื•ืจื›ื• ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅโ€ฌ
โ€ซื”ื•ื โ€ช.l=0.6mโ€ฌโ€ฌ
โ€ซื›ื— ืืคืงื™ ืงื‘ื•ืข ืฉื’ื•ื“ืœื• โ€ช P=20Nโ€ฌืžืชื—ื™ืœ ืœืคืขื•ืœ ืขืœ ื”ื‘ื•ืœ ื›ืืฉืจ ื”ื•ื ื ืžืฆื ื‘ืžื ื•ื—ื” ื‘ื ืงื•ื“ื” โ€ช ,Aโ€ฌื”ืžืจื•ื—ืงืช โ€ช l0=0.6mโ€ฌืžื”ืงื™ืจ ื”ืื ื›ื™ ืฉืืœื™ื•โ€ฌ
โ€ซืžื—ื•ื‘ืจ ื”ืงืฆื” ื”ืฉืžืืœื™ ืฉืœ ื”ืงืคื™ืฅโ€ช ,โ€ฌื›ื ืจืื” ื‘ืชืจืฉื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื‘ืขื•ื‘ืจื• ื‘ื ืงื•ื“ื” โ€ช Bโ€ฌื”ื ืžืฆืืช ื‘ืžืจื—ืง โ€ช d=0.25mโ€ฌืžื™ืžื™ืŸ ืœื ืงื•ื“ื” โ€ช?Aโ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื‘ื ืงื•ื“ื” โ€ช Bโ€ฌืžืคืกื™ืง ื”ื›ื— โ€ช Pโ€ฌืœืคืขื•ืœ ื•ื”ื’ื•ืฃ ืžืžืฉื™ืš ืœื‘ืฆืข ืชื ื•ื“ื•ืช ืืคืงื™ื•ืชโ€ช .โ€ฌืœืื™ื–ื” ืžืจื—ืง ืžื™ื ื™ืžืœื™ ืžื”ืงื™ืจ ื”ืื ื›ื™ ืžื’ื™ืข ื”ื‘ื•ืœ ื‘ืžื”ืœืšโ€ฌ
โ€ซืชื ื•ื“ื•ืชื™ื•?โ€ฌ
โ€ซโ€ช .3โ€ฌื›ืžื” ื–ืžืŸ ื—ื•ืœืฃ ืžืจื’ืข ื”ืคืกืงืช ืคืขื•ืœืช ื”ื›ื— โ€ช ,Pโ€ฌืขื“ ื”ืคืขื ื”ืจืืฉื•ื ื” ื‘ื” ืžื’ื™ืข ื”ื’ื•ืฃ ืืœ ื”ื ืงื•ื“ื” ื”ืงืจื•ื‘ื” ื‘ื™ื•ืชืจ ืืœ ื”ืงื™ืจ ื”ืื ื›ื™?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_003.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 1kgโ€ฌืงืฉื•ืจ ืžืฉื ื™ ืฆื“ื“ื™ื• ืœืฉื ื™ ืงืคื™ืฆื™ื ื–ื”ื™ื ืฉืงื‘ื•ืข ื”ืงืคื™ืฅ ืฉืœื”ื โ€ช 200N/mโ€ฌื”ื’ื•ืฃ ืžื•ืกื˜ โ€ช 20cmโ€ฌืฉืžืืœื” ืžื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœโ€ฌ
โ€ซื•ืžืฉื•ื—ืจืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ืขื”?โ€ฌ
โ€ซืžื”ื™ ื”ืื ืจื’ื™ื” ื”ื›ืœืœื™ืช ืฉืœ ื”ื’ื•ืฃ?โ€ฌ
โ€ซโ€ช?x(tโ€ฌโ€ฌ
โ€ซืชื•ืฃ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ื”ื’ื•ืฃ ืœื ืงื•ื“ื” ื”ื ืžืฆืืช โ€ช 5cmโ€ฌืžื™ืžื™ืŸ ืœื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_054.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช m=0.5kgโ€ฌื ืžืฆื ืขืœ ืžืฉื˜ื— ืื•ืคืงื™ ื—ืœืง ื•ืžื—ื•ื‘ืจ ืืœ ืงืฆื”ื• ืฉืœ ืงืคื™ืฅ ืื•ืคืงื™ ืฉืงื‘ื•ืขื• โ€ช.k=8N/mโ€ฌโ€ฌ
โ€ซืงืฆื”ื• ื”ืฉื ื™ ืฉืœ ื”ืงืคื™ืฅ ืžื—ื•ื‘ืจืจ ืœืงื™ืจ ืื ื›ื™โ€ช .โ€ฌื‘ืจื’ืข โ€ช t=0โ€ฌืžืขื ื™ืงื™ื ืœื’ื•ืฃ ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืช โ€ช v0=0.8m/Secโ€ฌื‘ื›ื™ื•ื•ืŸ ื”ื—ื™ื•ื‘ื™ ืฉืœ ืฆื™ืจ โ€ช,xโ€ฌโ€ฌ
โ€ซื›ืืฉืจ ื”ื’ื•ืฃ ื ืžืฆื ื‘ื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซโ€ช.5โ€ฌโ€ฌ
โ€ซืžื” ืžืฉืจืขืช ื”ืชื ื•ื“ื•ืช?โ€ฌ
โ€ซืจืฉื•ื ื‘ื™ื˜ื•ื™ื™ื ืžืคื•ืจืฉื™ื ืขื‘ื•ืจ ื”ืขืชืง ื”ื’ื•ืฃ ืžื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœ ื•ืขื‘ื•ืจ ืžื”ื™ืจื•ืชื• ื”ืจื’ืขื™ืช ื›ืคื•ื ืงืฆื™ื•ืช ืฉืœ ื”ื–ืžืŸโ€ช.โ€ฌโ€ฌ
โ€ซืจืฉื•ื ื‘ื™ื˜ื•ื™ื™ื ืžืคื•ืจืฉื™ื ืขื‘ื•ืจ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช )โ€ช Ek(tโ€ฌื•ืขื‘ื•ืจ ื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช ื”ืืœืกื˜ื™ืช)โ€ช.Ep(tโ€ฌโ€ฌ
โ€ซื”ืจืื” ืฉืกื›ื•ื ื”ืื ืจื’ื™ื•ืช ื”ื•ื ื’ื•ื“ืœ ืงื‘ื•ืขโ€ช.โ€ฌโ€ฌ
โ€ซื‘ื ื™ืกื•ื™ ืื—ืจ ืื•ืชื” ืžืขืจื›ืช ืชืœื•ื™ื” ื‘ืื•ืคืŸ ืื ื›ื™ ืžื”ืชืงืจื”โ€ช .โ€ฌื‘ืจื’ืข โ€ช ,t=0โ€ฌื›ืฉื”ื’ื•ืฃ ื‘ื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœโ€ช ,โ€ฌืžืขื ื™ืงื™ื ืœื• ืžื”ื™ืจื•ืช ื”ืชื—ืœืชื™ืชโ€ฌ
โ€ซโ€ช v0=0.8m/Secโ€ฌื”ืžื›ื•ื•ื ืช ืื ื›ื™ืช ืžืขืœื” )ื”ื›ื™ื•ื•ืŸ ื”ื—ื™ื•ื‘ื™ ืฉืœ ืฆื™ืจ (โ€ช .โ€ฌื—ื–ื•ืจ ืขืœ ืกืขื™ืฃ ื‘' ืขื‘ื•ืจ ื”ื ื™ืกื•ื™ ื”ื ื•ื›ื—ื™โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_055.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื‘ื•ืœ ืฉืžืกืชื• โ€ช M=0.5kgโ€ฌื ืžืฆื ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื—ืœืง ื•ืžื—ื•ื‘ืจ ืืœ ืงืฆื”ื• ื”ื™ืžื ื™ ืฉืœ ืงื‘ื™ืฅ ืฉืงื‘ื•ืขื• โ€ช .k=40N/mโ€ฌืื•ืจื›ื• ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅโ€ฌ
โ€ซื”ื•ื โ€ช.l=0.6mโ€ฌโ€ฌ
โ€ซื›ื— ืืคืงื™ ืงื‘ื•ืข ืฉื’ื•ื“ืœื• โ€ช P=20Nโ€ฌืžืชื—ื™ืœ ืœืคืขื•ืœ ืขืœ ื”ื‘ื•ืœ ื›ืืฉืจ ื”ื•ื ื ืžืฆื ื‘ืžื ื•ื—ื” ื‘ื ืงื•ื“ื” โ€ช ,Aโ€ฌื”ืžืจื•ื—ืงืช โ€ช l0=0.6mโ€ฌืžื”ืงื™ืจ ื”ืื ื›ื™ ืฉืืœื™ื•โ€ฌ
โ€ซืžื—ื•ื‘ืจ ื”ืงืฆื” ื”ืฉืžืืœื™ ืฉืœ ื”ืงืคื™ืฅโ€ช ,โ€ฌื›ื ืจืื” ื‘ืชืจืฉื™ืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .1โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื’ื•ืฃ ื‘ืขื•ื‘ืจื• ื‘ื ืงื•ื“ื” โ€ช Bโ€ฌื”ื ืžืฆืืช ื‘ืžืจื—ืง โ€ช d=0.25mโ€ฌืžื™ืžื™ืŸ ืœื ืงื•ื“ื” โ€ช?Aโ€ฌโ€ฌ
โ€ซโ€ช .2โ€ฌื‘ื ืงื•ื“ื” โ€ช Bโ€ฌืžืคืกื™ืง ื”ื›ื— โ€ช Pโ€ฌืœืคืขื•ืœ ื•ื”ื’ื•ืฃ ืžืžืฉื™ืš ืœื‘ืฆืข ืชื ื•ื“ื•ืช ืืคืงื™ื•ืชโ€ช .โ€ฌืœืื™ื–ื” ืžืจื—ืง ืžื™ื ื™ืžืœื™ ืžื”ืงื™ืจ ื”ืื ื›ื™ ืžื’ื™ืข ื”ื‘ื•ืœ ื‘ืžื”ืœืšโ€ฌ
โ€ซืชื ื•ื“ื•ืชื™ื•?โ€ฌ
โ€ซโ€ช .3โ€ฌื›ืžื” ื–ืžืŸ ื—ื•ืœืฃ ืžืจื’ืข ื”ืคืกืงืช ืคืขื•ืœืช ื”ื›ื— โ€ช ,Pโ€ฌืขื“ ื”ืคืขื ื”ืจืืฉื•ื ื” ื‘ื” ืžื’ื™ืข ื”ื’ื•ืฃ ืืœ ื”ื ืงื•ื“ื” ื”ืงืจื•ื‘ื” ื‘ื™ื•ืชืจ ืืœ ื”ืงื™ืจ ื”ืื ื›ื™?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_056.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 1kgโ€ฌืงืฉื•ืจ ืžืฉื ื™ ืฆื“ื“ื™ื• ืœืฉื ื™ ืงืคื™ืฆื™ื ื–ื”ื™ื ืฉืงื‘ื•ืข ื”ืงืคื™ืฅ ืฉืœื”ื โ€ช 200N/mโ€ฌื”ื’ื•ืฃ ืžื•ืกื˜ โ€ช 20cmโ€ฌืฉืžืืœื” ืžื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœโ€ฌ
โ€ซื•ืžืฉื•ื—ืจืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.1โ€ฌโ€ฌ
โ€ซโ€ช.2โ€ฌโ€ฌ
โ€ซโ€ช.3โ€ฌโ€ฌ
โ€ซโ€ช.4โ€ฌโ€ฌ
โ€ซืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ืขื”?โ€ฌ
โ€ซืžื”ื™ ื”ืื ืจื’ื™ื” ื”ื›ืœืœื™ืช ืฉืœ ื”ื’ื•ืฃ?โ€ฌ
โ€ซโ€ช?x(tโ€ฌโ€ฌ
โ€ซืชื•ืฃ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ื”ื’ื•ืฃ ืœื ืงื•ื“ื” ื”ื ืžืฆืืช โ€ช 5cmโ€ฌืžื™ืžื™ืŸ ืœื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_5_123.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซืžืกื” โ€ช m2=2 kgโ€ฌืงืฉื•ืจื” ืœืžืกื” โ€ช m1=1kgโ€ฌืข" ื™ ื—ื•ื˜ ื“ืงโ€ช .โ€ฌื”ืžืกื” โ€ช m1โ€ฌืงืฉื•ืจื” ืœืงืคื™ืฅ ืื ื›ื™ )ืจืื• ืฆื™ื•ืจ( ืฉืงื‘ื•ืข ืงืคื™ืฆื• โ€ช .k=100N/mโ€ฌื”ืžืขืจื›ืชโ€ฌ
โ€ซื ืžืฆืืช ื‘ืžื ื•ื—ื” ื•ืื– ื ืงืจืข ื”ื—ื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ืžื™ืงื•ื ื”ื”ืชื—ืœืชื™ ืฉืœ ื”ืชื ื•ืขื” ื™ื—ืกื™ืช ืœืžืฆื‘ ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ืžื™ืงื•ื ื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ ื™ื—ืกื™ืช ืœืžืฆื‘ ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅ ? ืžื”ื™ ื”ืืžืคืœื™ื˜ื•ื“ื” ืฉืœ ื”ืชื ื•ืขื” ื”ื”ืจืžื•ื ื™ืช ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžืฆืื• ื‘ื™ื˜ื•ื™ ืœืžื™ืงื•ื ื”ื’ื•ืฃ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ )ืžืจื’ืข ื”ื™ืงืจืข ื”ื—ื•ื˜(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_7_001.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื”ืจืžื•ื ื™ืชโ€ฌ
โ€ซื’ื•ืฃ ื ืข ื‘ืชื ื•ืขื” ื”ืจืžื•ื ื™ืช ืฉืžืฉืจืขืชื” โ€ช 10โ€ฌืก"ืž ื•ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœื” โ€ช 5โ€ฌืฉื ื™ื•ืชโ€ฌ
โ€ซื”ื’ื•ืฃ ื”ืชื—ื™ืœ ืœื ื•ืข ืžื”ื ืงื•ื“ื” ื”ืจื—ื•ืงื” ื‘ื™ื•ืชืจ ืžื ืงื•ื“ืช ืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆื ื‘ื™ื˜ื•ื™ ืœื”ืขืชืงโ€ช ,โ€ฌืžื”ื™ืจื•ืชโ€ช ,โ€ฌื•ืชืื•ืฆื” ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื ื”ืขืชืง ื•ื”ืžื”ื™ืจื•ืช ื‘ื–ืžืŸ โ€ชt=1โ€ฌโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_7_002.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื”ืจืžื•ื ื™ืชโ€ฌ
โ€ซืžื™ืงื•ืžื• ืฉืœ ื’ื•ืฃ ื ืชื•ืŸ ืข"ื™ ื”ื‘ื™ื˜ื•ื™โ€ฌ
โ€ซืโ€ช .โ€ฌื›ืชื•ื‘ ื‘ื™ื˜ื•ื™ ืœืžื”ื™ืจื•ืช ื•ืœืชืื•ืฆื” ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸโ€ฌ
โ€ซื‘โ€ช .โ€ฌืœืื—ืจ ืคืจืง ื–ืžืŸ โ€ช T/6โ€ฌื”ื’ื•ืฃ ื ืžืฆื ื‘ืขืจื›ื™ื ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซื—ืฉื‘ ืืช ื”ืžืฉืจืขืช โ€ช Aโ€ฌืืช ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ื•ืืช โ€ชX0โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื• ื•ืชืื•ืฆืชื• ืฉืœ ื”ื’ื•ืฃ ื‘ืขื•ื‘ืจื• ื‘ื ืงื•ื“ื”โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_7_003.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ืขื” ื”ืจืžื•ื ื™ืชโ€ฌ
โ€ซืขื‘ื•ืจ ืžื˜ื•ื˜ืœืช ื”ืจืžื•ื ื™ืช ื™ื“ื•ืข ืฉืื•ืจืš ื”ื—ื•ื˜ โ€ช 2โ€ฌืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซื‘ โ€ช t=0โ€ฌื™ื“ื•ืข ืฉโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžืฆื ืืช ื”ืืžืคืœื™ื˜ื•ื“ื” ืฉืœ ื”ื–ื•ื•ื™ืช ื•ืฉืœ ื”ื”ืขืชืงโ€ฌ
โ€ซื‘โ€ช .โ€ฌืจืฉื•ื ืžืฉื•ื•ืื•ืช ืขื‘ื•ืจโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_7_004.htmlโ€ฌโ€ฌ
โ€ซืงืคื™ืฅโ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 1โ€ฌืง"ื’ ืงืฉื•ืจ ืžืฉื ื™ ืฆื“ื“ื™ื• ืœืฉื ื™ ืงืคื™ืฆื™ื ื–ื”ื™ื ืฉืงื‘ื•ืข ื”ืงืคื™ืฅ ืฉืœื”ื โ€ช 200โ€ฌื ื™ื•ื˜ื•ืŸ ืœืžื˜ืจโ€ช.โ€ฌโ€ฌ
โ€ซื”ื’ื•ืฃ ืžื•ืกื˜ โ€ช 20โ€ฌืก"ืž ืฉืžืืœื” ืžื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ ื•ื ืขื–ื‘โ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชื“ื™ืจื•ืช ื”ืชื ื•ืขื”โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ื”ื›ืœืœื™ืช ืฉืœ ื”ื’ื•ืฃโ€ฌ
โ€ซื’โ€ช .โ€ฌืžืฆื ืืช )โ€ชx(tโ€ฌโ€ฌ
โ€ซื“โ€ช .โ€ฌืชื•ืš ื›ืžื” ื–ืžืŸ ื™ื’ื™ืข ื”ื’ื•ืฃ ืœื ืงื•ื“ื” ื”ื ืžืฆืืช โ€ช 5โ€ฌืก"ืž ืžื™ืžื™ืŸ ืœื ืงื•ื“ืช ืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซโ€ช19_Vibrations/e_19_8_121.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซืžืกื” ืฉืœ โ€ช 100โ€ฌื’ืจื ืงืฉื•ืจื” ืœื—ื•ื˜ ื‘ืื•ืจืš โ€ช 50โ€ฌืก"ืž ื”ืงืฉื•ืจ ืœืชืงืจื”โ€ช .โ€ฌืžืกื™ื˜ื™ื ืืช ื”ืžืกื” ื‘โ€ช 100-โ€ฌื™ื—ืกื™ืช ืœืื ืš )ืจืื• ืฆื™ื•ืจ( ื•ืžืฉื—ืจืจื™ื )ืจื’ืขโ€ฌ
โ€ซื”ืฉื™ื—ืจื•ืจ ื”ื•ื โ€ช.(t=0โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื”ืฉืชื›ื ืขื• ืฉืขื‘ื•ืจ ื–ื•ื•ื™ืช ืงื˜ื ื” ื–ื• ื”ืชื ื•ืขื” ื”ื™ื ื‘ืงื™ืจื•ื‘ ื”ืจืžื•ื ื™ืชโ€ช .โ€ฌืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžืฆืื• ืืช )โ€ช โ€“ q(tโ€ฌื”ื–ื•ื•ื™ืช )ื‘ืจื“ื™ืื ื™ื( ื‘ื–ืžืŸ โ€ช .tโ€ฌื’ื–ืจื• ืžื‘ื™ื˜ื•ื™ ื–ื” ืืช ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ื•ืืช ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌืชื•ืš ื›ืžื” ื–ืžืŸ ืชื’ื™ืข ื”ืžืฉืงื•ืœืช ืœื’ื•ื‘ื” ื”ื ืžื•ืš ื‘ื™ื•ืชืจ ?โ€ฌ
โ€ซื“โ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืชื” ื”ื–ื•ื•ื™ืชื™ืช ื‘ื ืงื•ื“ื” ื–ื• ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_8_122.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซื›ื“ื•ืจ ื‘ืื•ืœื™ื ื’ ื‘ืžืกื” โ€ช 3โ€ฌืง"ื’ ืžื•ื ื— ืขืœ ืจื™ืฆืคื” ื—ืœืงื” ื•ืงืฉื•ืจ ืœืงืคื™ืฅ ื›ืžื•ืจืื”โ€ช .โ€ฌืงื‘ื•ืข ื”ืงืคื™ืฅ โ€ช 111โ€ฌื ื™ื˜ื•ืŸ\ืžื˜ืจโ€ช .โ€ฌื›ื“ื•ืจ ื‘ืื•ืœื™ื ื’ ื–ื”ื” ื”ืžื’ื™ืขโ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืช โ€ช 10โ€ฌืžโ€ช/โ€ฌืฉ ืžืชื ื’ืฉ ื‘ื• ืืœืกื˜ื™ืช )ื–ืžืŸ ื”ื”ืชื ื’ืฉื•ืช ื–ื ื™ื—(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ื”ืืžืคืœื™ื˜ื•ื“ื” ืฉืœ ื”ืชื ื•ืขื” โ€ช? Aโ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ื–ืžืŸ ืžื—ื–ื•ืจ ื”ืชื ื•ืขื” ?โ€ฌ
โ€ซื’โ€ช.โ€ฌื›ืชื‘ื• ืืช )โ€ช โ€“ x(tโ€ฌืžื™ืงื•ื ื”ื›ื“ื•ืจ ื”ืงืฉื•ืจ ืžืจื’ืข ื”ืคื’ื™ืขื” ื•ืื™ืœืšโ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช.โ€ฌืชื•ืš ื›ืžื”ื–ืžืŸ ื™ื’ื™ืข ื”ื›ื“ื•ืจ ืœืžืจื—ืง ืฉืœ โ€ช A/2โ€ฌืžื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ )ื‘ืคืขื ื”ืจืืฉื•ื ื”( ?โ€ฌ
โ€ซโ€ช19_Vibrations/e_19_8_123.htmlโ€ฌโ€ฌ
โ€ซืชื ื•ื“ื•ืชโ€ฌ
โ€ซืžืกื” โ€ช m2=2 kgโ€ฌืงืฉื•ืจื” ืœืžืกื” โ€ช m1=1kgโ€ฌืข" ื™ ื—ื•ื˜ ื“ืงโ€ชm1 .โ€ฌืงืฉื•ืจื”ืœืงืคื™ืฅ ืื ื›ื™ )ืจืื• ืฆื™ื•ืจ( ืฉืงื‘ื•ืข ืงืคื™ืฆื• โ€ช .k=100N/mโ€ฌื”ืžืขืจื›ืช ื ืžืฆืืชโ€ฌ
โ€ซื‘ืžื ื•ื—ื” ื•ืื– ื ืงืจืข ื”ื—ื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ืžื™ืงื•ื ื”ื”ืชื—ืœืชื™ ืฉืœ ื”ืชื ื•ืขื” ื™ื—ืกื™ืช ืœืžืฆื‘ ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ืžื™ืงื•ื ื ืงื•ื“ืช ืฉื™ื•ื•ื™ ื”ืžืฉืงืœ ื™ื—ืกื™ืช ืœืžืฆื‘ ื”ืจืคื•ื™ ืฉืœ ื”ืงืคื™ืฅ ? ืžื”ื™ ื”ืืžืคืœื™ื˜ื•ื“ื” ืฉืœ ื”ืชื ื•ืขื” ื”ื”ืจืžื•ื ื™ืช ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžืฆืื• ื‘ื™ื˜ื•ื™ ืœืžื™ืงื•ื ื”ื’ื•ืฃ ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ืžืŸ )ืžืจื’ืข ื”ื™ืงืจืข ื”ื—ื•ื˜(โ€ช.โ€ฌโ€ฌ
โ€ซื“โ€ช.โ€ฌืžื“ื•ืข ื”ืชื ื•ืขื” ื”ื”ืจืžื•ื ื™ืช ื ืงืจืืช ืชื ื•ืขื” ื”ืจืžื•ื ื™ืช ืคืฉื•ื˜ื” ?โ€ฌ
19_Vibrations/e_19_8_124.html
โ€ซืชื ื•ื“ื•ืชโ€ฌ
:โ€ซ ืข" ืค ื”ืฆื™ื•ืจโ€ฌkA = kB = 500N/m โ€ซ ืœื”ืโ€ฌ,โ€ซ ืžื—ื•ื‘ืจื•ืช ืœืฉื ื™ ืงืคื™ืฆื™ื ื–ื”ื™ืโ€ฌMA = MB = 1 kg โ€ซ ืฉืžืกืชืŸโ€ฌB -โ€ซ ื•โ€ฌA โ€ซืฉืชื™ ืžืกื•ืช ื–ื”ื•ืชโ€ฌ
โ€ซ ืœืื—ืจ ื”ืฉื—ืจื•ืจโ€ฌ.โ€ซ ืฉืžืืœื” ื•ืžืฉื—ืจืจื™ื ืื•ืชื”โ€ฌx = 10 cm โ€ซ ืžืจื—ืง ืฉืœโ€ฌMA โ€ซ ืžื–ื™ื–ื™ื ืืช ื”ืžืกื”โ€ฌ.โ€ซื‘ืžืฆื‘ ื”ื”ืชื—ืœืชื™ ืฉื ื™ ื”ืงืคื™ืฆื™ื ื ืžืฆืื™ื ื‘ืžื ื•ื—ื”โ€ฌ
.โ€ซืžืชื ื’ืฉื•ืช ื”ืžืกื•ืช ื”ืชื ื’ืฉื•ืช ืคืœืกื˜ื™ืช ื•ื ืขื•ืช ื›ื’ื•ืฃ ืื—ื“โ€ฌ
? MB โ€ซ ื‘ืžืกื”โ€ฌMA โ€ซื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ืชืชื ื’ืฉ ื”ืžืกื”โ€ฌ.โ€ซืโ€ฌ
? โ€ซ ื‘ืขืช ื”ืคื’ื™ืขื”โ€ฌMA โ€ซืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื” ืฉืœโ€ฌ.โ€ซื‘โ€ฌ
? โ€ซื›ืžื” ื–ืžืŸ ื™ื—ืœื•ืฃ ืžื”ืคื’ื™ืขื” ื•ืขื“ ืœื ืงื•ื“ื” ื‘ื” ื™ื™ืขืฆืจื• ื”ืžืกื•ืชโ€ฌ.โ€ซื’โ€ฌ
? โ€ซืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœ ืชื ื•ืขืช ื”ืžืกื•ืช ืœืื—ืจ ื”ืคื’ื™ืขื”โ€ฌ.โ€ซื“โ€ฌ
โ€ซ ืฉืžืืœื” ืžื ืงื•ื“ืช ืฉื•ื•ื™โ€ฌcm 1 โ€ซ ืขื“ ืฉื ืงื•ื“ืช ื”ืžื’ืข ื‘ื™ืŸ ื”ืžืกื•ืช ืชื—ืœื•ืฃ ืขืœ ืคื ื™ ื ืงื•ื“ื” ื”ื ืžืฆืืชโ€ฌMA โ€ซื›ืžื” ื–ืžืŸ ื™ื—ืœื•ืฃ ืžืขืช ืฉื—ืจื•ืจ ื”ืžืกื”โ€ฌ.โ€ซื”โ€ฌ
? โ€ซื”ืžืฉืงืœโ€ฌ
? โ€ซืžื”ื• ื”ืžืจื—ืง ื”ืžืงืกื™ืžืœื™ ืฉืชื’ื™ืข ื ืงื•ื“ื” ื–ื• ืžื ืงื•ื“ืช ืฉื•ื•ื™ ื”ืžืฉืงืœโ€ฌ.โ€ซื•โ€ฌ
22_Rigid_body/e_22_1_011.html
Rigid Body
A disk of the mass
and radius
is connected to two parallel identical springs (
, ) as shown in the figure.
Find the frequency of rotational oscillations around the center of the disk.
22_Rigid_body/e_22_1_012.html
Rigid Body
A ball of the radius rotates with the angular velocity
around the horizontal axis passing through the centerof-mass. The ball is carefully put on a horizontal surface with the friction coefficient . Find
.
22_Rigid_body/e_22_1_013.html
Rigid Body
A cylinder of the radius is rolling without sliding inside a larger cylinder of the radius
figure (vertical cross-section).
as shown in the
a) Find the minimal angular frequency in the lowest point which allows to reach the highest point. b) Find the
frequency of small oscillations near the equilibrium.
22_Rigid_body/e_22_1_014.html
Rigid Body
A homogeneous cube is rotating around the axis passing through the center-of-mass. Describe qualitatively the
motion of the axis depending on the angle of the axis with the normal to the cube side.
22_Rigid_body/e_22_1_015.html
Rigid Body
A hollow cylinder and a solid cylinder of the same radius start rolling simultaneously without sliding down the
same slope from the same height. What is the ratio of the final velocities ? Which one comes to the end of the
slope earlier and what is the ratio of times ?
22_Rigid_body/e_22_1_016.html
Rigid Body
Two identical masses
connected by a massless rod of the length are moving on a circular orbit
around the Earth. The attraction force between the Earth and a point mass is
, where
is
the Earth mass and
is a universal constant. Find the frequency of small rotational oscillations of the system
(masses on the rod) around the center-of-mass.
22_Rigid_body/e_22_1_017.html
Rigid Body
A car engine is applying a torque to a wheel. The wheel mass is , radius is and the moment of inertia with
respect to the center is . The coefficient of the static friction with the road is . What is the maximum torque
which can be applied without making the wheel slide ?
22_Rigid_body/e_22_1_018.html
Rigid Body
A bobbin is moved by pulling a thread which is winded on the inner cylinder. The outer radius is , the inner
radius is , the bobbin mass is , the friction coefficient is . What is the maximal force
for which the
bobbin rolls without friction ? What is the bobbin velocity after it moves by the distance from the rest ?
22_Rigid_body/e_22_1_019.html
Rigid Body
Six identical point masses
,
are in the positions
,
,
,
. The anglular velocity vector
,
. Find
.
22_Rigid_body/e_22_1_020.html
Rigid Body
Three identical disks with the mass
and radius each are connected so that they have the common center
and their planes are mutually perpendicular. Find the moment of inertia relative to an arbitrary axis passing
through the center.
22_Rigid_body/e_22_1_021.html
Rigid Body
Two identical particles of the mass
are connected to the two ends of a rigid massless rod of the length .
The system initially rotates around the center-of-mass with the angular velocity . One of the particles
encounters a third one (with the same mass) at rest, which momentarily sticks to it. What is the angular velocity
of the rotation around the center-of-mass after the collision ? (No gravity.)
22_Rigid_body/e_22_1_022.html
Rigid Body
A homogeneous ball (mass , radius ) is struck by a horisontal force
in the point which is above the center
by the distance
. The time of force action is very small, but
is nonzero. Find the velocity of the centerof-mass and the angular velocity of the ball around the axis which goes through the center-of-mass, if a) there is
no friction with the floor, and b) if the friction prevents sliding. (
)
22_Rigid_body/e_22_1_023.html
Rigid Body
A massless dancer holds two identical masses ( ) at a distance
from the body
and spinning at a constant angular velocity
.
Suddenly the dancer move the masses to the distance of
from the body.
What will be the the new angular velocity?
22_Rigid_body/e_22_1_024.html
Rigid Body
Two bodies of masses
and
move under the action of their mutual gravitation.
Let
and
be the position vectors in a space-fixed coordinate system, and
Find the equation of motion for ,
and in the center-of-mass system.
.
22_Rigid_body/e_22_1_025.html
Rigid Body
Find the center of mass of a solid cone of mass
whose top radius is
and its height is
.
22_Rigid_body/e_22_1_026.html
Rigid Body
A stick of mass
and lenght
is initialy at rest in a vertical position on a frictionless table.
if the stick start falling, find the speed of the center of mass as a function of the angle that the stick makes
with the vertical.
22_Rigid_body/e_22_1_027.html
Rigid Body
Find the moment of inertia of a disk of radius
and mass
with respect to an axis that passing at the edge of the disk and
perpendicular to its plane.
22_Rigid_body/e_22_1_028.html
Rigid Body
A thin cylinder of radius mass
is rolling inside a biger cylinder of radius
1. Find the kinetic energy of the small cylinder.
2. Find the frequency of small oscillations.
.
22_Rigid_body/e_22_1_029.html
Rigid Body
Find the center of mass for the following bodies:
1. A paraboloid
between
and
2. A disk of radius
with a uniform surface density
from the center of the first circle.
with a uniform density .
which has a circular hole of radius
at a distance
22_Rigid_body/e_22_1_030.html
Rigid Body
Find the moment of inertia for the following bodies:
1. A cylinder of mass
radius
and height
rotating about its symmetry axis.
2. A cylinder of mass
radius
and height
rotating about an axis parallel to the symmetry axis and
tangent to the surface.
3. A thin square of side
with its diagonal along
-axis (mass
22_Rigid_body/e_22_1_031.html
Rigid Body
Prove the following moment of inertia for the following bodies:
). Find
,
and
.
22_Rigid_body/e_22_1_032.html
Rigid Body
A billiard ball ( mass
, radius ) is placed on a table with kinetic friction coefficient .
At
the ball is struck (for an infinitesimal duration
such that
is nonzero) horizontally by a cue at
a height from the table and start moving in a velocity .
1. What is the final speed of the center of mass of the ball?
2. At what height (from the center of mass) one has to strike the ball so that rolling motion starts immediately?
22_Rigid_body/e_22_1_033.html
Rigid Body
A uniform thin rigid rod of mass
is supported by two rotating rollers
whose axes are separated by a fixed distance . The rod is initially placed
at rest asymmetrically.
1. Assume that the rollers rotate in opposite directions. The coefficient
of kinetic friction between the bar and the rollers is . Write the
equation of motion of the bar and solve for the displacement
of the center of the bar from roller (1),
assuming
and
.
2. Now consider the case in which the directions of rotation of the rollers
are reversed, calculate the displacement
,assuming
and
.
22_Rigid_body/e_22_1_034.html
Rigid Body
A uniform cylinder starts to slide without friction on a slope (angle
with the horizon).
When the cylinder passes a distance
the slope gets rough with a friction coefficient .
What will be the speed of the cylinder when the slipping ends?
22_Rigid_body/e_22_1_035.html
Rigid Body
A uniform rod of mass
and lenght is free to move in a vertically about an axis that pass
at a point
from the top. A bullet of mass
hits the rod horizontally at the top with a speed .
What is the maximal angle for which the rod will rotate?
22_Rigid_body/e_22_1_036.html
Inertia Tensor
Find the inertia tensor of a uniform thin board with dimensions a*b.
22_Rigid_body/e_22_1_037.html
Moment of Inertia
1) Find the moment of inertia tensor of a uniform empty cylinder of radius R, length L and mass M with respect
to it's main symmetry axis.
2) The same, but for a full cylinder.
22_Rigid_body/e_22_1_038.html
Moment of Inertia
A uniform full cylinder of radius R, length L and mass M is taken, and 4 holes are drilled into it.
Each hole has a radius of a/3, and it's center is located a distance of a/2 from the main symmetry axis. The holes
are drilled symmetricaly, so as to form a cross.
Find the moment of inertia tensor of this new cylinder respect to it's main symmetry axis.
22_Rigid_body/e_22_1_039.html
Rigid body - conservation laws
A bar of length L and mass M is lying on a frictionless table. A ball with mass m and velocity v hits the bar
perpendicularly at a distance d from the bar's center. Assume the collision is fully elastic, and find m such that
after the collision only the bar moves.
22_Rigid_body/e_22_1_040.html
Physical Pendulum
Show how we can measure
timer.
using an arbitrary rigid body (for which we now the center of mass), a ruler and a
22_Rigid_body/e_22_1_041.html
โ€ซืคืจืกืฆื™ื”โ€ฌ
โ€ซโ€ช .โ€ฌื”ืžืขืจื›ืช ืžืกืชื•ื‘ื‘ืช ืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–โ€ฌ
โ€ซืžืขืจื›ืช ื‘ื ื•ื™ื” ืžืฉืชื™ ืžืกื•ืช ื ืงื•ื“ืชื™ื•ืชโ€ช , ,โ€ฌื”ืžื•ื—ื–ืงื•ืช ืขืœ ื™ื“ื™ ืžื•ื˜ ื—ืกืจ ืžืกื” ื‘ืื•ืจืšโ€ฌ
โ€ซโ€ช ,โ€ฌื›ืš ืฉื”ื–ื•ื™ืช ื‘ื™ืŸ ื”ืžื•ื˜ ืœืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ื™ื โ€ช .โ€ฌืžืฆืื• ืืช ืžื•ืžื ื˜ ื”ื›ื•ื— ืฉื™ืฉ ืœืกืคืง ื›ื“ื™ ืœืงื™ื™ื ืืช ื”ืชื ื•ืขื”โ€ฌ
โ€ซื”ืžื•ื˜โ€ช ,โ€ฌื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืช ืงื‘ื•ืขื”โ€ฌ
โ€ซื”ืกื™ื‘ื•ื‘ื™ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_1_042.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ื•ืœ ืœืœื ื”ื—ืœืงื”โ€ฌ
โ€ซ)ืคื™ืกื™ืงื” โ€ช 1โ€ฌืœืคื™ืกื™ืงืื™ืโ€ช ,2/03/2006 ,โ€ฌืžื•ืขื“ ื‘'โ€ช(4 ,โ€ฌโ€ฌ
โ€ซืขืœ ืงืจืฉ ืื•ืคืงื™ ืžื—ื•ืกืคืก ื ืžืฆื ื’ืœื™ืœ ืื—ื™ื“ ื‘ืžืจื—ืง ืžื”ืงืฆื” ื”ื™ืžื ื™โ€ช .โ€ฌืžืชื—ื™ืœื™ื ืœื”ื ื™ืข ืืช ื”ืงืจืฉ ื‘ืชืื•ืฆื”โ€ฌ
โ€ซื”ื’ืœื™ืœ ื›ืืฉืจ ื”ื•ื ืžื’ื™ืข ืœืงืฆื”โ€ช .โ€ฌืื™ืŸ ื”ื—ืœืงื”โ€ช.โ€ฌโ€ฌ
โ€ซืฉืžืืœื”โ€ช .โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื ืขโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_1_043.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซ)ืคื™ืกื™ืงื” โ€ช 1โ€ฌืœืคื™ืกื™ืงืื™ืโ€ช ,25/2/2007 ,โ€ฌืžื•ืขื“ ื‘'โ€ช(5 ,โ€ฌโ€ฌ
โ€ซื”ืขื•ื‘ืจ ื‘ืžืจื—ืง ืžื”ืืžืฆืขโ€ฌ
โ€ซืžืขืจื›ืช ืžื•ืจื›ื‘ืช ืžืžื•ื˜ ืื—ื™ื“ ื‘ืขืœ ืื•ืจืš ื•ืžืกื” โ€ช ,โ€ฌืืฉืจ ื™ื›ื•ืœ ืœื”ืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ืื•ืคืงื™โ€ฌ
โ€ซื”ื ื’ื“ื™ ืฉืœ ื”ืžื•ื˜ ืžื—ื•ื‘ืจ ืœืงืคื™ืฅ ืื ื›ื™ ื‘ืขืœ ืงื‘ื•ืข ืงืคื™ืฅ โ€ช .โ€ฌื‘ืฉื™ื•ื•ื™ ืžืฉืงืœ ื”ืžื•ื˜ ื ืžืฆื ื‘ืžืฆื‘ ืื•ืคืงื™โ€ช .โ€ฌืžืฆืื• ืืช ื–ืžืŸ ื”ืžื—ื–ื•ืจโ€ฌ
โ€ซืฉืœ ืชื ื•ื“ื•ืช ืงื˜ื ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื”ืงืฆื”โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_021.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซื”ืจืื”โ€ช/โ€ฌื™ ื›ื™ ืžืจื›ื– ื”ืžืกื” ืฉืœ ื—ืจื•ื˜ ื‘ืขืœ ื’ื•ื‘ื”โ€ฌ
โ€ซื•ืฆืคื™ืคื•ืช ืžืกื” ืื—ื™ื“ื” ื ืžืฆื ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซืžื‘ืกื™ืกื•โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_022.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซืื‘ ื•ื‘ื ื• ื ืžืฆืื™ื ื‘ืงืฆื” ื”ืฉืžืืœื™ ืฉืœ ืกื™ืจื” ื”ื ืžืฆืืช ื‘ืžื ื•ื—ื”โ€ช .โ€ฌื‘ืจื’ืข ืžืกื•ื™ื™ื ืฆื•ืขื“ ื”ื‘ืŸ ืืœ ืžืจื›ื– ื”ืกื™ืจื”โ€ฌ
โ€ซื•ื”ืื‘ ืฆื•ืขื“ ืืœ ืงืฆื” ื”ื™ืžื ื™ ืฉืœ ื”ืกื™ืจื”โ€ช .โ€ฌื‘ื›ืžื” ืชื ื•ืข ื”ืกื™ืจื” ื‘ื™ื—ืก ืœืžื™ืงื•ืžื” ื”ื”ืชื—ืœืชื™?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืžืกืช ื”ืกื™ืจื”โ€ฌ
โ€ซืžืกืช ื”ืื‘โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืžืกืช ื”ื‘ืŸโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืื•ืจืš ื”ืกื™ืจื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_023.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซืืฉืจ ืงืฆื”ื• ื”ืฉืžืืœื™ ืžื•ื ื— ื‘ืจืืฉื™ืช?โ€ฌ
โ€ซืโ€ช .โ€ฌื”ื™ื›ืŸ ืžืจื›ื– ื”ืžืกื” ืฉืœ ืžื•ื˜ ื‘ืื•ืจืš ื‘ืขืœ ืฆืคื™ืคื•ืช ืžืกื” ืื•ืจื›ื™ืชโ€ฌ
โ€ซืืฉืจ ืงื“ืงื•ื“ื” ืžื•ื ื— ื‘ืจืืฉื™ืช?โ€ฌ
โ€ซื‘ืขืœืช ืฆืคื™ืคื•ืช ืžืกื” ืฉื˜ื—ื™ืชโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ื™ื›ืŸ ืžืจื›ื– ื”ืžืกื” ืฉืœ ื’ื–ืจื” ื‘ืขืœืช ืจื“ื™ื•ืก ื•ื–ื•ื™ืชโ€ฌ
โ€ซืžืชืงื‘ืœืช ื‘ืกืขื™ืฃ ื‘' ื”ืชื•ืฆืื” ืฉืœ ืกืขื™ืฃ ื'? ื”ืกื‘ืจโ€ช/โ€ฌื™โ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌื”ืื ื‘ื’ื‘ื•ืœโ€ฌ
โ€ซื•โ€ช? -โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌืžื”ื ื”ื™ื—ื™ื“ื•ืช ืฉืœโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_024.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซืžืขืœ ื”ืื•ืคืงโ€ช .โ€ฌื‘ื ืงื•ื“ื” ืžืกื•ื™ื™ืžืช ื‘ืžืกืœื•ืœื• ืžืชืคื•ืฆืฅ ื”ืคื’ื– ืœืฉื ื™ ืจืกื™ืกื™ืโ€ฌ
โ€ซืคื’ื– ื ื•ืจื” ื‘ืžื”ื™ืจื•ืช ื‘ื–ื•ื™ืชโ€ฌ
โ€ซืžื ืงื•ื“ืช ื”ื™ืจื™โ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืžื ืงื•ื“ืช ื”ื™ืจื™โ€ฌ
โ€ซโ€ช .โ€ฌื”ืจืกื™ืก ื”ืจืืฉื•ืŸ ืคื•ื’ืข ื‘ืงืจืงืข ื‘ืžืจื—ืงโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื‘ืขืœื™ ืžืกื•ืชโ€ฌ
โ€ซืคื•ื’ืข ื”ืจืกื™ืก ื”ืฉื ื™โ€ช ,โ€ฌืื ื ืชื•ืŸ ืฉืฉื ื™ ื”ืจืกื™ืกื™ื ืคื•ื’ืขื™ื ื‘ืงืจืงืข ื‘ื• ื–ืžื ื™ืช?โ€ฌ
โ€ซืโ€ช .โ€ฌืคืชื•ืจโ€ช/โ€ฌื™ ืžืฉื™ืงื•ืœื™ ืžืจื›ื– ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืคืชื•ืจโ€ช/โ€ฌื™ ืžืฉื™ืงื•ืœื™ ืชื ืขโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_041.htmlโ€ฌโ€ฌ
โ€ซื—ืฉื‘ ืืช ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื’ืœื™ืœ ืื—ื™ื“ ืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื– ื”ื’ืœื™ืœโ€ฌ
โ€ซโ€ช.โ€ฌื—ืฉื‘ ืืช ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืฆื•ืจื” ื‘ืื™ื•ืจ ื“ืจืš ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื– ื”ืžืกื” ื•ืžืงื‘ื™ืœ ืœืฆื™ืจ ื”ื’ืœื™ืœโ€ฌ
โ€ซโ€ช.โ€ฌื ืชื•ืŸ ื›ื™ ื”ืฆืคื™ืคื•ืช ืื—ื™ื“ื” โ€ช L.โ€ฌืื•ืจืš ื”ื’ืœื™ืœ โ€ช R/2.โ€ฌื•ืจื“ื™ื•ืก ื”ืงื“ื— โ€ช Rโ€ฌืจื“ื™ื•ืก ื”ื’ืœื™ืœ ื”ื•ืโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_042.htmlโ€ฌโ€ฌ
โ€ซื”ืžื‘ื ื” ื”ืžืชื•ืืจ ื‘ืื™ื•ืจ ืžื•ืจื›ื‘ ืžื•ืจื›ื‘ ืžืฉืœื•ืฉื” ื’ืœื’ืœื™ื ื–ื”ื™ื ืขืœโ€ช-โ€ฌื’ื‘ื™ ืžืฉื˜ื— ืื•ืคืงื™ืฅืžื”ื ืžืงื“ืžื™ ื”ื—ื™ื›ื•ืš ื”ืžื™ื ื™ืžืœื™ื™ื ื‘ื™ืŸ ื’ืœื™ืœ ืœื’ืœื™ืœ ื•ื‘ื™ืŸโ€ฌ
โ€ซื”ื’ืœื™ืœ ืœืžืฉื˜ื— ืขืœโ€ช-โ€ฌืžื ืช ืฉื”ืžื‘ื ื” ื™ืฉืืจ ื‘ืฉ"ืž ืกื˜ื˜ื™ ืจืžื–โ€ช :โ€ฌื™ืฉ ืœื‘ื—ื•ืจ ื‘ืงืคื™ื“ื” ืืช ื ืงื•ื“ืช ื”ืฆื™ืจโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_043.htmlโ€ฌโ€ฌ
โ€ซืžื˜ืคืก ื”ืจื™ื ื”ืฉื•ืงืœ โ€ช 55โ€ฌืง" ื’ื› ื ืข ืœืื•ืจืš ื—ืจื™ืฅ ืฉืจื•ื—ื‘ื• โ€ช 20โ€ฌืก"ืž ืข" ื™ ื›ืš ืฉื”ื•ื ืžื•ืฉืš ืขื ื™ื“ื™ื• ืืช ื“ื•ืคืŸ ื”ื—ืจื™ืฅ ื•ื“ื•ื—ืฃ ืขื ืจื’ืœื™ื• ืืช ื”ืงื™ืจโ€ฌ
โ€ซืฉืžื ื’ื“โ€ช .โ€ฌืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืžื˜ืคืก ื ืžืฆื ื‘ืžืจื—ืง โ€ช 40โ€ฌืก"ืž ืžืงืฆื” ื”ื—ืจื™ืฅโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ื™ื“ื™ื™ื ืœืกืœืข ื”ื•ื โ€ช 0.4โ€ฌื•ืžืงื“ืโ€ฌ
โ€ซโ€ช.โ€ฌื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ื ืขืœื™ื™ื ืœืกืœืข ื”ื•ื โ€ช1.2โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ื›ื•ื— ื”ืžื™ื ื™ืžืœื™ ืฉืขืœ ื”ืžื˜ืคืก ืœื“ื—ื•ืฃ ืขื ื”ืจื’ืœื™ื™ื ื•ืœืžืฉื•ืš ืขื ื”ื™ื“ื™ื™ื ืขืœโ€ช-โ€ฌืžื ืช ืฉื™ื”ื™ื” ื™ืฆื™ื‘โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ื”ืžืจื—ืง ื‘ื™ืŸ ื”ื™ื“ื™ื™ื ืœืจื’ืœื™ื™ื ืฉืœ ื”ืžื˜ืคืกโ€ช ,โ€ฌืขื‘ื•ืจ ื”ื›ื•ื— ื”ืžื•ืคืขืœ ื‘ืกืขื™ืฃ ื”ืงื•ื“ืโ€ฌ
โ€ซื’โ€ช .โ€ฌืื ื™ื•ืจื“ ื’ืฉื ื•ื”ืกืœืข ื ืจื˜ื‘โ€ช ,โ€ฌืžืงื“ืžื™ ื”ื—ื™ื›ื•ืš ืงื˜ื ื™ืโ€ช .โ€ฌื›ื™ืฆื“ ื”ื“ื‘ืจ ื™ืฉืคื™ืข ืขืœ ืชื•ืฆืื•ืช ื”ืกืขื™ืคื™ื ื”ืงื•ื“ืžื™ืโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_2_044.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซื—ืฉื‘ื• ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ื’ื•ืคื™ื ื”ื‘ืื™ืโ€ช ,โ€ฌื›ื•ืœื ื‘ืขืœื™ ืฆืคื™ืคื•ืช ืžืกื” ืื—ื™ื“ื”โ€ช :โ€ฌืโ€ช .โ€ฌื—ืฆื™ ืงืฉืช ืฉืœ ืžืขื’ืœโ€ช .โ€ฌื‘โ€ช .โ€ฌื—ืฆื™ ืžืขื’ืœโ€ช .โ€ฌื’โ€ช .โ€ฌืกืคื™ืจื” ืฉื”ื•ื—ืกืจื”โ€ฌ
โ€ซโ€ช.โ€ฌืžืžื ื” ืกืคื™ืจื” ื‘ืขืœืช ื—ืฆื™ ืžื”ืจื“ื™ื•ืก ืฉืžื™ืงื•ืžื” ื ืžืฆื ื‘ืžืจื—ืง ื—ืฆื™ ืจื“ื™ื•ืก ืžืžืจื›ื– ื”ืžืขื’ืœโ€ฌ
โ€ซโ€ช1โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_222.htmlโ€ฌโ€ฌ
โ€ซื”ืชื ื’ืฉื•ืชโ€ฌ
โ€ซื‘ืžืขื‘ื“ื” ื—ืœืงื™ืง ื‘ืขืœ ืžืกื” โ€ช 3.16โ€ฌืง"ื’ ื ืข ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 15.6โ€ฌืžื˜ืจ ืœืฉื ื™ื™ื” ืฉืžืืœื” ื•ืžืชื ื’ืฉ ื—ื–ื™ืชื™ืช ื‘ื—ืœืงื™ืง ื”ื ืข ื™ืžื™ื ื” ื‘ืžื”ื™ืจื•ืช โ€ช12.2โ€ฌโ€ฌ
โ€ซื•ืžืกืชื• โ€ช 2.84โ€ฌืง"ื’โ€ช .โ€ฌืžืฆื ืืช ืžื”ื™ืจื•ืช ืžืจื›ื– ื”ืžืกื” ืื—ืจื™ ื”ื”ืชื ื’ืฉื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_223.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ื”ืžืกื” ืฉืœ ื—ืจื•ื˜โ€ฌ
โ€ซืžืฆื ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื—ืจื•ื˜ ืฉืžืกืชื• โ€ช Mโ€ฌื’ื•ื‘ื”ื• โ€ช Hโ€ฌื•ืจื“ื™ื•ืก ื‘ืกื™ืกื• โ€ช.Rโ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_224.htmlโ€ฌโ€ฌ
โ€ซืžืฆื ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื—ืจื•ื˜ ืฉืžืกืชื• โ€ช Mโ€ฌื’ื•ื‘ื”ื• โ€ช Hโ€ฌื•ืจื“ื™ื•ืก ื‘ืกื™ืกื• โ€ช.Rโ€ฌื“ืจืš ืžืฉืคืš )ืงื‘ื•ืข ื‘ืžืงื•ื ( ืฉื•ืคื›ื™ื ื—ื•ืœโ€ช ,โ€ฌื‘ืงืฆื‘ ืงื‘ื•ืข ืฉืœโ€ฌ
โ€ซืœืฉื ื™ื™ื” ืขืœ ืกืจื˜ ื ืข ืฉืžืกืชื• โ€ช Mโ€ฌืง"ื’โ€ช .โ€ฌืžื” ื’ื•ื“ืœ โ€ช Fโ€ฌืฉืœ ื”ื›ื•ื— ืฉื™ืฉ ืœื”ืคืขื™ืœ ืขืœ ื”ืกืจื˜ ืขืœ ืžื ืช ืœื”ื‘ื˜ื™ื— ืฉืžื”ื™ืจื•ืชื• ืชื™ืฉืืจ ืงื‘ื•ืขื”?โ€ฌ
โ€ซืง "ื’โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_225.htmlโ€ฌโ€ฌ
โ€ซืžืจื›ื– ืžืกื”โ€ฌ
โ€ซืžื”ื• ืžืจื›ื– ื”ืžืกื” ืฉืœ ื—ืฆื™ ืžืขื’ืœ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_226.htmlโ€ฌโ€ฌ
โ€ซืžื•ืžื ื˜ ื”ืชืžื“โ€ฌ
โ€ซืžื”ื• ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื’ืœื™ืœ ื‘ืขืœ ืžืกื” โ€ช ,mโ€ฌื’ื•ื‘ื” โ€ช ,hโ€ฌืจื“ื™ื•ืก ื‘ืกื™ืก โ€ช Rโ€ฌื”ืžืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื—ืง โ€ช dโ€ฌืžืžืจื›ื– ืžืกืชื•?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_227.htmlโ€ฌโ€ฌ
โ€ซืžื•ื‘ืจื™ื ื•ืคื™ืกื™ืงื”โ€ฌ
โ€ซื›ืืฉืจ ืžืขืœื™ื ืกืคื” ื‘ืžืขืœื” ื”ืžื“ืจื’ื•ืชโ€ช ,โ€ฌืžื™ ืžืชืืžืฅ ื™ื•ืชืจโ€ช ,โ€ฌื–ื” ืฉืœืžื˜ื” ืื• ื–ื” ืฉืœืžืขืœื”?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_228.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื ืชื•ืŸ ืžื•ื˜ ื‘ืขืœ ืื•ืจืš โ€ชdโ€ฌื•ืžืกื” โ€ช m1โ€ฌื”ืžื•ื ื— ื‘ืื•ืคืŸ ืื•ืคืงื™ ืข"ื’ ืฉื•ืœื—ืŸโ€ช ,โ€ฌื›ืืฉืจ ื”ืžื•ื˜ ืžื—ื•ื‘ืจ ื‘ืงืฆื”ื• ื”ืื—ื“ ืœื“ื•ืคืŸ ื”ืฉื•ืœื—ืŸ ื‘ืขื–ืจืช ืฆื™ืจโ€ช .โ€ฌื‘ืจื’ืขโ€ฌ
โ€ซืžืกื•ื™ื ืคื•ื’ืข ื‘ืงืฆื” ื”ืžื•ื˜ ื›ื“ื•ืจ ื‘ืขืœ ืžืกื” โ€ช m2โ€ฌื•ืžื”ื™ืจื•ืช โ€ช .V0โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ืงื•ื•ื™ืช ืฉืœ ื”ื›ื“ื•ืจ ืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืช )ื”ืืœืกื˜ื™ืช(โ€ช ,โ€ฌื•ืžื”ื™โ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื•ื˜?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_3_229.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื ืชื•ืŸ ื›ื“ื•ืจ ื‘ืขืœ ืžืกื” ืฉืœ โ€ช 3.2โ€ฌืง"ื’ ื•ืจื“ื™ื•ืก โ€ช 12โ€ฌืก"ืžโ€ช ,โ€ฌื”ืžืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจื• ื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืช ืฉืœ โ€ช 15โ€ฌืจื“ื™ืืŸ ืœืฉื ื™ื™ื” ื ื’ื“ ื›ื™ื•ื•ืŸ ื”ืฉืขื•ืŸโ€ช .โ€ฌื›ื“ื•ืจโ€ฌ
โ€ซโ€ช .โ€ฌืœืื—ืจ ื›ืžื” ื–ืžืŸ ื™ืชื—ื™ืœ ื”ื›ื“ื•ืจ ืœื”ืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื”? ืžื”โ€ฌ
โ€ซื–ื” ืžื•ื ื— ืขืœ ืžืฉื˜ื— ืืฉืจ ื‘ื™ื ื• ืœื‘ื™ืŸ ื”ื›ื“ื•ืจ ืงื™ื™ื ืžืงื“ื ื—ื™ื›ื•ืš ืฉืœโ€ฌ
โ€ซืชื”ื™ื” ืžื”ื™ืจื•ืชื•?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_001.htmlโ€ฌโ€ฌ
โ€ซื—ื™ืฉื•ื‘ ืžืจื›ื– ืžืกื”โ€ฌ
โ€ซื ืชื•ื ื•ืช ื”ืฆื•ืจื•ืช ื”ืคืฉื•ื˜ื•ืช ื”ื‘ืื•ืชโ€ช:โ€ฌโ€ฌ
โ€ซื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืžืจื›ื– ื”ืžืกื” ื”ื’ื•ืคื™ื ื”ื "ืœโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_002.htmlโ€ฌโ€ฌ
โ€ซืžื•ืžื ื˜ ื”ืชืžื“ โ€ช2Dโ€ฌโ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื˜ืจื™ื– ืขื‘ื•ืจ ื”ืฆื™ืจื™ื ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซื( ื”ืฆื™ืจ ื”ืกื™ื‘ื•ื‘ )ืฆื™ืจ โ€ช (zโ€ฌืขื•ื‘ืจ ื“ืจืš ืžืจื›ื– ื”ืžืกื”โ€ช:โ€ฌโ€ฌ
โ€ซ)ื”ืฆื™ืจ ื ื™ืฆื‘ ืœืžื™ืฉื•ืจ ืฉืœ ื”ื˜ืจื™ื–(โ€ฌ
โ€ซื‘( ื”ืฆื™ืจ ืขื•ื‘ืจ ื“ืจืš ื”ื‘ืกื™ืก )ืžืงื‘ื™ืœ ืœืฆื™ืจ โ€ช:(yโ€ฌโ€ฌ
โ€ซ)ื”ืฆื™ืจ ื ืžืฆื ื‘ืชื•ืš ื”ืžื™ืฉื•ืจ ืฉืœ ื”ื˜ืจื™ื–(โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_004.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ืœืช ืžืกื™ื‘ื™ืชโ€ฌ
โ€ซื ืชื•ื ื” ืงืœื™ืคื” ื›ื“ื•ืจื™ืช ื”ืžืกืชื•ื‘ื‘ืช ืกื‘ื™ื‘ ืฆื™ืจ ืžืจื›ื– ืžืกื” ืขืœ ื’ื‘ื™ ืžื™ืกื‘ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌืขืœ ื”ืงืœื™ืคื” ืžืœื•ืคืฃ ื—ื‘ืœ ืงืœ ื”ืขื•ื‘ืจ )ืœืœื ื”ื—ืœืงื”!( ื“ืจืšโ€ฌ
โ€ซื’ืœื’ืœืชโ€ฌ
โ€ซ)ื‘ืขืœืช ืžืกื”!( ื•ืžื—ื•ื‘ืจ ืœื’ื•ืฃ โ€ช m2โ€ฌื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื( ืžื” ืžื”ื™ืจื•ืชื• ืฉืœ ื’ื•ืฃ โ€ช m2โ€ฌืœืื—ืจ ืฉืขื‘ืจ ื’ื•ื‘ื” โ€ช?hโ€ฌโ€ฌ
โ€ซื‘( ืžื” ื”ื™ื ืชืื•ืฆืชื•?โ€ฌ
โ€ซื’( ืžื” ื”ืžืชื™ื—ื•ืช ื‘ื—ื‘ืœ )ื‘ื—ืœืงื™ื• ื”ืฉื•ื ื™ื(?โ€ฌ
โ€ซื ืชื•ื ื™ื ื ื•ืกืคื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืžืกืช ื”ืงืœื™ืคื” โ€“ โ€ช , m1โ€ฌืจื“ื™ื•ืก ื”ืงืœื™ืคื” โ€“โ€ช.Rโ€ฌื ื™ืชืŸ ืœืชืืจ ืืช ื”ื’ืœื’ืœืช ื›โ€ช 3-โ€ฌื’ืœื™ืœื™ื ื”ื–ื”ื™ื ื‘ืžืกืชื โ€ช ,m3โ€ฌื›ืืฉืจ ื”ืžื‘ื ื” ื”ื•ื ืฉืฉื ื™ ื”ื’ืœื™ืœื™ืโ€ฌ
โ€ซื”ื—ื™ืฆื•ื ื™ื™ื ื”ื ื‘ืขืœื™ ืจื“ื™ื•ืก ื’ื“ื•ืœ โ€ช r3โ€ฌื•ื”ื’ืœื™ืœโ€ฌ
โ€ซืฉื‘ื ื™ื”ื ื”ื•ื ื‘ืขืœ ืจื“ื™ื•ืก โ€ช.r2โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_005.htmlโ€ฌโ€ฌ
โ€ซืžื•ืžื ื˜ื™ื ื•ืงื˜ื™ืœืช ื“ืจืงื•ื ื™ืโ€ฌ
โ€ซื“ืจื™ื™ืง ืงื•ื˜ืœ ื”ื“ืจืงื•ื ื™ื ืžื˜ื™ื™ืœ ื‘ื˜ื™ืจื”โ€ช .โ€ฌืœืคืชืขโ€ช ,โ€ฌื‘ืงืฆื” ื”ืžืกื“ืจื•ืŸ )ื‘ืžืจื—ืง โ€ช 30โ€ฌืžื˜ืจื™ื( ืจื•ืื” ื“ืจื™ื™ืง ื“ืจืงื•ืŸโ€ช .โ€ฌืžื™ื™ื“ ืคื•ื ื” ื“ืจื™ื™ืง ื•ื ืžืœื˜ ืœื—ื“ืจโ€ฌ
โ€ซื”ืงืจื•ื‘ ื‘ื™ื•ืชืจ )ื”ื•ื ืœื ื”ื™ื” ืงื•ื˜ืœ ื›"ื› ืžื•ืฆืœื—(โ€ช ,โ€ฌืฉื“ืœืชื” ืคืชื•ื—ื” ื‘โ€ช .90º-โ€ฌื“ืจื™ื™ืง ืžืชืืžืฅ ืœืกื’ื•ืจ ืืช ื”ื“ืœืช ื‘ืžื”ื™ืจื•ืช โ€ช ,โ€ฌืœื›ืŸ ื”ื•ื ืžืคืขื™ืœ ืืช ื›ืœโ€ฌ
โ€ซื›ื•ื—ื• )โ€ช (F=650Nโ€ฌื‘ืงืฆื” ื”ื“ืœืช )ืจื•ื—ื‘ ื”ื“ืœืช ืžื˜ืจ ื•ื—ืฆื™(โ€ช .โ€ฌื‘ื ื•ืกืฃ ื™ื“ื•ืข ื›ื™ ื’ื•ื‘ื” ื”ื“ืœืช โ€ช 3โ€ฌืžื˜ืจื™ืโ€ช ,โ€ฌืžืกืชื” โ€ช kg 850โ€ฌื•ืžื”ื™ืจื•ืช ื”ื“ืจืงื•ืŸ โ€ช45โ€ฌโ€ฌ
โ€ซืงืž"ืฉโ€ช.โ€ฌโ€ฌ
โ€ซื”ืื ื“ืจื™ื™ืง ื™ื”ืคื•ืš ืœืืจื•ื—ืช ืฆื”ืจื™ื™ื )ื”ืื ื”ื•ื ื™ืกืคื™ืง ืœืกื’ื•ืจ ืืช ื”ื“ืœืช ืœืคื ื™ ืฉื”ื“ืจืงื•ืŸ ื™ื’ื™ืข(? )ื”ื ื—ื™ื™ื”โ€ช :โ€ฌื™ืฉ ืœื—ืฉื‘ ืืช ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœโ€ฌ
โ€ซื”ื“ืœืช(โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_006.htmlโ€ฌโ€ฌ
โ€ซื˜ืจื™ื‘ื•ืฉื”โ€ฌ
โ€ซื”ื˜ืจื™ื‘ื•ืฉื” )โ€ช (Trebuchetโ€ฌื”ื™ื ืžื›ื•ื ืช ืžืฆื•ืจ ืžื™ืžื™ ื”ื‘ื™ื ื™ื™ืโ€ช .โ€ฌื‘ื™ื›ื•ืœืชื” ื”ื™ื” ืœื”ืขื™ืฃ ืกืœืขื™ื ื‘ืžืฉืงืœ ืขืฉืจื•ืช ืง"ื’ ืœืžืจื—ืง ื›ืžื” ืžืื•ืช ืžื˜ืจื™ืโ€ฌ
โ€ซื‘ื“ื™ื•ืง ืจื‘โ€ช.โ€ฌโ€ฌ
โ€ซื( ืชืืจโ€ช/โ€ฌื™ ืืช ืื•ืคืŸ ืคืขื•ืœืชื” ืฉืœ ื”ื˜ืจื™ื‘ื•ืฉื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืœืžื” ื”ืงืœื™ืข ืžื—ื•ื‘ืจ ื‘ื—ื‘ืœ ืœืžื•ื˜ ื”ืชื ื•ืคื” ื•ืœื ื ืžืฆื ื™ืฉื™ืจื•ืช ืขืœื™ื•?โ€ฌ
โ€ซื’( ืขืœื™ืš ืœืชื›ื ืŸ ื˜ืจื™ื‘ื•ืฉื” ) ืœืžืฆื•ื ืืช ื”ืžืกืช ื”ืชื ื•ืคื” โ€ช (Mโ€ฌื”ืžืกื•ื’ืœืช ืœื”ืขื™ืฃ ืกืœืข ื‘ืžืฉืงืœ โ€ช 40โ€ฌืง"ื’ ืœืžืจื—ืง โ€ช 300โ€ฌืžื˜ืจื™ืโ€ช .โ€ฌื‘ื™ื“ืš ืžื•ื˜ ืชื ื•ืคื”โ€ฌ
โ€ซื‘ืื•ืจืš โ€ช 28โ€ฌืžื˜ืจื™ื )ื”ื™ื—ืก ื”ืจืฆื•ื™ ื‘ื™ืŸ ื”ื—ืœืง ื”ืงืฆืจ ืœืืจื•ืš ื”ื•ื โ€ช (6:1โ€ฌื‘ืžืฉืงืœ โ€ช 100โ€ฌืง"ื’โ€ช .โ€ฌื‘ื ื•ืกืฃโ€ช ,โ€ฌื™ื“ื•ืข ืฉื”ื’ื•ื‘ื” ื”ืชื—ืœืชื™ ืฉืœ ื”ืงืœืข ื’ื“ื•ืœ ืคื™ โ€ช4โ€ฌโ€ฌ
โ€ซืžื”ื’ื•ื‘ื” ื”ืกื•ืคื™ ืžืกืช ื”ืชื ื•ืคื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_007.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ื•ืœ ื›ื“ื•ืจื™ืโ€ฌ
โ€ซื ืชื•ื ื™ื ืฉื ื™ ื›ื“ื•ืจื™ื ื‘ืขืœื™ ืžืกื” ื–ื”ื” โ€ช ,mโ€ฌื”ืžืชื’ืœื’ืœื™ื ืœืœื ื”ื—ืœืงื” ื‘ืžื”ื™ืจื•ืช โ€ช .vโ€ฌื‘ื ืงื•ื“ื” ืžืกื•ื™ืžืช ื›ืœ ืื—ื“ ืžื”ื›ื“ื•ืจื™ื ื ืชืงืœ ื‘ืžื“ืจื•ืŸโ€ช .โ€ฌื”ืื—ื“โ€ฌ
โ€ซื—ืœืง ืœื’ืžืจื™ ื•ื”ืฉื ื™ ืœืโ€ช .โ€ฌืžื™ ืžื‘ื™ืŸ ื”ื›ื“ื•ืจื™ื ื™ืขืœื” ื’ื‘ื•ื” ื™ื•ืชืจ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_008.htmlโ€ฌโ€ฌ
โ€ซื”ืชื ื’ืฉื•ื™ื•ืช ื‘ื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื ืชื•ืŸ ืžื•ื˜ ืื•ืคืงื™ ื”ืžื•ื ื— โ€ช m1โ€ฌืขืœ ืฉื•ืœื—ืŸ ื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌื’ื•ืฃ ื ืงื•ื“ืชื™ ื‘ืขืœ ืžืกื” โ€ช m2โ€ฌืคื•ื’ืข ื‘ื• ื‘ืžืจื—ืง โ€ช dโ€ฌืžืžืจื›ื–ื• ื‘ื”ืชื ื’ืฉื•ืช ืืœืกื˜ื™ืช ืœื—ืœื•ื˜ื™ืŸโ€ช.โ€ฌโ€ฌ
โ€ซื( ืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื• ื”ืงื•ื•ื™ืช ื•ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื•ื˜? )ื”ืžื”ื™ืจื•ืช ื”ืกื•ืคื™ืช ืฉืœ โ€ช m2โ€ฌื”ื™ื ืœื โ€ช(!0โ€ฌโ€ฌ
โ€ซื‘( ื”ื ื—โ€ช/โ€ฌื™ ื›ื™ ื”ื”ืชื ื’ืฉื•ืช ื”ื™ื ืคืœืกื˜ื™ืช ืœื—ืœื•ื˜ื™ืŸโ€ช .โ€ฌืžื” ืชื”ื™ื” ืžื”ื™ืจื•ืชื• ื”ืงื•ื•ื™ืช ื•ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื•ื˜?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_009.htmlโ€ฌโ€ฌ
โ€ซื‘ื™ืœื™ืืจื“โ€ฌ
โ€ซื ืชื•ืŸ ื›ื“ื•ืจ ื‘ื™ืœื™ืืจื“ ื”ื ืžืฆื ื‘ืžื ื•ื—ื”โ€ช .โ€ฌื”ื›ื“ื•ืจ ืžืงื‘ืœ ืžื›ื” ื—ื“ื” ืžืกื˜ืงื” ื”ื ืžืฆืืช ื‘ืžืฆื‘ ืื•ืคืงื™โ€ช ,โ€ฌื‘ื’ื•ื‘ื” โ€ช hโ€ฌืžืžืจื›ื– ื”ื›ื“ื•ืจโ€ช .โ€ฌืฆื•ืจืช ืžื›ื” ืกืคืฆื™ืคื™ืชโ€ฌ
โ€ซื–ื• ื™ื•ืฆืจืช ืืคืงื˜ ื”ื ืงืจื โ€œโ€ช :โ€Forward Englishโ€ฌื”ื›ื“ื•ืจ ืžืชื—ื™ืœ ืœื ื•ืข ื‘ืžื”ื™ืจื•ืช โ€ช ,V0โ€ฌื•ื‘ื’ืœืœ ื”ืืคืงื˜ ื”ื•ื ืžื’ื™ืข ื‘ืกื•ืคื• ืฉืœ ืชื”ืœื™ืš ืœืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืกื•ืคื™ืชโ€ฌ
โ€ซื”ื•ื›ื—โ€ช/โ€ฌื™ ื›ื™โ€ฌ
โ€ซโ€ช ,โ€ฌื›ืืฉืจ โ€ช Rโ€ฌื”ื•ื ืจื“ื™ื•ืก ื”ื›ื“ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_010.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ื•ืœ ื‘ืžื“ืจื•ืŸโ€ฌ
โ€ซื ืชื•ืŸ ื’ืœื™ืœ ื—ืœื•ืœ ื”ืžืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื” ืžืจืืฉ ืžื™ืฉื•ืจ ืžืฉื•ืคืข ื‘ื’ื•ื‘ื” โ€ช .Hโ€ฌืœืื—ืจ ืฉื”ื’ืœื™ืœ ืžื’ื™ืข ืœืงืฆื” ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข ื”ื•ื ื ื•ืคืœ ื‘ื ืคื™ืœื”โ€ฌ
โ€ซื—ื•ืคืฉื™ืช ืœืื•ืจืš ื’ื•ื‘ื” โ€ช .hโ€ฌื‘ืฉืœื‘ ื–ื” ื”ื’ืœื™ืœ ืžื’ื™ืข ืœืžื™ืฉื•ืจ ืžืฉื•ืคืข ื‘ื–ื•ื•ื™ืช โ€ช ฮธโ€ฌื”ืฉื•ื•ื” ืœื–ื•ื•ื™ืช ื”ืจื’ืขื™ืช ื‘ื” ื”ื•ื ื ืžืฆื ื‘ืจื’ืข ื”ื’ืขืชื• ืœืžื™ืฉื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื( ืื™ืš ืชื™ืจืื” ืชื ื•ืขืช ื”ื’ืœื™ืœ ืขืœ ื”ืžื™ืฉื•ืจ ื”ืžืฉื•ืคืข ื”ืฉื ื™ โ€“ ื”ืื ื”ื•ื ื™ื—ืœื™ืง ืื• ื™ืชื’ืœื’ืœ?โ€ฌ
โ€ซื‘( ื”ืื ืชื ื•ืขื” ื–ื• ืชื™ืฉืืจ ื™ืฆื™ื‘ื”? ืื ืœืโ€ช ,โ€ฌืœื›ืžื” ื–ืžืŸ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_011.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ื•ืœ ืคื—ื™ืชโ€ฌ
โ€ซืคื—ื™ืช ืจื™ืงื” ืฉืžืกืชื” โ€ช m1โ€ฌืžืชื’ืœื’ืœืช ืœืœื ื”ื—ืœืงื” ืขืœ ืžื™ืฉื•ืจ ืื•ืคืงื™ ื‘ืžื”ื™ืจื•ืช โ€ช .vโ€ฌื”ืคื—ื™ืช ืžืœืื” ื‘ืงืจื— ื™ื‘ืฉ )ืžื•ืฆืง( ืฉืžืกืชื• โ€ช .m2โ€ฌื‘ืžืจื›ื–ื™โ€ฌ
โ€ซื”ื‘ืกื™ืกื™ื ืฉืœ ื”ืคื—ื™ืช ื™ืฉ ื—ื•ืจื™ื ืงื˜ื ื™ื ืฉื“ืจื›ื ืžืชืื“ื” ื”ืงืจื— ื”ื™ื‘ืฉโ€ช .โ€ฌื”ื ื—โ€ช/โ€ฌื™ ืฉื”ืงืจื— ืžืชืื“ื” ืžื”ืจ ืžืื•ื“โ€ช.โ€ฌโ€ฌ
โ€ซื( ืื™ืœื• ื—ื•ืงื™ ืฉื™ืžื•ืจ ืžืชืงื™ื™ืžื™ื ื‘ืžืฉืš ื”ืชื ื•ืขื”? ื ืžืงโ€ช/โ€ฌื™โ€ฌ
โ€ซื‘( ืžื” ื”ื™ื ืžื”ื™ืจื•ืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืคื—ื™ืช ืžื™ื“ ืœืื—ืจ ืฉื›ืœ ื”ืงืจื— ื”ื™ื‘ืฉ ื”ืชืื“ื”?โ€ฌ
โ€ซื’( ืžื” ื”ื™ื ื”ืžื”ื™ืจื•ืช ื”ืกื™ื‘ื•ื‘ื™ืช ืฉืœ ื”ืคื—ื™ืช ื‘ืื•ืชื• ื–ืžืŸ?โ€ฌ
โ€ซื“( ื”ืื ื‘ืื•ืชื• ื–ืžืŸ ืชื ื•ืขืชื” ื ืฉืืจืช ื›ื’ืœื’ื•ืœ ืœืœื ื”ื—ืœืงื”?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_012.htmlโ€ฌโ€ฌ
โ€ซื—ืจื•ื˜ ืžืชื’ืœื’ืœโ€ฌ
โ€ซื ืชื•ืŸ ื—ืจื•ื˜ ื‘ืขืœ ืจื“ื™ื•ืก โ€ช ,Rโ€ฌืžืกื” โ€ช ,Mโ€ฌื–ื•ื•ื™ืช ืคืชื™ื—ื” โ€ช 2ฮธโ€ฌื•ื’ื•ื‘ื” โ€ช .hโ€ฌื”ื—ืจื•ื˜ ืžืชื’ืœื’ืœ ืขืœ ื’ื‘ื™ ืฉื•ืœื—ืŸโ€ช ,โ€ฌื›ืืฉืจ ื‘ืกื™ืกื• ืžืื•ื ืš ืœืฉื•ืœื—ืŸ ื›ืžืชื•ืืจ ื‘ืื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ื—ืจื•ื˜ ื”ื™ื โ€ช .โ„ฆโ€ฌื‘ื˜ืโ€ช/โ€ฌื™ ืืช ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื”ื›ืœืœื™ืช ืฉืœ ื”ืงื•ื ื•ืก ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ืคืจืžื˜ืจื™ื ื”ื "ืœโ€ช.โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌืฆืจื™ืš ืœื”ืฉืชืžืฉ ื‘ืขื•ื‘ื“ื” ืฉื”ื—ืจื•ื˜ ืžืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื”โ€ช ,โ€ฌื›ืœื•ืžืจ ืฉื™ืฉ ื”ืชืืžื” ื‘ื™ืŸ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ื—ืจื•ื˜ ืœืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืชโ€ฌ
โ€ซืฉืœ ื”ื‘ืกื™ืก )โ€ช:(ฯ‰โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_017.htmlโ€ฌโ€ฌ
โ€ซื’ืฉืจ ืขื ืžืฉืื™ืชโ€ฌ
โ€ซื’ืฉืจ ืฉืื•ืจื›ื•โ€ฌ
โ€ซืžืฉืื™ืช ืฉืžืกืชื”โ€ฌ
โ€ซื ืชืžืš ื‘ื›ืœ ืงืฆื” ื›ืžืชื•ืืจ ื‘ืฆื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซื•ืžืกืชื•โ€ฌ
โ€ซื ืžืฆืืชโ€ฌ
โ€ซืžืงืฆื” ื”ื’ืฉืจโ€ช.โ€ฌโ€ฌ
โ€ซืžื”ื ื”ื›ื•ื—ื•ืช ื”ืคื•ืขืœื™ื ื‘ื›ืœ ืื—ืช ืžื ืงื•ื“ื•ืช ื”ืชืžื™ื›ื” ืฉื‘ืงืฆื•ื•ืช ื”ื’ืฉืจ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_018.htmlโ€ฌโ€ฌ
โ€ซืžื•ื˜ ื•ืขืœื™ื• ืžืกื•ืชโ€ฌ
โ€ซื‘ืžื™ืงื•ืžื™ื ืฉืžืชื•ืืจื™ื ื‘ืฆื™ื•ืจโ€ช .โ€ฌื”ืžื•ื˜ ื ืฉืขืŸ ื‘ืžื ื•ื—ื” ื‘ืฉืชื™ ื ืงื•ื“ื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื•ืื•ืจืš ืชื•ืžืš ืฉื ื™ ืžืกื•ืชโ€ฌ
โ€ซืžื•ื˜ ืื—ื™ื“ ืขื ืžืกื”โ€ฌ
โ€ซืขื‘ื•ืจ ืื™ื–ื” ืขืจืš ืฉืœ โ€ช xโ€ฌื”ืžื•ื˜ ื™ื”ื™ื” ืžืื•ื–ืŸ ื›ื•ืœื• ืขืœ ื ืงื•ื“ืช ืžืฉืขืŸ โ€ช) Pโ€ฌื›ื— ื”ื ื•ืจืžืœ ื‘ื ืง' โ€ช Oโ€ฌื™ื”ื™ื” ืฉื•ื•ื” ืœืืคืก(?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_019.htmlโ€ฌโ€ฌ
โ€ซื”ื‘ื•ื˜ื™ืง ืฉืœ ืœื•ืœื•โ€ฌ
โ€ซืฉืœื˜ ื‘ืฆื•ืจืช ื—ืฆื™ ืกืคื™ืจื” ื‘ืขืœืช ืงื•ื˜ืจโ€ฌ
โ€ซื•ืฆืคื™ืคื•ืช ืžืกื” ืื—ื™ื“ื” ื ืชืœื” ืžืฉื ื™ ื—ื•ื˜ื™ื ื›ืžืชื•ืืจ ื‘ืฆื™ื•ืจโ€ช .โ€ฌืžื”ืŸ ื”ืžืชื™ื—ื•ื™ื•ืช ื‘ื—ื•ื˜ื™ื?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_020.htmlโ€ฌโ€ฌ
โ€ซืžื•ืžื ื˜โ€ฌ
โ€ซื—ื‘ืœ ื—ืกืจ ืžืกื” ืื™ื ืกื•ืคื™ ืžืœื•ืคืฃ ืกื‘ื™ื‘ ืฉื ื™ ื’ืœื™ืœื™ื ื‘ืขืœื™โ€ฌ
โ€ซื‘ื”ืชืืžื”โ€ช.โ€ฌโ€ฌ
โ€ซื•โ€ฌ
โ€ซื•ืจื“ื™ื•ืกื™ืโ€ฌ
โ€ซื•โ€ฌ
โ€ซืžืกื•ืชโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื ืชืื•ืฆื•ืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืžืกื•ืช?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื” ื”ืชืื•ืฆื•ืช ื”ืจื“ื™ืืœื™ื•ืช ืฉืœ ื”ืžืกื•ืช?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_021.htmlโ€ฌโ€ฌ
โ€ซื’ืœื’ื•ืœโ€ฌ
โ€ซื“ื™ืกืงื” ืžืœืื” ื‘ืขืœืช ืจื“ื™ื•ืกโ€ฌ
โ€ซื‘ืžื•ืจื“ ืฉื™ืคื•ืข ื‘ื–ื•ื•ื™ืช โ€ช.โ€ฌโ€ฌ
โ€ซืžืฆื ืืช ืชืื•ืฆืช ืžืจื›ื– ื”ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื•ืžืกื”โ€ฌ
โ€ซืžืชื’ืœื’ืœืช ืœืœื ื”ื—ืœืงื”โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_022.htmlโ€ฌโ€ฌ
โ€ซื›ื“ื•ืจ ื’ืœื’ืœืช ื•ืžืกื”โ€ฌ
โ€ซื™ื›ื•ืœ ืœื”ืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ืื ื›ื™ ืœืจืฆืคื” ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–ื•โ€ช ,โ€ฌื‘ื”ืฉืคืขืช ื—ื‘ืœ ื”ื›ืจื•ืšโ€ฌ
โ€ซื•ืจื“ื™ื•ืกื•โ€ฌ
โ€ซื›ื“ื•ืจ ืฉืžืกืชื•โ€ฌ
โ€ซื•ืจื“ื™ื•ืกื”โ€ฌ
โ€ซืขืœื™ื•โ€ช .โ€ฌื”ื—ื‘ืœ ืงืฉื•ืจ ื“ืจืš ื’ืœื’ืœืช )ื’ืœื™ืœ ืžืœื( ืฉืžืกืชื”โ€ฌ
โ€ซื”ื™ื›ื•ืœื” ืœื”ืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ืกื™ื‘ื•ื‘ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–ื” ืœืชื™ื‘ื” ืฉืžืกืชื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌื›ืžื• ื›ืŸ ื ืชื•ืŸ ืฉื”ื—ื‘ืœ ืื™ื ื• ืžื—ืœื™ืง ืขืœ ื”ื›ื“ื•ืจ ื•ืขืœ ื”ื’ืœื’ืœืชโ€ฌ
โ€ซื ืชื•ืŸ ื›ื™ ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ืฉืœ ื›ื“ื•ืจ ื‘ื™ื—ืก ืœืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–ื•โ€ฌ
โ€ซื•ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื’ืœื™ืœ ืžืœื ื‘ื™ื—ืก ืœืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–ื• ื”ื•ืโ€ฌ
โ€ซื”ื•ืโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌืโ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืชื• ื”ื–ื•ื™ืชื™ืช ืฉืœ ื”ื›ื“ื•ืจโ€ฌ
โ€ซโ€ช.โ€ฌื‘โ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืชื” ื”ื–ื•ื™ืชื™ืช ืฉืœ ื”ื’ืœื’ืœืชโ€ฌ
โ€ซโ€ช.โ€ฌื’โ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ืžืกื”โ€ฌ
โ€ซโ€ช.โ€ฌื“โ€ช .โ€ฌืžื”ื™ ื”ืžืชื™ื—ื•ืช ื‘ื›ืœ ืื—ื“ ืžื”ื—ื‘ืœื™ืโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_023.htmlโ€ฌโ€ฌ
โ€ซืžืกื” ื’ืœื’ืœืช ื•ืงืคื™ืฅโ€ฌ
โ€ซืžืกื” ืงืฉื•ืจื” ื‘ื—ื•ื˜ ื—ืกืจ ืžืกื” ืœืงืคื™ืฅ ืขื ืงื‘ื•ืข ืงืคื™ืฅโ€ฌ
โ€ซื•ืฆื•ืจื” ื“ื™ืกืงื” ื‘ืจื“ื™ื•ืกโ€ฌ
โ€ซื”ื—ื‘ืœ ื ืžืฆื ื‘ืžื’ืข ืขื ื’ืœื’ืœืช ืฉืžืกืชื”โ€ฌ
โ€ซโ€ช.โ€ฌื”ื™ื›ื•ืœื” ืœื”ืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ืกื™ื‘ื•ื‘ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื–ื”โ€ฌ
โ€ซโ€ช.โ€ฌื‘ืžื”ืœืš ื”ืชื ื•ืขื” ืœื ืžื—ืœื™ืง ื”ื—ื‘ืœ ืขืœ ื’ื‘ื™ ื”ื’ืœื’ืœืชโ€ฌ
โ€ซโ€ช.โ€ฌืžืฉื—ืจืจื™ื ืืช ื”ืžืกื” ืžืžื ื•ื—ื” ื›ืืฉืจ ื”ืงืคื™ืฅ ืจืคื•ื™โ€ฌ
โ€ซโ€ช .โ€ฌืžืฆื ืžื”ื™ืจื•ืช ื”ืžืกื” ืœืื—ืจ ืฉื™ืจื“ื” ืžืจื—ืงโ€ฌ
โ€ซืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื“ื™ืกืงื” ืžืœืื” ื‘ื™ื—ืก ืœืฆื™ืจ ืกื™ื‘ื•ื‘ ื‘ืžืจื›ื–ื” ื”ื•ืโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_024.htmlโ€ฌโ€ฌ
โ€ซืื™ื–ื•ืŸ ืกืจื’ืœโ€ฌ
โ€ซืชื•ืœื™ื ื‘ืฆื“ ืื—ื“ ืžืฉืงื•ืœืช ืฉืžืกืชื”โ€ฌ
โ€ซืขืœ ืกืจื’ืœ ื—ืกืจ ืžืกื” ืฉืื•ืจื›ื•โ€ฌ
โ€ซื•ื‘ืฆื“ ื”ืฉื ื™ ืžืฉืงื•ืœืช ืฉืžืกืชื”โ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช.โ€ฌืโ€ช .โ€ฌื‘ืื™ื–ื” ืžืจื—ืง ืžื”ืžืฉืงื•ืœืช ื”ืจืืฉื•ื ื” ื™ืฉ ืœืžืงื ืืช ื ืงื•ื“ืช ื”ืžืฉืขืŸ ืขืœ ืžื ืช ืฉื”ืกืจื’ืœ ื™ืฉืืจ ืžืื•ื–ืŸโ€ฌ
โ€ซโ€ช.โ€ฌื‘โ€ช .โ€ฌื ื—ืœื™ืฃ ืืช ื”ืกืจื’ืœ ื—ืกืจ ื”ืžืกื” ื‘ืกืจื’ืœ ืื—ื™ื“ ืฉืžืกืชื•โ€ฌ
โ€ซโ€ช.โ€ฌืžืฆื ืืช ื ืงื•ื“ืช ื”ืžืฉืขืŸ ื›ืขืชโ€ฌ
โ€ซโ€ช ,โ€ฌืฉื‘ืฆื“ื• ื”ืื—ื“ ื”ืžืกื”โ€ฌ
โ€ซื•ื‘ืฆื“ื• ื”ืฉื ื™ ืื™ืŸ ืžืกื”โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_025.htmlโ€ฌโ€ฌ
โ€ซืกื•ืœื ืžื—ืœื™ืงโ€ฌ
โ€ซื•ืื•ืจื›ื• โ€ช ,โ€ฌื ืฉืขืŸ ืขืœ ืงื™ืจ ืื ื›ื™โ€ฌ
โ€ซืกื•ืœื ืฉืžืกืชื•โ€ฌ
โ€ซโ€ช .โ€ฌื—ืกืจ ื—ื™ื›ื•ืšโ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ืกื•ืœื ืœืจืฆืคื” ื”ื•ืโ€ฌ
โ€ซื‘ื” ื ื™ืชืŸ ืœื”ืขืžื™ื“ ื”ืกื•ืœื ื‘ืฉื™ื•ื•ื™ ืžืฉืงืœโ€ฌ
โ€ซืžื”ื™ ื”ื–ื•ื™ืช ื”ืžื™ื ื™ืžืœื™ืชโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_026.htmlโ€ฌโ€ฌ
โ€ซื‘ื™ืœื™ืืจื“โ€ฌ
โ€ซื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืงืœ ื‘ื™ืœื™ืืจื“ ื—ื•ื‘ื˜ ื‘ื›ื“ื•ืจ ื‘ื™ืœื™ืืจื“ ืฉืžืกืชื•โ€ฌ
โ€ซืžืขืœ ืžืจื›ื– ื”ื›ื“ื•ืจ ื‘ื• ืขืœื™ื• ืœื—ื‘ื•ื˜ ืขืœ ืžื ืช ืฉื”ื›ื“ื•ืจ ื™ืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื”โ€ฌ
โ€ซืžื”ื• ื”ื’ื•ื‘ื” ื”ืžื™ื ื™ืžืœื™โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_027.htmlโ€ฌโ€ฌ
โ€ซืชืื•ืฆืช ื”ืžืกื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌื ืชื•ื ื” ื”ืžืขืจื›ืช ื‘ืฆื™ื•ืจโ€ช ,โ€ฌืžืฆื ืืช ืชืื•ืฆืช ื›ืœ ื”ืžืกื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“ ื•ื›ืŸ ื ืชื•ืŸ ืฉื”ื—ื‘ืœ ื”ื•ื ื—ืกืจ ืžืกื” ื•ื”ื•ื ื ืข ืขืœ ื’ื‘ื™ ื”ื’ืœื™ืœ ืœืœื ื”ื—ืœืงื”โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_028.htmlโ€ฌโ€ฌ
โ€ซืฆืขื™ื“ื” ืขืœ ืกื—ืจื—ืจื”โ€ฌ
โ€ซืžืฉื˜ื— ืกื‘ื•ื‘ ืขื’ื•ืœ ืžืกืชื•ื‘ื‘ ืขืœ ืฆื™ืจ ืื ื›ื™ ืงื‘ื•ืข ื•ืžืฉืœื™ืโ€ฌ
โ€ซโ€ช .โ€ฌืกื™ื‘ื•ื‘ ืื—ื“ ื‘ ืฉื ื™ื•ืชโ€ช .โ€ฌืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืžืฉื˜ื— ืกื‘ื™ื‘ ืฆื™ืจื• ื”ื•ืโ€ฌ
โ€ซื”ืขื•ืžื“ ื‘ืชื—ื™ืœื” ื‘ืžืจื›ื– ื”ืžืฉื˜ื— ืฆื•ืขื“ ืœืื•ืจืš ื”ืจื“ื™ื•ืกโ€ฌ
โ€ซโ€ช.โ€ฌืื“ื ืฉืžืกืชื•โ€ฌ
โ€ซื›ืžื” ืฉื ื™ื•ืช ื™ืืจืš ืกื™ื‘ื•ื‘ ืฉืœ ื”ืžืฉื˜ื— ื›ืืฉืจ ื”ืื“ื ื ืžืฆื ื‘ืžืจื—ืง ืžืžืจื›ื–ื•โ€ฌ
โ€ซืžื” ื”ืขื‘ื•ื“ื” ืฉื”ืื“ื ื”ืฉืงื™ืข ื‘ื”ืœื™ื›ืชื•โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_029.htmlโ€ฌโ€ฌ
โ€ซืžื—ืœื™ืงื™ ืงืจื— ืžืกืชื•ื‘ื‘ื™ืโ€ฌ
โ€ซืฉื ื™ ืžื—ืœื™ืงื™ื ืขืœ ื”ืงืจื— ื‘ืขืœ ืื•ืชื” ืžืกื” ื ืขื™ื ืื—ื“ ืœืงืจืืช ื”ืฉื ื™โ€ฌ
โ€ซื‘ืงื•ื•ื™ื ืžืงื‘ื™ืœื™ื ืฉื”ืžืจื—ืง ื‘ื™ื ื ื”ื•ื โ€ช .โ€ฌืื—ื“ ืžื”ื ื ืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื•ื”ื•ื ืžื—ื–ื™ืง ื‘ื™ื“ื• ื‘ืงืฆื”ื• ืฉืœ ืžื•ื˜ ืื•ืคืงื™ ื—ืกืจ ืžืกื” ื‘ืื•ืจืšโ€ฌ
โ€ซื•ืชื•ืคืก ื‘ืžื•ื˜ ื›ืืฉืจ ื”ื•ื ื ืคื’ืฉ ืื™ืชื•โ€ฌ
โ€ซโ€ช.โ€ฌื•ื”ืžื—ืœื™ืง ื”ืฉื ื™ ื ืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌืโ€ช .โ€ฌืžื” ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ืœืื—ืจ ืชืคื™ืกืช ื”ืžื•ื˜โ€ฌ
โ€ซืื—ื“ ืžื”ืฉื ื™โ€ฌ
โ€ซโ€ช.โ€ฌื›ืขืช ื”ืžื—ืœื™ืงื™ื ืžื•ืฉื›ื™ื ื‘ืžื•ื˜ ืขื“ ืืฉืจ ื”ื ื‘ืžืจื—ืงโ€ฌ
โ€ซโ€ช.โ€ฌื‘โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ื”ื—ื“ืฉื”โ€ฌ
โ€ซโ€ช.โ€ฌื’โ€ช .โ€ฌืžื” ื”ืขื‘ื•ื“ื” ืฉื ื“ืจืฉื” ื‘ืฉื‘ื™ืœ ืœืขื‘ื•ืจ ืœืžืฆื‘ ื–ื”โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_030.htmlโ€ฌโ€ฌ
โ€ซืชื ืข ื–ื•ื•ื™ืชื™โ€ฌ
โ€ซืžื—ื•ื‘ืจื™ื ืขืœ ื™ื“ื™ ืžื•ื˜ ืงืฉื™ื— ื•ื—ืกืจ ืžืกื” ื‘ืื•ืจืšโ€ฌ
โ€ซโ€ช ,โ€ฌื”ื’ื•ืคื™ืโ€ฌ
โ€ซืžืชืงืจื‘โ€ฌ
โ€ซื•ืžืกืชื•ื‘ื‘ื™ื ืกื‘ื™ื‘ ืžืจื›ื– ื”ืžืกื” ื‘ืžื”ื™ืจื•ืช ื–ื•ื•ื™ืชื™ืช โ€ช .โ€ฌื”ื’ื•ืฃโ€ฌ
โ€ซโ€ช .โ€ฌืืœ ื”ืžืขืจื›ืช ื‘ืžื”ื™ืจื•ืช ื‘ื–ื•ื™ืช โ€ช ,โ€ฌื‘ื™ื—ืก ืœื›ื™ื•ื•ืŸ ื”ื—ื™ื•ื‘ื™ ืฉืœ ืฆื™ืจ ื”โ€ฌ
โ€ซื•ื ื“ื‘ืง ืืœื™ื• ื‘ืจื’ืข ืฉื”ืžื•ื˜ ืžืงื‘ื™ืœ ืœืฆื™ืจโ€ฌ
โ€ซืžืชืงืจื‘ ืœื’ื•ืฃโ€ฌ
โ€ซโ€ช .โ€ฌื”ื’ื•ืฃโ€ฌ
โ€ซ?ืโ€ช .โ€ฌืžื” ืžื”ื™ืจื•ืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืžืขืจื›ืช ื›ื•ืœื”โ€ฌ
โ€ซ?ื‘โ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืชโ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื• ื”ืฉื™ื ื•ื™ ื”ื›ืœืœื™ ื‘ืื ืจื’ื™ื” ืฉืœ ื”ืžืขืจื›ืชโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_031.htmlโ€ฌโ€ฌ
โ€ซืงื•ืฃ ืžื˜ืคืก ืขืœ ืกื•ืœืโ€ฌ
โ€ซื”ืกื•ืœื ื‘ืื™ื•ืจโ€ช ,โ€ฌื‘ืžืกื” ืฉืœ โ€ช2โ€ฌืง"ื’โ€ช ,โ€ฌื ืฉืขืŸ ืขืœ ื”ืงื™ืจ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 30โ€ฌืžืขืœื•ืชโ€ช .โ€ฌืžืงื“ื ื”ื—ื™ื›ื•ืš ื‘ื™ืŸ ื”ืกื•ืœื ืœืงื™ืจ ื”ื•ื โ€ช 0.5โ€ฌื•ื‘ื™ืŸ ื”ืกื•ืœื ืœืจืฆืคื” โ€ช.0.34โ€ฌโ€ฌ
โ€ซืงื•ืฃ ื‘ืžืกื” โ€ช10โ€ฌืง"ื’ ืžืชื—ื™ืœ ืœื˜ืคืก ืขืœ ื”ืกื•ืœื ื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช .โ€ฌื›ืืฉืจ ื”ืงื•ืฃ ืžื’ื™ืข ืœืžืจื—ืง ืžืกื•ื™ื™ื ืžื‘ืกื™ืก ื”ืกื•ืœืโ€ช ,โ€ฌื”ืกื•ืœื ืžื—ืœื™ืงโ€ช .โ€ฌืžืฆืื• ืืชโ€ฌ
โ€ซโ€ช.โ€ฌื”ื™ื—ืก ื‘ื™ืŸ ืžืจื—ืง ื–ื” ื•ื‘ื™ืŸ ืื•ืจืš ื”ืกื•ืœืโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_130.htmlโ€ฌโ€ฌ
โ€ซืžื•ืžื ื˜ ื”ืชืžื“ ืฉืœ ืกืคื™ืจื”โ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘ ืžื•ืžื ื˜ ื”ืชืžื“ ืฉืœ ืกืคื™ืจื” )ื›ื“ื•ืจ ื—ืœื•ืœ( ื‘ืขืœืช ืฆืคื™ืคื•ืช ืžืกื” ืื—ื™ื“ื”โ€ช ,โ€ฌืจื“ื™ื•ืก โ€ช Rโ€ฌื•ืžืกื” โ€ช Mโ€ฌืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื“ืจืš ืžืจื›ื– ื”ืžืกื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื—ืฉื‘ ืžื•ืžื ื˜ ื”ืชืžื“ ืฉืœ ืื•ืชื” ืกืคื™ืจื” ืกื‘ื™ื‘ ืฆื™ืจ ื”ืžืฉื™ืง ืœืกืคื™ืจื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_140.htmlโ€ฌโ€ฌ
โ€ซืžื•ืœืงื•ืœืช ื—ืžืฆืŸโ€ฌ
โ€ซืžื•ืœืงื•ืœืช ื—ืžืฆืŸโ€ฌ
โ€ซืžืกืช ื›ืœ ืื˜ื•ื ื”ื™ืโ€ฌ
โ€ซืžืกืชื•ื‘ื‘ืช ื‘ืžื™ืฉื•ืจโ€ฌ
โ€ซืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื– ื‘ื™ืŸ ื”ืžื•ืœืงืœื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซื•ื‘ื˜ืžืคืจื˜ื•ืจืช ื”ื—ื“ืจ ื”ืžืจื—ืง ื‘ื™ืŸ ื”ืื˜ื•ืžื™ื ื”ื•ืโ€ฌ
โ€ซืœื”ืชื™ื™ื—ืก ื›ืืœ ืžืกื” ื ืงื•ื“ืชื™ืช(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘ ืืช ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืžื•ืœืงื•ืœื” ืกื‘ื™ื‘ ืฆื™ืจโ€ฌ
โ€ซื‘โ€ช .โ€ฌื ืชื•ืŸ ืฉื”ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ืกื‘ื™ื‘ ืฆื™ืจ ื”ื™ืโ€ฌ
โ€ซ)ืœืื˜ื•ื ื ื™ืชืŸโ€ฌ
โ€ซ)ืฉื™ื ืœื‘ ืฉื”ื‘ืขื™ื” ื”ื™ื ื‘ื“ื™ื“ื”(โ€ฌ
โ€ซโ€ช ,โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื›ืชื•ืฆืื” ืžื”ืกื™ื‘ื•ื‘?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_150.htmlโ€ฌโ€ฌ
โ€ซื›ื“ื•ืจ ื‘ื™ื™ืกื‘ื•ืœโ€ฌ
โ€ซื ืข ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซืžืจื›ื– ื”ืžืกื” ืฉืœ ื›ื“ื•ืจ ื‘ื™ื™ืกื‘ื•ืœ ื‘ืขืœ ืจื“ื™ื•ืกโ€ฌ
โ€ซื”ื›ื“ื•ืจ ืžืกืชื•ื‘ื‘ ืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจ ื“ืจืš ืžืจื›ื– ื”ืžืกื” ืฉืœื• ื‘ืžื”ื™ืจื•ืช ื–ื•ื™ืชื™ืชโ€ฌ
โ€ซโ€ช,โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื—ืฉื‘ ืืช ื”ื™ื—ืก ื‘ื™ืŸ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื›ืชื•ืฆืื” ืžืžื”ื™ืจื•ืช ืžืจื›ื– ื”ืžืกื” ืœื‘ื™ืŸ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื›ืชื•ืฆืื” ืžื”ืกื™ื‘ื•ื‘โ€ช.โ€ฌโ€ฌ
โ€ซื”ืชื™ื™ื—ืก ืœื›ื“ื•ืจ ื›ืืœ ื›ื“ื•ืจ ืžืœืโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_4_160.htmlโ€ฌโ€ฌ
โ€ซืžื—ื•ื’ื™ ื”ืฉืขื•ืŸ ืฉืœ ื”ื‘ื™ื’ ื‘ืŸโ€ฌ
โ€ซื”ื’ื“ืœื™ื ืฉืœ ืžื—ื•ื’ื™ ื”ืฉืขื•ืŸ ืฉืœ ื”ืฉืขื•ืŸ ื”ืžืคื•ืจืกื "ื‘ื™ื’ ื‘ืŸ" ื”ืโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซื•ื”ืžืกื•ืช ืฉืœื”ื ื”ืŸโ€ฌ
โ€ซื•โ€ช-โ€ฌโ€ฌ
โ€ซโ€ช .โ€ฌื—ืฉื‘ ืืช ืกืš ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ื›ืชื•ืฆืื” ืžื”ืกื™ื‘ื•ื‘ ืฉืœ ื”ืžื—ื•ื’ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซื”ืžื—ื•ื’ ื”ืงืฆืจ ืžื™ื™ืฆื’ ืืช ื”ืฉืขื” ื•ื”ืืจื•ืš ืžื™ื™ืฆื’ ื“ืงื”โ€ช.โ€ฌโ€ฌ
โ€ซื ื™ืชืŸ ืœื”ืชื™ื™ื—ืก ืœืžื—ื•ื’ื™ื ื›ืืœ ืžื•ื˜ื•ืช ืขื ืžื•ืžื ื˜ ื”ืชืžื“โ€ฌ
โ€ซื ื™ืชืŸ ืœื”ื ื™ื— ืฉืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ื•ื ื‘ืงืฆื” ืฉืœ ื”ืžื—ื•ื’โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_002.htmlโ€ฌโ€ฌ
โ€ซืคื™ืฆื•ื— ืื’ื•ื–ื™ืโ€ฌ
โ€ซืื’ื•ื– ืžืœืš ื ืฉื‘ืจ ื›ืืฉืจ ืžืฉื ื™ ืฆื™ื“ื™ื• ืคื•ืขืœ ืขืœื™ื• ื›ื•ื— ืฉืœ โ€ช 46โ€ฌื ื™ื•ื˜ื•ืŸโ€ช .โ€ฌืžื”ื• ื”ื›ื•ื— ืฉื™ืฉ ืœื”ืคืขื™ืœ ืขืœ ืงืฆื•ืช ื”ื™ื“ื™ื•ืช ืฉืœ ื”ืžืคืฆื— ืขืœ ืžื ืช ืœืคืฆื—ื•?โ€ฌ
โ€ซื”ืื ื”ืฉื™ืžื•ืฉ ื‘ืžืคืฆื— ืขื“ื™ืฃ ื‘ืื•ืคืŸ ืžืฉืžืขื•ืชื™ ื‘ื™ื—ืก ืœืฉื™ื˜ื” ื”ืžืกื•ืจืชื™ืช ืฉืœ ืœื”ื›ื•ืช ื‘ืื’ื•ื– ื‘ืื‘ืŸ ืื• ื‘ืื’ืจื•ืฃ?โ€ช ...โ€ฌืžื™ืžื“ื™ ื”ืžืคืฆื— ื ืชื•ื ื™ื ื‘ืฆื™ื•ืจโ€ฌ
โ€ซโ€ช46-9โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_081.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื’ื“ืœ ืคื™ื–ื” ื”ื•ื ื‘ื’ื•ื‘ื” ืฉืœ โ€ช 55โ€ฌืžื˜ืจ ื•ืงื•ื˜ืจื• โ€ช 7โ€ฌืžื˜ืจโ€ช .โ€ฌืจืืฉ ื”ืžื’ื“ืœ ื ืžืฆื ื‘ืžืจื—ืง ืฉืœ โ€ช 4.5โ€ฌืžื˜ืจ ืžื”ืื ืš ืœืงืจืงืขโ€ช) .โ€ฌื ื ื™ื— ื›ื™( ื”ืžื’ื“ืœ ืœื ื ื•ืคืœโ€ฌ
โ€ซืžื›ื™ื•ื•ืŸ ืฉืžืจื›ื– ื”ืžืกื” ืฉืœื• ื”ื•ื ืžืขืœ ื‘ืกื™ืก ื”ืžื’ื“ืœ )ืœืฆื•ืจืš ื”ืชืจื’ื™ืœ ื ืชื™ื™ื—ืก ืœืžื’ื“ืœ ื›ื’ืœื™ืœ(โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ืžืจื—ืง ื”ื ื•ืกืฃ ืฉื™ืฉ ืœื”ื–ื™ื– ืืช ืจืืฉ ื”ืžื’ื“ืœ ืžื”ืื ืš ื‘ื›ื“ื™ ืฉื™ื™ืคื•ืœ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื” ืชื”ื™ื” ื–ื•ื•ื™ืช ื”ื ื˜ื™ื™ื” ืฉืœ ื”ืžื’ื“ืœ ื‘ืžืงืจื” ื–ื”?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_082.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื˜ืคืก ื”ืจื™ื ื”ืฉื•ืงืœ โ€ช 55โ€ฌืง" ื’ื› ื ืข ืœืื•ืจืš ื—ืจื™ืฅ ืฉืจื•ื—ื‘ื• โ€ช w=0.2mโ€ฌืข" ื™ ื›ืš ืฉื”ื•ื ืžื•ืฉืš ืขื ื™ื“ื™ื• ืืช ื“ื•ืคืŸ ื”ื—ืจื™ืฅ ื•ื“ื•ื—ืฃ ืขื ืจื’ืœื™ื• ืืชโ€ฌ
โ€ซื”ืงื™ืจ ืฉืžื ื’ื“โ€ช .โ€ฌืžืจื›ื– ื”ืžืกื” ืฉืœ ื”ืžื˜ืคืก ื ืžืฆื ื‘ืžืจื—ืง ืฉืœ โ€ช d=0.4mโ€ฌืžืงืฆื” ื”ื—ืจื™ืฅโ€ช .โ€ฌื ืชื•ืŸ ื›ื™ ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ื™ื“ื™ื™ื ืœืกืœืข ื”ื•ืโ€ฌ
โ€ซโ€ช m1=0.4โ€ฌื•ืžืงื“ื ื”ื—ื™ื›ื•ืš ื”ืกื˜ื˜ื™ ื‘ื™ืŸ ื”ื ืขืœื™ื™ื ืœืกืœืข ื”ื•ื โ€ช) .m2=1.2โ€ฌืฉื™ื˜ืช ื˜ื™ืคื•ืก ื–ื• ื ืงืจืืช โ€ช.(laybackโ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ื›ื•ื— ื”ืžื™ื ื™ืžืœื™ ืฉืขืœ ื”ืžื˜ืคืก ืœื“ื—ื•ืฃ ืขื ื”ืจื’ืœื™ื™ื ื•ืœืžืฉื•ืš ืขื ื”ื™ื“ื™ื™ื ืขืœโ€ช-โ€ฌืžื ืช ืฉื™ื”ื™ื” ื™ืฆื™ื‘?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ื”ืžืจื—ืง ื‘ื™ืŸ ื”ื™ื“ื™ื™ื ืœืจื’ืœื™ื™ื ืฉืœ ื”ืžื˜ืคืก )โ€ช ,(hโ€ฌืขื‘ื•ืจ ื”ื›ื•ื— ื”ืžื•ืคืขืœ ื‘ืกืขื™ืฃ ื”ืงื•ื“ื?โ€ฌ
โ€ซื’โ€ช .โ€ฌืื ื™ื•ืจื“ ื’ืฉื ื•ื”ืกืœืข ื ืจื˜ื‘โ€ช ,โ€ฌืžืงื“ืžื™ ื”ื—ื™ื›ื•ืš ืงื˜ื ื™ืโ€ช .โ€ฌื›ื™ืฆื“ ื”ื“ื‘ืจ ื™ืฉืคื™ืข ืขืœ ืชื•ืฆืื•ืช ื”ืกืขื™ืคื™ื ื”ืงื•ื“ืžื™ื?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_083.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื™ืœื“ ืฉืžืกืชื• โ€ช 30โ€ฌืง"ื’ ืขื•ืžื“ ื‘ืงืฆื” ืงืจื•ืกืœื” ื ื™ื™ื—ืช ืฉืžืกืชื” โ€ช 100โ€ฌืง"ื’ ื•ืจื“ื™ื•ืกื” โ€ช 2โ€ฌืžื˜ืจโ€ช .โ€ฌืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืงืจื•ืกืœื” ืกื‘ื™ื‘ ืฆื™ืจ ื”ืขื•ื‘ืจโ€ฌ
โ€ซื‘ืžืจื›ื–ื” ื”ื•ื โ€ช .kg*m2 150โ€ฌื”ื™ืœื“ ืชื•ืคืก ื›ื“ื•ืจ ื‘ืžืกื” ืฉืœ โ€ช 1โ€ฌืง"ื’ ื”ื ื–ืจืง ืืœื™ื• ืžื—ื‘ืจื•โ€ช .โ€ฌืจื’ืข ืœืคื ื™ ืฉื”ื›ื“ื•ืจ ื ืชืคืก ืžื”ื™ืจื•ืชื• ื”ื™ื โ€ช 12โ€ฌืžื˜ืจโ€ฌ
โ€ซืœืฉื ื™ื” ื•ื›ื™ื•ื•ื ื” ื ื•ื˜ื” ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช 37โ€ฌืžืขืœื•ืช ืœืžืฉื™ืง ืœืงืจื•ืกืœื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื” ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืงืจื•ืกืœื” ืœืื—ืจ ืชืคื™ืกืช ื”ื›ื“ื•ืจ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื” ื”ืžื”ื™ืจื•ืช ื”ืžืฉื™ืงื™ืช ืฉืœ ื”ื™ืœื“ ืœืื—ืจ ืชืคื™ืกืช ื”ื›ื“ื•ืจ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_084.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืฉืชื™ ื ืฉื™ื ืžื—ืœื™ืงื•ืช ืขืœ ื”ืงืจื—โ€ช .โ€ฌืœื›ืœ ืื—ืช ืžืกื” ืฉืœ โ€ช 50โ€ฌืง"ื’โ€ช .โ€ฌื”ืŸ ืžืชืงืจื‘ื•ืช ืื—ืช ื›ืœืคื™ ื”ืฉื ื™ื” ื‘ืžื”ื™ืจื•ืช ืฉืœ โ€ช 1.4โ€ฌืžื˜ืจ ืœืฉื ื™ื” )ื›ืœ ืื—ืช(โ€ฌ
โ€ซื‘ืžืกืœื•ืœื™ื ืžืงื‘ื™ืœื™ื ืฉื”ืžืจื—ืง ื‘ื™ื ื™ื”ื ื”ื•ื โ€ช 3โ€ฌืžื˜ืจโ€ช .โ€ฌืžื—ืœื™ืงื” ืื—ืช ื ื•ืฉืืช ืžื•ื˜ ื‘ืขืœ ืžืกื” ื–ื ื™ื—ื”โ€ช .โ€ฌื”ืžื—ืœื™ืงื” ื”ืฉื ื™ื” ืื•ื—ื–ืช ื‘ืงืฆื” ื”ืžื•ื˜ ื›ืืฉืจโ€ฌ
โ€ซื”ืŸ ื—ื•ืœืคื•ืช ื–ื• ืœื™ื“ ื–ื•โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืชืืจโ€ช/โ€ฌื™ ื‘ืฆื•ืจื” ื›ืžื•ืชื™ืช ืืช ืชื ื•ืขืช ื”ืžื—ืœื™ืงื•ืช ืœืื—ืจ ืฉืฉืชื™ื”ืŸ ืื•ื—ื–ื•ืช ื‘ืžื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ืฉืœ ื”ืžื—ืœื™ืงื•ืช ?โ€ฌ
โ€ซื’โ€ช.โ€ฌื‘ืฉืœื‘ ื”ื‘ื ื”ืžื—ืœื™ืงื•ืช ืžืชืงืจื‘ื•ืช ืœืื•ืจืš ื”ืžื•ื˜ ืขื“ ืฉื”ืžืจื—ืง ื‘ื™ื ื™ื”ืŸ ื”ื•ื ืžื˜ืจโ€ช .โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื”ื–ื•ื•ื™ืชื™ืช ืฉืœื”ืŸ ื›ืขืช?โ€ฌ
โ€ซื“โ€ช.โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ืฉืœื”ืŸ?โ€ฌ
โ€ซื”โ€ช.โ€ฌื”ืกื‘ืจโ€ช/โ€ฌื”ืกื‘ื™ืจื™ ืžืฉื™ืงื•ืœื™ ืื ืจื’ื™ื” ื›ื™ืฆื“ ื”ืื ืจื’ื™ื” ื”ืงื™ื ื˜ื™ืช ืขืœืชื”?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_090.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฆืคื™ื“ โ€ช -โ€ฌืกื˜ื˜ื™ืงื”โ€ฌ
โ€ซืžื•ื˜ ืฉืžืกืชื•โ€ช mโ€ฌื•ืื•ืจื›ื• โ€ช Lโ€ฌืžื—ื•ื‘ืจ ื‘ืขื–ืจืช ื—ื•ื˜ ืœืงื™ืจ ืœื ื—ืœืง ื›ืžื•ืจืื”โ€ช .โ€ฌื ืชื•ืŸ โ€ช . 700=bโ€ฌืžื” ืฆืจื™ืš ืœื”ื™ื•ืช ืžืงื“ื ื”ื—ื™ื›ื•ืš ื›ื“ื™ ืฉื”ืžืงืœ ืœื ื™ื—ืœื™ืงโ€ฌ
โ€ซ)ืชืฉื•ื‘ื” ืžืกืคืจื™ืช!( ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_091.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฆืคื™ื“ โ€ช -โ€ฌืกื˜ื˜ื™ืงื”โ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 5โ€ฌืง"ื’ ืชืœื•ื™ ืขืœ ืžื•ื˜ ืื—ื™ื“ ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช โ€ช 600โ€ฌืžื”ืงื™ืจโ€ช .โ€ฌื”ืžื•ื˜โ€ช ,โ€ฌืฉืžืกืชื• โ€ช ,mโ€ฌืžื•ื—ื–ืง ืข"ื™ ื—ื‘ืœ ื”ื™ื•ืฆืจ ืขื ื”ืงื™ืจ ื–ื•ื•ื™ืช ื‘ืช โ€ช.450โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžืชื™ื—ื•ืช โ€ช Tโ€ฌื‘ื—ื‘ืœ ื–ื”โ€ช ,โ€ฌืื ื™ื“ื•ืข ืฉโ€ช? m=2kg -โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ืจืื• ืฉื›ืืฉืจ โ€ช mโ€ฌืฉื•ืืฃ ืœืืคืกโ€ช ,โ€ฌื”ื›ื•ื— ืฉืžืคืขื™ืœ ื”ืžื•ื˜ ืขืœ ื ืงื•ื“ืช ื—ื™ื‘ื•ืจ ื”ื—ื•ื˜ื™ื ืžื›ื•ื•ืŸ ืœืื•ืจืš ื”ืžื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซ)ื”ืขืจื” โ€ช :โ€ฌื›ื•ื— ื–ื” ืžืืคืก ืืช ื”ืฉืงื•ืœ ืฉืœ โ€ช Tโ€ฌื•โ€ช.('T -โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_092.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฆืคื™ื“ โ€ช -โ€ฌื“ื™ืžื ื™ืงื”โ€ฌ
โ€ซื›ื•ืจื›ื™ื ื—ื•ื˜ ืคืขืžื™ื ืื—ื“ื•ืช ืกื‘ื™ื‘ ื’ืœื™ืœ ืฉืžืกืชื• โ€ช mโ€ฌื•ืจื“ื™ื•ืกื• โ€ช . rโ€ฌืืช ืงืฆื” ื”ื—ื•ื˜ ืžื—ื–ื™ืงื™ื ืงื‘ื•ืข ื•ืžืฉื—ืจืจื™ื ืืช ื”ื’ืœื™ืœ ืœืœื ืžื”ื™ืจื•ืชโ€ฌ
โ€ซื”ืชื—ืœืชื™ืชโ€ช .โ€ฌื”ื—ื•ื˜ ื ืฉืœืฃ ืืš ืื™ื ื• ืžื—ืœื™ืง ื›ืืฉืจ ื”ื’ืœื™ืœ ื ื•ืคืœ ื›ืฆืขืฆื•ืข ื™ื•โ€ช-โ€ฌื™ื•โ€ช .โ€ฌื—ืฉื‘ ืืช ืชืื•ืฆืช ื”ื’ืœื™ืœ ื•ื”ืžืชื™ื—ื•ืช ื‘ื—ื•ื˜ ืืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื”ื’ืœื™ืœ ื”ื™ื ื• ืžืงืฉื™โ€ฌ
โ€ซื‘โ€ช .โ€ฌื”ื’ืœื™ืœ ื”ื ื• ื—ืœื•ืœ ื“ืง ื“ืคื ื•ืชโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_093.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฆืคื™ื“ โ€ช -โ€ฌื“ื™ื ืžื™ืงื”โ€ฌ
โ€ซืžื•ื˜ ืขืฅ ืฉืื•ืจื›ื• โ€ช 2โ€ฌืžื˜ืจ ื•ืžืกืชื• โ€ช 5โ€ฌืง"ื’ ื ื™ืฆื‘ ืื ื›ื™ืช ื›ืฉื”ื•ื ืžื—ื•ื‘ืจ ืœืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื‘ืงืฆื”ื•โ€ช .โ€ฌื™ื•ืจื™ื ืœืชื•ืš ื”ืžื•ื˜ ืงืœื™ืข ืฉืžืกืชื• โ€ช 10โ€ฌื’ืจื ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช 400โ€ฌืžื˜ืจ ืœืฉื ื™ื” ื”ื ื— ื›ื™ ื”ืงืœื™ืข ืžืชื ื”ื’ ื›ืžื• ืžืกื” ื ืงื•ื“ืชื™ืชโ€ช .โ€ฌื‘ืงืœื™ืข ื ืชืงืข ื‘ืžื•ื˜ โ€ช 10โ€ฌืกื ื˜ื™ืžื˜ืจ ืžืขืœ ืงืฆื”ื•โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ื ืžื”ื™ืจื•ืช ื”ื–ื•ื™ืชื™ืช ืฉืœ ื”ืžื•ื˜ ืœืื—ืจ ืฉื ืชืงืข ื‘ื• ื”ื›ื“ื•ืจ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื™ื ื”ื–ื•ื™ืช ื”ืžืงืžื™ืžืืœื™ืช ืืœื™ื” ื™ื’ื™ืข ื”ืžื•ื˜?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_5_094.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืฆืคื™ื“ โ€ช -โ€ฌืกื˜ื˜ื™ืงื”โ€ฌ
โ€ซื’ื•ืฃ ืฉืžืกืชื• โ€ช 5โ€ฌืง"ื’ ืชืœื•ื™ ืขืœ ืžื•ื˜ ืื—ื™ื“ ื”ื ื˜ื•ื™ ื‘ื–ื•ื•ื™ืช โ€ช 600โ€ฌืžื”ืงื™ืจโ€ช .โ€ฌื”ืžื•ื˜โ€ช ,โ€ฌืฉืžืกืชื• โ€ช ,mโ€ฌืžื•ื—ื–ืง ืข"ื™ ื—ื‘ืœ ื”ื™ื•ืฆืจ ืขื ื”ืงื™ืจ ื–ื•ื•ื™ืช ื‘ืช โ€ช.450โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ื”ืžืชื™ื—ื•ืช โ€ช Tโ€ฌื‘ื—ื‘ืœ ื–ื”โ€ช ,โ€ฌืื ื™ื“ื•ืข ืฉโ€ช? m=2kg -โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌืžื”ื• ื”ื›ื— ืฉืžืคืขื™ืœ ื”ืงื™ืจ ืขืœ ื”ืžื•ื˜?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_012.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื ื•ืข ืžื—ื•ื‘ืจ ืœื’ืœื’ืœืช ืžืจื›ื–ื™ืช ื”ื‘ื ื•ื™ื” ืžื’ืœื™ืœื™ ื‘ืจื“ื™ื•ืก โ€ช R/2โ€ฌื•ืžืกื” โ€ช ,mโ€ฌืืœื™ื• ืžื•ืฆืžื“ื™ื ืžืฉื ื™ ืฆื™ื“ื™ื• ืฉืชื™ ื“ืกืงื™ื•ืช ืื—ื™ื“ื•ืช ื‘ืขืœื•ืช ืžืกื” โ€ชmโ€ฌโ€ฌ
โ€ซื•ืจื“ื™ื•ืก โ€ช .Rโ€ฌื›ืืฉืจ ื”ืžื ื•ืข ืžืกืชื•ื‘ื‘ ื•ืžืกื•ื‘ื‘ ืืช ื”ื’ืœื’ืœืชโ€ช ,โ€ฌืžืชืœืคืฃ ื›ื‘ืœ ื—ืกืจ ืžืกื” ืขืœ ื”ื’ืœื™ืœ )ืœืœื ื”ื—ืœืงื”(โ€ช .โ€ฌืขืœ ื”ื›ื‘ืœ ืชืœื•ื™ื” ืžืกื” ืฉืœ โ€ช.3.5mโ€ฌโ€ฌ
โ€ซืžื”ื• ืžื•ืžื ื˜ ื”ื›ื•ื— ืฉืœ ื”ืžื ื•ืข ื”ื“ืจื•ืฉ ื›ื“ื™ ืœื”ืขืœื•ืช ืืช ื”ืžืกื” ื‘ืชื ืื™ื ื”ื‘ืื™ืโ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌื‘ืชืื•ืฆื” ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ืžื”ื™ืจื•ืช ืงื‘ื•ืขื”โ€ช.โ€ฌโ€ฌ
โ€ซื’โ€ช .โ€ฌืคืชืจื• ืืช ืกืขื™ืฃ ื‘' ื‘ืขื–ืจืช ืื ืจื’ื™ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_092.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžืฆืโ€ช/โ€ฌื™ ืืช ืžืจื›ื– ื”ืžืกื” ืฉืœ ืžืขืจื›ืช ื”ืžื•ื˜ื•ืช ื”ื‘ืื”โ€ช:โ€ฌโ€ฌ
โ€ซ]โ€ชL1 = 1.2 [m] m1 = 12 [kgโ€ฌโ€ฌ
โ€ซ]โ€ชL2 = 1.5 [m] m2 = 15 [kgโ€ฌโ€ฌ
โ€ซ]โ€ชL3 = 1.8 [m] m3 = 18 [kgโ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_093.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื ืชื•ื ื” ื“ื™ืกืงื” ืฉืงื•ื˜ืจื” โ€ช 2R = 1 mโ€ฌื•ื‘ื” ื—ื•ืจ ืžืขื’ืœื™ ืฉืงื•ื˜ืจื• โ€ช .2r = 0.5 mโ€ฌื”ืชืคืœื’ื•ืช ื”ืžืกื” ื‘ื“ื™ืกืงื” ื”ื™ื ืื—ื™ื“ื”โ€ช .โ€ฌืžืฆืโ€ช/โ€ฌื™ ืืช ื ืงื•ื“ืช ืžืจื›ื–โ€ฌ
โ€ซื”ืžืกื” ืฉืœ ื”ื“ื™ืกืงื”โ€ช) .โ€ฌืจืžื–โ€ช :โ€ฌื”ืชื™ื™ื—ืกื• ืœื—ื•ืจ ื›ืืœ ื“ื™ืกืงื” ืขื ืžืกื” ืฉืœื™ืœื™ืช(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_094.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื™ืœื“ ืฉืžืกืชื• โ€ช [kg] 40โ€ฌื”ื•ืœืš ืขืœ ืงื•ืจื” ืฉืื•ืจื›ื” โ€ช 2โ€ฌืžื˜ืจื™ืโ€ช .โ€ฌื•ืžืกืชื” โ€ช .[kg] 20โ€ฌื”ืงื•ืจื” ืชืœื•ื™ื” ื‘ืื•ื•ื™ืจ ืขืœ ื™ื“ื™ ืฉื ื™ ื—ื•ื˜ื™ืโ€ช ,โ€ฌื”ืžืชื™ื—ื•ืช ื”ืžืงืกื™ืžืœื™ืชโ€ฌ
โ€ซืฉืœ ื”ื—ื•ื˜ ื”ื™ืžื ื™ ื”ื™ื โ€ช [N] 350โ€ฌื•ืฉืœ ื”ื—ื•ื˜ ื”ืฉืžืืœื™ โ€ช .[N] 400โ€ฌืžื”ื• ื”ืชื—ื•ื ื‘ื• ื™ื›ื•ืœ ืœืฆืขื•ื“ ื”ื™ืœื“ ื‘ื‘ื™ื˜ื—ื” ืขืœ ื”ืงื•ืจื” ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_095.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื‘ืชื—ืจื•ืช ืจื™ืฆื” ื‘ืžืกืœื•ืœ ืžืขื’ืœื™โ€ช ,โ€ฌืฉื ืขืจื›ื” ืœืงืจืืช ื”ืื•ืœื™ืžืคื™ืื“ื” )ื‘ื™ื™ื’'ื™ืŸ โ€ช (2008โ€ฌื‘ืฉื›ื•ื ื” ื“'โ€ช ,โ€ฌืœืคื ื™ ื”ื”ืงืคื” ื”ืื—ืจื•ื ื” ืžื•ื‘ื™ืœ ื”ืืฆืŸ ื”ื ื•ื“ืข ืจื‘โ€ฌ
โ€ซืฉื•ื“ื“ ืขืœ ืคื ื™ ื”ืืฆืŸ ื”ืžืคื•ืจืกื ืจื‘ ืฉื•ื˜ืจ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช) ฯ€ /8โ€ฌื›ืœื•ืžืจ ืœืจื‘ ืฉื•ื˜ืจ ื ื•ืชืจื” ื”ืงืคื” ืฉืœื™ืžื” ื•ืจื‘ ืฉื•ื“ื“ ื”ืงืคื” ืคื—ื•ืช โ€ช .(ฯ€ /8โ€ฌื™ื“ื•ืข ื›ื™โ€ฌ
โ€ซืจื“ื™ื•ืกื• ืฉืœ ื”ืžืกืœื•ืœ ื”ืžืขื’ืœื™ ื‘ืฉื›ื•ื ื” ื”ื•ื โ€ช [m] 100โ€ฌื•ื›ืŸ ืฉืžื”ื™ืจื•ืชื ืฉืœ ืจื‘ ืฉื•ื“ื“ ื•ืจื‘ ืฉื•ื˜ืจ ื”ื™ื โ€ช.[km/h] 54โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืขื• ืจื‘ ืฉื•ื“ื“ ื•ืจื‘ ืฉื•ื˜ืจ ืœืงื• ื”ืกื™ื•ื ื‘ืชื ืื™ื ืืœื• ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ื”ืชื—ืฉื‘ ื‘ืขื•ื‘ื“ื” ื›ื™ ืจื‘ ืฉื•ื“ื“ ื”ื•ื ืžืขืฉืŸ ื›ื‘ื“โ€ช ,โ€ฌื•ืžืชื—ื™ืœ ืœื”ืื˜ ืœืงืจืืช ื”ืกื™ื•ืโ€ช ,โ€ฌืžื” ืชื”ื™ื” ื”ืชืื•ื˜ื” )ืชืื•ืฆื” ืฉืœื™ืœื™ืช( ื”ื–ื•ื•ื™ืชื™ืช ื”ืงื‘ื•ืขื”โ€ฌ
โ€ซื”ืžื™ื ื™ืžืœื™ืช ืฉืชื’ืจื•ื ืœืžื”ืคืš ื‘ืžื™ืจื•ืฅ ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืื ื™ืคืชื™ืข ืจื‘ ืฉื•ื“ื“ ื•ื™ืžืฉื™ืš ื‘ืื•ืชื• ืงืฆื‘ ืขื“ ื”ืกื™ื•ื ืžื” ืชื”ื™ื” ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ื”ืžื™ื ื™ืžืœื™ืช ืฉืœ ืจื‘ ืฉื•ื˜ืจ ื›ืš ืฉื™ืขืงื•ืฃ ืืช ื™ืจื™ื‘ื• ื”ืžืจโ€ฌ
โ€ซื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช ฯ€ /10โ€ฌืžืงื• ื”ืกื™ื•ื?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_096.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืื ืžืงื™ืคื™ื ืืช ื›ื“ื•ืจ ื”ืืจืฅ ื‘ื—ื‘ืœ ืœืื•ืจืš ืงื• ื”ืžืฉื•ื•ื” )ื•ืžื”ื“ืงื™ื ื”ื™ื˜ื‘( ื•ืื—" ื› ืžื•ืกื™ืคื™ื ืœืื•ืจื›ื• ืฉืœ ื”ื—ื‘ืœ โ€ช 1โ€ฌืžื˜ืจ ื•ืฉื•ื‘ ืžืกื“ืจื™ื ืื•ืชื•โ€ฌ
โ€ซื‘ืฆื•ืจื” ืžืขื’ืœื™ืช ืœืื•ืจืš ืงื• ื”ืžืฉื•ื•ื”โ€ช ,โ€ฌื”ืื ื‘ืจื•ื•ื— ืฉื ื•ืฆืจ ื™ื•ื›ืœื• ืœืขื‘ื•ืจ ืฉืคืŸ ? ื—ืชื•ืœ?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_113.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื•ื˜ ืฆืจ ื•ืื—ื™ื“ ืฉืื•ืจื›ื• โ€ช Lโ€ฌื•ืžืกืชื• โ€ช Mโ€ฌืขื•ืžื“ ืขืœ ืฉื•ืœื—ืŸ ืื•ืคืงื™ ื—ืœืงโ€ช .โ€ฌืžื˜ื‘ืข ืงื˜ืŸ ืฉืžืกืชื• โ€ช mโ€ฌืžื—ืœื™ืง ืขืœ ื”ืฉื•ืœื—ืŸ ื•ืžืชื ื’ืฉ ื‘ืชื—ืชื™ืช ื”ืžื•ื˜โ€ฌ
โ€ซื”ืชื ื’ืฉื•ืช ืืœืกื˜ื™ืช ืœื—ืœื•ื˜ื™ืŸโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืื™ืœื• ื’ื“ืœื™ื ืคื™ืกื™ืงืœื™ื™ื ื ืฉืžืจื™ื ื‘ืชื”ืœื™ืšโ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืื ื”ืžื˜ื‘ืข ื ืฉืืจ ื‘ืžื ื•ื—ื” ืœืื—ืจ ื”ื”ืชื ื’ืฉื•ืชโ€ช ,โ€ฌืžื”ื• ื™ื—ืก ื”ืžืกื•ืช ื‘ื™ืŸ ื”ืžื˜ื‘ืข ืœืžื•ื˜โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_114.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืจ' ื™ื“ื™ื“ื ื• ื”ื•ื•ืชื™ืงโ€ช ,โ€ฌื ื•ื”ื’ ืœืฉื—ืง ื‘ื™ืœื™ืืจื“ ืžื“ื™ ื™ื•ื ื—ืžื™ืฉื™โ€ช .โ€ฌื›ืืฉืจ ื”ื’ื™ืข ืชื•ืจื• ืœื”ื›ื•ืชโ€ช ,โ€ฌื ื™ืกื” ืจ' ืœื—ืฉื‘ ื”ื™ื›ืŸ ื ื™ืชืŸ ืœื”ื›ื•ืช ื‘ื›ื“ื•ืจ ืžื›ื” ืื•ืคืงื™ืช ื›ืšโ€ฌ
โ€ซืฉื”ื›ื“ื•ืจ ื™ืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื” ืžื™ื“ ืœืื—ืจ ื”ืžื›ื”โ€ช .โ€ฌืขื–ืจื• ืกื˜ื•ื“ื ื˜ื™ื ื™ืงืจื™ื ืœืจ' ืœืžืฆื•ื ืืช ื”ื ืงื•ื“ื” ื”ื ื›ื•ื ื”โ€ช.โ€ฌโ€ฌ
โ€ซืชืฉื•ื‘ื” ืกื•ืคื™ืชโ€ช 2/5R :โ€ฌืžืขืœ ื”ืฆื™ืจ ื”ืขื•ื‘ืจ ื‘ืžืจื›ื– ื”ื›ื“ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_115.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืœืื—ืจ ืฉื ื™ืกื” ื•ืœื ื”ืฆืœื™ื— ืœื”ื›ื•ืช ื‘ื“ื™ื•ืง ื‘ื ืงื•ื“ื” ื”ืžื‘ื•ืงืฉืชโ€ช ,โ€ฌืืœื ื“ื•ื•ืงื ื‘ื’ื•ื‘ื” โ€ช Rโ€ฌืžื”ืฉื•ืœื—ืŸโ€ช ,โ€ฌืจืื” ืจ' ื›ื™ืฆื“ ื”ื›ื“ื•ืจ ืžืชื’ืœื’ืœ ื•ืžื—ืœื™ืง ืืšโ€ฌ
โ€ซืœืื—ืจ ืžืจื—ืง ืžืกื•ื™ื ื”ืชื’ืœื’ืœ ื”ื›ื“ื•ืจ ืœืœื ื”ื—ืœืงื”โ€ช .โ€ฌืžื”ื• ื”ืžืจื—ืง ื”ืื•ืคืงื™ ืืฉืจ ืขื‘ืจ ื”ื›ื“ื•ืจ ืขื“ ืฉื”ืชื’ืœื’ืœ ืœืœื ื”ื—ืœืงื” ?โ€ฌ
โ€ซื”ื ื—โ€ช/โ€ฌื™ ื›ื™ ืžืงืžื™ ื”ื—ื™ื›ื•ืš ืฉื•ื•ื™ื โ€ช µ s = µ k = µ :โ€ฌื•ื”ืžื”ื™ืจื•ืช ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ื›ื“ื•ืจ ื”ื™ื โ€ช.v0โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_314.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืงื•ืจื” ืฉืžืกืชื” โ€ช m=10kgโ€ฌื•ืื•ืจื›ื” โ€ช L=0.5 mโ€ฌืžื•ื—ื–ืงืช ืœืงื™ืจ ืข"ื™ ืฆื™ืจโ€ช .โ€ฌืžืฉื—ืจืจื™ื ืืช ื”ืงื•ืจื” ืžืžื ื•ื—ื”โ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืงื•ืจื” ื™ื—ืกื™ืช ืœืฆื™ืจ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžืฆืื• ื‘ื™ื˜ื•ื™ ืœืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ื›ืคื•ื ืงืฆื™ื” ืฉืœ ื”ื–ื•ื•ื™ืช โ€ช.bโ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ืŸ ื”ืชืื•ืฆื•ืช ื”ืžืฉื™ืงื™ื•ืชโ€ช aA ,โ€ฌื•โ€ช aB -โ€ฌื‘ื ืงื•ื“ื•ืช โ€ช Aโ€ฌื•โ€ช B-โ€ฌื›ืืฉืจ โ€ช? b=500โ€ฌื ืชื•ืŸ โ€ช? OA=0.15m, OB=0.4m :โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_315.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื˜ืงื” ืขืฉื•ื™ื” ืžืงืจืฉ ื‘ืขืœ ืฆืคื™ืคื•ืช ืžืกื” ืฉื˜ื—ื™ืช ืฉืœโ€ฌ
โ€ซื‘ืฆื•ืจืช ื“ื™ืกืงื” ื‘ืจื“ื™ื•ืก โ€ช ,[cm] 15โ€ฌื•ืžื•ื˜ ื—ื“ ืžืžื“ื™ ื‘ืื•ืจืš โ€ช[cm] 20โ€ฌโ€ฌ
โ€ซื•ืžืกื” ืฉืœ โ€ช 50โ€ฌื’ืจืโ€ช ,โ€ฌืขืœ ืคื™ ื”ืฆื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืžื˜ืงื” ื›ืืฉืจ ืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ื•ื ื ืงื•ื“ืช ื”ื—ื™ื‘ื•ืจ ืฉืœ ื”ืžื•ื˜ ืœื“ื™ืกืงื”?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื˜ืงื” ื›ืืฉืจ ืžื•ืคืขืœ ื‘ืงืฆื” ื”ืžื•ื˜ ื›ื•ื— ืฉืœโ€ฌ
โ€ซื”ืชืขืœืžื• ืžื›ื•ื— ื”ื›ื•ื‘ื“ ื‘ืกืขื™ืฃ ื–ื”โ€ช.โ€ฌโ€ฌ
โ€ซืื ื›ื™ืช ืœืžื™ืฉื•ืจ ื”ืžื•ื˜ ื•ื”ื“ื™ืกืงื” )ืข" ืค ื”ืฆื™ื•ืจ(โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_316.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซืžื˜ืงื” ืขืฉื•ื™ื” ืžืงืจืฉ ื‘ืขืœ ืฆืคื™ืคื•ืช ืžืกื” ืฉื˜ื—ื™ืช ืฉืœโ€ฌ
โ€ซื‘ืฆื•ืจืช ื“ื™ืกืงื” ื‘ืจื“ื™ื•ืก โ€ช ,[cm] 15โ€ฌื•ืžื•ื˜ ื—ื“ ืžืžื“ื™ ื‘ืื•ืจืš โ€ช[cm] 20โ€ฌโ€ฌ
โ€ซื•ืžืกื” ืฉืœ โ€ช 50โ€ฌื’ืจืโ€ช ,โ€ฌืขืœ ืคื™ ื”ืฆื™ื•ืจโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ืžื•ืžื ื˜ ื”ื”ืชืžื“ ืฉืœ ื”ืžื˜ืงื” ื›ืืฉืจ ืฆื™ืจ ื”ืกื™ื‘ื•ื‘ ื”ื•ื ื ืงื•ื“ืช ื”ื—ื™ื‘ื•ืจ ืฉืœ ื”ืžื•ื˜ ืœื“ื™ืกืงื”?โ€ฌ
โ€ซืื ื›ื™ืช ืœืžื™ืฉื•ืจ ื”ืžื•ื˜ ื•ื”ื“ื™ืกืงื” )ืข" ืค ื”ืฆื™ื•ืจ(โ€ช.โ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ืฉืœ ื”ืžื˜ืงื” ื›ืืฉืจ ืžื•ืคืขืœ ื‘ืงืฆื” ื”ืžื•ื˜ ื›ื•ื— ืฉืœโ€ฌ
โ€ซื”ืชืขืœืžื• ืžื›ื•ื— ื”ื›ื•ื‘ื“ ื‘ืกืขื™ืฃ ื–ื”โ€ช.โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_394.htmlโ€ฌโ€ฌ
โ€ซื’ื•ืฃ ืงืฉื™ื—โ€ฌ
โ€ซื‘ืชื—ืจื•ืช ืจื™ืฆื” ื‘ืžืกืœื•ืœ ืžืขื’ืœื™โ€ช ,โ€ฌืฉื ืขืจื›ื” ืœืงืจืืช ื”ืื•ืœื™ืžืคื™ืื“ื” )ื‘ื™ื™ื’'ื™ืŸ โ€ช (2008โ€ฌื‘ืฉื›ื•ื ื” ื“'โ€ช ,โ€ฌืœืคื ื™ ื”ื”ืงืคื” ื”ืื—ืจื•ื ื” ืžื•ื‘ื™ืœ ื”ืืฆืŸ ื”ื ื•ื“ืข ืจื‘โ€ฌ
โ€ซืฉื•ื“ื“ ืขืœ ืคื ื™ ื”ืืฆืŸ ื”ืžืคื•ืจืกื ืจื‘ ืฉื•ื˜ืจ ื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช) ฯ€ /8โ€ฌื›ืœื•ืžืจ ืœืจื‘ ืฉื•ื˜ืจ ื ื•ืชืจื” ื”ืงืคื” ืฉืœื™ืžื” ื•ืจื‘ ืฉื•ื“ื“ ื”ืงืคื” ืคื—ื•ืช โ€ช .(ฯ€ /8โ€ฌื™ื“ื•ืข ื›ื™โ€ฌ
โ€ซืจื“ื™ื•ืกื• ืฉืœ ื”ืžืกืœื•ืœ ื”ืžืขื’ืœื™ ื‘ืฉื›ื•ื ื” ื”ื•ื โ€ช [m] 100โ€ฌื•ื›ืŸ ืฉืžื”ื™ืจื•ืชื ืฉืœ ืจื‘ ืฉื•ื“ื“ ื•ืจื‘ ืฉื•ื˜ืจ ื”ื™ื โ€ช.[km/h] 54โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌื›ืขื‘ื•ืจ ื›ืžื” ื–ืžืŸ ื™ื’ื™ืขื• ืจื‘ ืฉื•ื“ื“ ื•ืจื‘ ืฉื•ื˜ืจ ืœืงื• ื”ืกื™ื•ื ื‘ืชื ืื™ื ืืœื• ?โ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ื”ืชื—ืฉื‘ ื‘ืขื•ื‘ื“ื” ื›ื™ ืจื‘ ืฉื•ื“ื“ ื”ื•ื ืžืขืฉืŸ ื›ื‘ื“โ€ช ,โ€ฌื•ืžืชื—ื™ืœ ืœื”ืื˜ ืœืงืจืืช ื”ืกื™ื•ืโ€ช ,โ€ฌืžื” ืชื”ื™ื” ื”ืชืื•ื˜ื” )ืชืื•ืฆื” ืฉืœื™ืœื™ืช( ื”ื–ื•ื•ื™ืชื™ืช ื”ืงื‘ื•ืขื”โ€ฌ
โ€ซื”ืžื™ื ื™ืžืœื™ืช ืฉืชื’ืจื•ื ืœืžื”ืคืš ื‘ืžื™ืจื•ืฅ ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืื ื™ืคืชื™ืข ืจื‘ ืฉื•ื“ื“ ื•ื™ืžืฉื™ืš ื‘ืื•ืชื• ืงืฆื‘ ืขื“ ื”ืกื™ื•ื ืžื” ืชื”ื™ื” ื”ืชืื•ืฆื” ื”ื–ื•ื•ื™ืชื™ืช ื”ืžื™ื ื™ืžืœื™ืช ืฉืœ ืจื‘ ืฉื•ื˜ืจ ื›ืš ืฉื™ืขืงื•ืฃ ืืช ื™ืจื™ื‘ื• ื”ืžืจโ€ฌ
โ€ซื‘ื–ื•ื•ื™ืช ืฉืœ โ€ช ฯ€ /10โ€ฌืžืงื• ื”ืกื™ื•ื?โ€ฌ
โ€ซโ€ช22_Rigid_body/e_22_8_395.htmlโ€ฌโ€ฌ
โ€ซโ€ช2. Figure 9-40 shows the lines of action and the points of application of two forces about the origin 0, allโ€ฌโ€ฌ
โ€ซโ€ชvectors being in the plane of the figure.โ€ฌโ€ฌ
โ€ซโ€ชImagine these forces to be acting on a rigid body pivoted about an axis through 0 and perpendicular to the planeโ€ฌโ€ฌ
โ€ซโ€ชof the figure.โ€ฌโ€ฌ
โ€ซโ€ช(a) Find an expression for the magnitude of the resultant torque on the body.โ€ฌโ€ฌ
โ€ซโ€ช(b) If r1 = 1.30 m, r2 = 2.15 m, F1 = 4.20 N, F2= 4.90 N, 1 = 75.0 , and 2 = 58.0 ,โ€ฌโ€ฌ
โ€ซ?โ€ชwhat are the magnitude and direction of the resultant torqueโ€ฌโ€ฌ
22_Rigid_body/e_22_8_396.html
3. Redraw Fig. 9-40 under the following transformations:
(a) F -F,
(b) r -r, and
(c) F -F and r -r, in each case showing the new direction of the torque. Check for consistency with the
right-hand rule.
22_Rigid_body/e_22_8_397.html
9. What is the torque about the origin on a particle located at x = 1.5 m, y = -2.0 m, Z = 1.6 m due to a force
F = (3.5 N)i - (2.4 N)j + (4.3 N)k?
Express your result in unit vector notation.
22_Rigid_body/e_22_8_398.html
10. A particle is located at r = (0.54 m)i + (-0.36 m)j + (0.85 m)k. A constant force of magnitude 2.6 N acts on
the particle.
Find the components of the torque about the origin when the force acts in
(a) the positive x direction and
(b) the negative z direction.
22_Rigid_body/e_22_8_399.html
12. Three particles are attached to a thin rod of length 1.00 m and negligible mass that pivots about the origin in
the xy plane.
Particle 1 (mass 52 g) is attached a distance of 27 cm from the origin, particle 2 (35 g) is at 45 cm, and particle
3 (24 g) at 65 cm.
(a) What is the rotational inertia of the assembly?
(b) If the rod were instead pivoted about the center of mass of the assembly, what would be the rotational
inertia?
22_Rigid_body/e_22_8_400.html
13. Two thin rods of negligible mass are rigidly attached at their ends to form a 90° angle. The rods rotate in the
xy plane with the joined ends forming the pivot at the origin.
A particle of mass 75 g is attached to one rod a distance of 42 cm from the origin, and a particle of mass 30 g is
attached to the other rod a distance of 65 cm from the origin.
(a) What is the rotational inertia of the assembly?
(b) How would the rotational inertia change if the particles were both attached to one rod at the given distances
from the origin?
22_Rigid_body/e_22_8_401.html
15. A helicopter rotor blade is 7.80 m long and has a mass of 110 kg.
(a) What force is exerted on the bolt attaching the blade to the rotor axle when the rotor is turning at 320
rev/min?
(Hint: For this calculation the blade can be considered to be a point mass at the center of mass. Why?)
(b) Calculate the torque that must be applied to the rotor to bring it to full speed from rest in 6.70 s. Ignore air
resistance.
(The blade cannot be considered to be a point mass for this calculation. Why not? Assume the distribution of a
uniform rod.)
22_Rigid_body/e_22_8_402.html
22. A certain nut is known to require forces of 46 N exerted on it from both sides to crack it.
What forces F will be required when it is placed in the nutcracker shown in Fig. 9-46?
22_Rigid_body/e_22_8_403.html
23. The leaning Tower of Pisa (see Fig. 9-47) is 55 m high and 7.0 m in diameter.
The top of the tower is displaced 4.5 m from the vertical. Treating the tower as a uniform, circular cylinder,
(a) What additional displacement, measured at the top, will bring the tower to the verge of toppling?
(b) What angle with the vertical will the tower make at that moment? (The current rate of movement of the top
is 1 mmlyear.)
22_Rigid_body/e_22_8_404.html
24. A cube stays at rest on a horizontal table when a small horizontal force is applied perpendicular to and at
the center of an upper edge.
The force is now steadily increased. Does the cube slide or topple first?
The coefficient of static friction between the surfaces is equal to 0.46.
22_Rigid_body/e_22_8_405.html
28. A diver of weight 582 N stands at the end of a uniform 4.48-m diving board of weight 142 N.
The board is attached by two pedestals 1.55 m apart, as shown in Fig. 9-48.
Find the tension (or compression) in each of the two pedestals.
22_Rigid_body/e_22_8_406.html
29. What minimum force F applied horizontally at the axle of the wheel in Fig. 9-49 is necessary to raise the
wheel over an obstac1e of height h?
Take r as the radius of the wheel and W as its weight.
22_Rigid_body/e_22_8_407.html
31. One end of a uniform beam weighing 52.7 1b and 3.12 ft long is attached to a wall with a hinge.
The other end is supported by a wire making equal angles of 27.0° with the beam and wall (see Fig. 9-51).
(a) Find the tension in the wire.
(b) Compute the horizontal and vertical components of the force on the hinge.
22_Rigid_body/e_22_8_408.html
19. A uniform disk of radius R and mass M is spinning with angular speed
surface;
the coefficient of kinetic friction between disk and surface is k.
(a) Find the frictional torque on the disk.
(b) How long will it take for the disk to come to rest?
o. It is placed on a flat horizontal
22_Rigid_body/e_22_8_409.html
20. A hoop rolling down an inclined plane of inclination angle keeps pace with a block sliding down the
same plane.
Show that the coefficient of kinetic friction between the block and the plane is given by k =1/2 tan .
22_Rigid_body/e_22_8_410.html
21. A uniform sphere rolls down an incline.
(a) What must be the incline angle if the linear acceleration of the center of the sphere is to be O.133g?
(b) For this angle, what would be the acceleration of a frictionless block sliding down the incline?
22_Rigid_body/e_22_8_411.html
22. A solid cylinder of length L and radius R has a weight W. Two cords are wrapped around the cylinder, one
near each end, and the cord ends are attached to hooks on the ceiling.
The cylinder is held horizontally with the two cords exactly vertical and is then released (Fig. 9-66). Find
(a) the tension in each cord as they unwind and
(b) the linear acceleration of the cylinder as it falls.
22_Rigid_body/e_22_8_412.html
23. Show that a cylinder will slip on an inclined plane of inclination angle
if the coefficient of static friction
between plane and cylinder is less than 1/3 tan .
22_Rigid_body/e_22_8_413.html
24. A uniform disk, of mass M and radius R, lies on one side initially at rest on a frictionless horizontal surface.
A constant force F is than applied tangentially at its perimeter by means of a string wrapped around its edge.
Describe the subsequent (rotational and translational) motion of the disk.
22_Rigid_body/e_22_8_414.html
25. A sphere, a cylinder, and a hoop (each of radius R and mass M) start from rest and roll down the same
incline.
(a) Which object gets to the bottom first?
(b) Does your answer depend on the mass or radius of the objects? Explain.
22_Rigid_body/e_22_8_415.html
9. A sanding disk with rotational inertia 1.22 X 10-3 kg m2 is attached to an electric drill whose motor
delivers a torque of 15.8 N m. Find
(a) the angular momentum and
(b) the angular speed of the disk 33.0 ms after the motor is turned on. (1 ms = 10-3 s)
22_Rigid_body/e_22_8_416.html
10. A wheel of radius 24.7 cm, moving initially at 43.3 m/s, rolls to a stop in 225 m. Calculate
(a) its linear acceleration and
(b) its angular acceleration.
(c) The wheel's rotational inertia is 0.155 kg m2. Calculate the torque exerted by rolling friction on the wheel.
22_Rigid_body/e_22_8_417.html
13. A uniform stick has a mass of 4.42 kg and a length of 1.23 m. It is initially lying flat at rest on a frictionless
horizontal surface
and is struck perpendicularly by a puck (rubber disk) imparting a horizontal impulsive force of impulse 12.8 N
s at a distance of 46.4 cm from the center.
Determine the subsequent motion of the stick.
22_Rigid_body/e_22_8_418.html
18. In a lecture demonstration, a toy train track is mounted on a large wheel that is free to turn with negligible
friction about a vertical axis; see Fig. 10-25.
A toy train of mass m is placed on the track and, with the system initially at rest, the electrical power is turned
on.
The train reaches a steady speed v with respect to the track. What is the angular velocity of the wheel, if its
mass is M and its radius R?
(Neglect the mass of the spokes (wires) of the wheel.)
22_Rigid_body/e_22_8_419.html
24. A girl of mass 50.6. kg stands on the edge of a frictionless merry-go-round of mass 827 kg and radius 3.72
m that is not moving.
She throws a l.13-kg rock in a horizontal direction that is tangent to the outer edge of the merry-go-round.
The speed of the rock, relative to the ground, is 7.82 m/s. Calculate
(a) the angular speed of the merry-go-round and
(b) the linear speed of the girl after the rock is thrown. Assume that the merry-go-round is a uniform disk.
22_Rigid_body/e_22_8_420.html
26. A top is spinning at 28.6 rev/s about an axis making an angle of 34.0° with the vertical.
Its mass is 492 g and its rotational inertia is 5.12 x 10-4 kg m2. The center of mass is 3.88 cm from the pivot
point.
The spin is clockwise as seen from above.
Find the magnitude (in rev/s) and direction of the angular velocity of precession.
22_Rigid_body/e_22_8_421.html
3. To get a billiard ball to roll without sliding from the start, the cue must hit the ball not at the center
(that is, a height above the table equal to the ball's radius R) but exactly at a height 2R/5 above the center.
Prove this result.
22_Rigid_body/e_22_8_422.html
5. A billiard ball, initially at rest, is given a sharp impulse by a cue. The cue is held horizontally a distance h
above the centerline as in Fig. 10-28.
The ball leaves the cue with a speed vo and, because of its "forward English" (spin), eventually acquires a final
speed of 9vo/7.
Show that h = 4R/5, where R is the radius of the ball.
22_Rigid_body/e_22_8_423.html
8. A uniform fiat disk of mass M and radius R rotates about a horizontal axis through its center with angular
speed o.
(a) What is its angular momentum?
(b) A chip of mass m breaks off the edge of the disk at an instant such that the chip rises vertically above the
point at which it broke off (Fig. 10-29).
How high above the point does it rise before starting to fall?
(c) What is the final angular speed of the broken disk?
โ€ซโ€ช22_Rigid_body/e_25_4_001.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื”โ€ฌ
โ€ซื‘ืื•ืคืŸ ืจื’ื™ืœ ื›ื•ื— ื”ืžืฉื™ื›ื” ื‘ื™ืŸ ืฉื ื™ ื’ื•ืคื™ื ืขื•ืœื” ืขื ื”ืงื˜ื ืช ื”ืžืจื—ืง ื‘ื ื™ื”ืโ€ช.โ€ฌโ€ฌ
โ€ซืื‘ืœโ€ช...โ€ฌโ€ฌ
โ€ซื”ื•ื›ื—โ€ช/โ€ฌื™ ื›ื™ ืขื‘ื•ืจ ื”ืชืฆื•ืจื” ื”ื‘ืื” ืงื™ื™ื ื˜ื•ื•ื— ืžืจื—ืงื™ื ืžืกื•ื™ื ื‘ื• ื›ื•ื— ื”ื›ื•ื‘ื“ ื™ื•ืจื“ ืขื ื”ืงื˜ื ืช ื”ืžืจื—ืง ื‘ื ื™ื”ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_25_4_002.htmlโ€ฌโ€ฌ
โ€ซื’ื™ืื•ืชโ€ฌ
โ€ซื( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื›ื•ื— ื”ื›ื‘ื™ื“ื” ื”ืคื•ืขืœ ืขืœ ื”ืžื™ื ืขืœ ื™ื“ื™ ื”ืฉืžืฉ ื•ืขืœ ื™ื“ื™ ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืœืื•ืจ ื”ืชื•ืฆืื” ืฉืœ ืกืขื™ืฃ ื'โ€ช ,โ€ฌืžื“ื•ืข ืžื™ื™ื—ืกื™ื ืืช ืชื•ืคืขืช ื”ื’ื™ืื•ืช ื‘ืขื™ืงืจ ืœื™ืจื—? )ืจืžื–โ€ช :โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื™ื—ืก ื‘ื™ืŸ ื›ื•ื— ื”ื’ืื•ืช ืฉืœ ื”ืฉืžืฉ ื•ืฉืœโ€ฌ
โ€ซื”ื™ืจื—(โ€ฌ
โ€ซโ€ช22_Rigid_body/e_25_4_003.htmlโ€ฌโ€ฌ
โ€ซื”ื’ืขื” ืœื™ืจื—โ€ฌ
โ€ซืื™ื–ื• ืžื”ื™ืจื•ืช ืฆืจื™ืš ืœื”ืขื ื™ืง ืœื˜ื™ืœ ืข"ืž ืฉื™ื’ื™ืข ืžื›ื“ื•"ื” ืœื™ืจื—?โ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌื”ืฉืืœื” ืœื ืคืฉื•ื˜ื” ื›ืคื™ ืฉื”ื™ื ื ืจืื™ืช!โ€ฌ
โ€ซโ€ช22_Rigid_body/e_25_4_004.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื”โ€ฌ
โ€ซืกืคื™ื ืช ื—ืœืœ ื ืžืฆืืช ื‘ืžืกืœื•ืœ ืžืขื’ืœื™ ื ืžื•ืš ืกื‘ื™ื‘ ื›ื•ื›ื‘ ืœื›ืช ื‘ืžืขืจื›ืช ื›ื•ื›ื‘ื™ื ืฉื”ืชื’ืœืชื” ืœืื—ืจื•ื ื”โ€ช .โ€ฌื”ื˜ื‘ื— ื”ื›ื ื™ืก ืืช ืืจื•ื—ืช ื”ืขืจื‘ ืœืชื ื•ืจโ€ช,โ€ฌโ€ฌ
โ€ซื•ื›ื™ื•ื•ืŸ ืืช ื”ืฉืขื•ืŸ ืœโ€ช 45-โ€ฌื“ืงื•ืชโ€ช .โ€ฌืœืื—ืจ ืฉืขื‘ืจื• โ€ช 45โ€ฌื”ื“ืงื•ืช ื”ื˜ื‘ื— ืฉื ืœื‘ ืœื›ืš ืฉื”ืกืคื™ื ื” ื ืžืฆืืช ื‘ื“ื™ื•ืง ื‘ืื•ืชื• ืžืงื•ื ื‘ืžืกืœื•ืœื” ืฉื‘ื• ื”ื™ื™ืชื”โ€ฌ
โ€ซื‘ืขืช ืฉื”ื•ื ื”ื›ื ื™ืก ืืช ื”ืืจื•ื—ื” ืœืชื ื•ืจโ€ช .โ€ฌื›ืืฉืจ ืฆื™ื™ืŸ ื–ืืช ื‘ืื–ื ื™ ื”ืฆื•ื•ืชโ€ช ,โ€ฌื”ืชืจื’ืฉ ื”ืงืฆื™ืŸ ื”ื˜ื›ื ื™ ืฉืœ ื”ืกืคื™ื ื” ื•ืงืจืโ€ช " :โ€ฌื–ื” ืจืง ื‘ืžืขื˜ ื™ื•ืชืจ ืžื—ืฆื™โ€ฌ
โ€ซืชืงื•ืคืช ืžืกืœื•ืœื” ืฉืœ ืชื—ื ืช ื”ื—ืœืœ ื”ื ืžืฆืืช ื‘ืžืกืœื•ืœ ื ืžื•ืš ืกื‘ื™ื‘ ื›ื“ื•"ื”!"โ€ช .โ€ฌืœืžืฉืžืข ื”ืงืฆื™ืŸ ื”ื˜ื›ื ื™ ืงื ืžื™ื™ื“ ื”ืงืคื˜ืŸ ื•ื”ื•ืจื” ืขืœ ื”ื›ื ืช ืฆื•ื•ืช ื ื—ื™ืชื”โ€ฌ
โ€ซืžืฉื•ื ืฉืœื˜ืขื ืชื• ื”ื›ื•ื›ื‘ ืขืฉื•ื™ ืคืœื˜ื™ื ื” ื˜ื”ื•ืจื”! ืื™ืš ื”ื’ื™ืข ื”ืงืคื˜ืŸ ืœืžืกืงื ืชื•?โ€ฌ
โ€ซืฆืคื™ืคื•ืช ื”ืคืœื˜ื™ื ื” โ€“ โ€ชgr/cm3 21.4โ€ฌโ€ฌ
โ€ซืฆืคื™ืคื•ืช ื›ื“ื•"ื” โ€“ โ€ชgr/cm3 5.3โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌืื™ืŸ ืงืฉืจ ืœื˜ื‘ื—!โ€ฌ
โ€ซโ€ช22_Rigid_body/e_25_4_005.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื”โ€ฌ
โ€ซื( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืฆืคื™ืคื•ืช ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืชืื•ืฆืช ื”ื›ื•ื‘ื“ ืขืœ ืคื ื™ ื”ื™ืจื—โ€ช ,โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื‘ืขื–ืจืช ื˜ื•ืจ ื˜ื™ื™ืœื•ืจ )ืกื“ืจ โ€ช (Iโ€ฌืขื“ ื›ืžื” ื”ื™ื ืžืฉืชื ื” ื”ื’ื•ื‘ื” ื›ื‘ื™ื“ื” ืžืคื ื™ ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘ื›ืžื” ืชืฉืชื ื” ืชืฉื•ื‘ืชืš ืื ืชืงื—โ€ช/โ€ฌื™ ืกื“ืจ โ€ช ?IIโ€ฌืžื” ื”ื™ื ื”ืฉื’ื™ืื” ื™ื—ืกื™ืช ืœื—ื™ืฉื•ื‘ ื”ืžื“ื•ื™ืง?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช22_Rigid_body/e_25_4_006.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื” ื•ืชื ื•ืขื” ื”ืจืžื•ื ื™ืชโ€ฌ
โ€ซืžื ื”ืจื” ืฆืจื” ื•ื—ืœืงื” ื ื—ืคืจื” ืœืื•ืจืš ืงื•ื˜ืจื• ืฉืœ ื›ื“ื•"ื” )ื'(โ€ช .โ€ฌืขื•ื–ื‘ื™ื ื‘ืคืชื—ื” ื›ื“ื•ืจโ€ช .โ€ฌื‘ื”ื ื—ื” ืฉืฆืคื™ืคื•ืช ื›ื“ื•"ื” ืงื‘ื•ืขื”โ€ช ,โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™โ€ช:โ€ฌโ€ฌ
โ€ซื( ืชื ื•ืขืช ื”ื›ื“ื•ืจ ื”ื™ื ืชื”"ืคโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื”ืื ืชื ื•ืขืชื• ืชื”ื™ื” ืชื”"ืค ื’ื ืขื‘ื•ืจ ืžืงืจื™ื ื‘' ื•ื’'?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_1_011.htmlโ€ฌโ€ฌ
โ€ซโ€ชGravityโ€ฌโ€ฌ
โ€ซโ€ช. The star has aโ€ฌโ€ฌ
โ€ซโ€ชand radiusโ€ฌโ€ฌ
โ€ซโ€ชWhat force acts on a star inside a spherically symmetric galaxy of the massโ€ฌโ€ฌ
โ€ซโ€ชmassโ€ฌโ€ฌ
โ€ซโ€ชand is at binthe radiusโ€ฌโ€ฌ
โ€ซโ€ชfrom the center of the galaxy.โ€ฌโ€ฌ
โ€ซโ€ช25_Gravity/e_25_1_012.htmlโ€ฌโ€ฌ
โ€ซโ€ชGravityโ€ฌโ€ฌ
โ€ซโ€ชA binary stellar system consists of two identical stars rotating around the center-of-mass of the system onโ€ฌโ€ฌ
โ€ซโ€ชcircular orbits. The period of rotationโ€ฌโ€ฌ
โ€ซโ€ชand the velocity of the stars are known. Find the masses and theโ€ฌโ€ฌ
โ€ซโ€ชdistance between the stars.โ€ฌโ€ฌ
โ€ซโ€ช25_Gravity/e_25_1_013.htmlโ€ฌโ€ฌ
Gravity
Saturn rings consist of football ball size particles which are moving on circular orbits around the planet. What is
the maximal ratio of the ring width to its inner radius if the velocities at the inner and outer edge should not
differ by more than 0.5\% ?
25_Gravity/e_25_1_014.html
Gravity
Three identical stars with the mass
are rotating so that they forman equilateral triangle (side length ). What
is the angular velocity ? What is the ratio
? Is this configuration stable ?
25_Gravity/e_25_1_015.html
Gravity
A particle is moving along the axis of a homogeneous ring (mass
, radius ). The particle velocity at
infinity is zero. What is its velocity when it passes through the center of the ring ?
25_Gravity/e_25_1_016.html
Gravity
The space between the two concentric spheres with the radii and ,
constant density . Find the gravitational field as a function of radius
, is filled with a matter with the
in the whole space.
25_Gravity/e_25_1_017.html
Gravity
A binary system consists of two stars with the masses
period of the orbital motion.
and
. The distance between them is
. Find the
25_Gravity/e_25_4_001.html
โ€ซื›ื‘ื™ื“ื”โ€ฌ
.โ€ซื‘ืื•ืคืŸ ืจื’ื™ืœ ื›ื•ื— ื”ืžืฉื™ื›ื” ื‘ื™ืŸ ืฉื ื™ ื’ื•ืคื™ื ืขื•ืœื” ืขื ื”ืงื˜ื ืช ื”ืžืจื—ืง ื‘ื ื™ื”ืโ€ฌ
...โ€ซืื‘ืœโ€ฌ
:โ€ซื™ ื›ื™ ืขื‘ื•ืจ ื”ืชืฆื•ืจื” ื”ื‘ืื” ืงื™ื™ื ื˜ื•ื•ื— ืžืจื—ืงื™ื ืžืกื•ื™ื ื‘ื• ื›ื•ื— ื”ื›ื•ื‘ื“ ื™ื•ืจื“ ืขื ื”ืงื˜ื ืช ื”ืžืจื—ืง ื‘ื ื™ื”ืโ€ฌ/โ€ซื”ื•ื›ื—โ€ฌ
25_Gravity/e_25_4_002.html
โ€ซื’ื™ืื•ืชโ€ฌ
โ€ซื( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื›ื•ื— ื”ื›ื‘ื™ื“ื” ื”ืคื•ืขืœ ืขืœ ื”ืžื™ื ืขืœ ื™ื“ื™ ื”ืฉืžืฉ ื•ืขืœ ื™ื“ื™ ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ืœืื•ืจ ื”ืชื•ืฆืื” ืฉืœ ืกืขื™ืฃ ื'โ€ช ,โ€ฌืžื“ื•ืข ืžื™ื™ื—ืกื™ื ืืช ืชื•ืคืขืช ื”ื’ื™ืื•ืช ื‘ืขื™ืงืจ ืœื™ืจื—? )ืจืžื–โ€ช :โ€ฌื—ืฉื‘โ€ช/โ€ฌื™ ืืช ื”ื™ื—ืก ื‘ื™ืŸ ื›ื•ื— ื”ื’ืื•ืช ืฉืœ ื”ืฉืžืฉ ื•ืฉืœโ€ฌ
โ€ซื”ื™ืจื—(โ€ฌ
โ€ซโ€ช25_Gravity/e_25_4_003.htmlโ€ฌโ€ฌ
โ€ซื”ื’ืขื” ืœื™ืจื—โ€ฌ
โ€ซืื™ื–ื• ืžื”ื™ืจื•ืช ืฆืจื™ืš ืœื”ืขื ื™ืง ืœื˜ื™ืœ ืข"ืž ืฉื™ื’ื™ืข ืžื›ื“ื•"ื” ืœื™ืจื—?โ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌื”ืฉืืœื” ืœื ืคืฉื•ื˜ื” ื›ืคื™ ืฉื”ื™ื ื ืจืื™ืช!โ€ฌ
โ€ซโ€ช25_Gravity/e_25_4_004.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื”โ€ฌ
โ€ซืกืคื™ื ืช ื—ืœืœ ื ืžืฆืืช ื‘ืžืกืœื•ืœ ืžืขื’ืœื™ ื ืžื•ืš ืกื‘ื™ื‘ ื›ื•ื›ื‘ ืœื›ืช ื‘ืžืขืจื›ืช ื›ื•ื›ื‘ื™ื ืฉื”ืชื’ืœืชื” ืœืื—ืจื•ื ื”โ€ช .โ€ฌื”ื˜ื‘ื— ื”ื›ื ื™ืก ืืช ืืจื•ื—ืช ื”ืขืจื‘ ืœืชื ื•ืจโ€ช,โ€ฌโ€ฌ
โ€ซื•ื›ื™ื•ื•ืŸ ืืช ื”ืฉืขื•ืŸ ืœโ€ช 45-โ€ฌื“ืงื•ืชโ€ช .โ€ฌืœืื—ืจ ืฉืขื‘ืจื• โ€ช 45โ€ฌื”ื“ืงื•ืช ื”ื˜ื‘ื— ืฉื ืœื‘ ืœื›ืš ืฉื”ืกืคื™ื ื” ื ืžืฆืืช ื‘ื“ื™ื•ืง ื‘ืื•ืชื• ืžืงื•ื ื‘ืžืกืœื•ืœื” ืฉื‘ื• ื”ื™ื™ืชื”โ€ฌ
โ€ซื‘ืขืช ืฉื”ื•ื ื”ื›ื ื™ืก ืืช ื”ืืจื•ื—ื” ืœืชื ื•ืจโ€ช .โ€ฌื›ืืฉืจ ืฆื™ื™ืŸ ื–ืืช ื‘ืื–ื ื™ ื”ืฆื•ื•ืชโ€ช ,โ€ฌื”ืชืจื’ืฉ ื”ืงืฆื™ืŸ ื”ื˜ื›ื ื™ ืฉืœ ื”ืกืคื™ื ื” ื•ืงืจืโ€ช " :โ€ฌื–ื” ืจืง ื‘ืžืขื˜ ื™ื•ืชืจ ืžื—ืฆื™โ€ฌ
โ€ซืชืงื•ืคืช ืžืกืœื•ืœื” ืฉืœ ืชื—ื ืช ื”ื—ืœืœ ื”ื ืžืฆืืช ื‘ืžืกืœื•ืœ ื ืžื•ืš ืกื‘ื™ื‘ ื›ื“ื•"ื”!"โ€ช .โ€ฌืœืžืฉืžืข ื”ืงืฆื™ืŸ ื”ื˜ื›ื ื™ ืงื ืžื™ื™ื“ ื”ืงืคื˜ืŸ ื•ื”ื•ืจื” ืขืœ ื”ื›ื ืช ืฆื•ื•ืช ื ื—ื™ืชื”โ€ฌ
โ€ซืžืฉื•ื ืฉืœื˜ืขื ืชื• ื”ื›ื•ื›ื‘ ืขืฉื•ื™ ืคืœื˜ื™ื ื” ื˜ื”ื•ืจื”! ืื™ืš ื”ื’ื™ืข ื”ืงืคื˜ืŸ ืœืžืกืงื ืชื•?โ€ฌ
โ€ซืฆืคื™ืคื•ืช ื”ืคืœื˜ื™ื ื” โ€“ โ€ชgr/cm3 21.4โ€ฌโ€ฌ
โ€ซืฆืคื™ืคื•ืช ื›ื“ื•"ื” โ€“ โ€ชgr/cm3 5.3โ€ฌโ€ฌ
โ€ซืจืžื–โ€ช :โ€ฌืื™ืŸ ืงืฉืจ ืœื˜ื‘ื—!โ€ฌ
โ€ซโ€ช25_Gravity/e_25_4_005.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื”โ€ฌ
โ€ซื( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืฆืคื™ืคื•ืช ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื—ืฉื‘โ€ช/โ€ฌื™ ืืช ืชืื•ืฆืช ื”ื›ื•ื‘ื“ ืขืœ ืคื ื™ ื”ื™ืจื—โ€ช ,โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื‘ืขื–ืจืช ื˜ื•ืจ ื˜ื™ื™ืœื•ืจ )ืกื“ืจ โ€ช (Iโ€ฌืขื“ ื›ืžื” ื”ื™ื ืžืฉืชื ื” ื”ื’ื•ื‘ื” ื›ื‘ื™ื“ื” ืžืคื ื™ ื”ื™ืจื—โ€ช.โ€ฌโ€ฌ
โ€ซื‘ื›ืžื” ืชืฉืชื ื” ืชืฉื•ื‘ืชืš ืื ืชืงื—โ€ช/โ€ฌื™ ืกื“ืจ โ€ช ?IIโ€ฌืžื” ื”ื™ื ื”ืฉื’ื™ืื” ื™ื—ืกื™ืช ืœื—ื™ืฉื•ื‘ ื”ืžื“ื•ื™ืง?โ€ฌ
โ€ซื ืชื•ื ื™ืโ€ช:โ€ฌโ€ฌ
โ€ซโ€ช25_Gravity/e_25_4_006.htmlโ€ฌโ€ฌ
โ€ซื›ื‘ื™ื“ื” ื•ืชื ื•ืขื” ื”ืจืžื•ื ื™ืชโ€ฌ
โ€ซืžื ื”ืจื” ืฆืจื” ื•ื—ืœืงื” ื ื—ืคืจื” ืœืื•ืจืš ืงื•ื˜ืจื• ืฉืœ ื›ื“ื•"ื” )ื'(โ€ช .โ€ฌืขื•ื–ื‘ื™ื ื‘ืคืชื—ื” ื›ื“ื•ืจโ€ช .โ€ฌื‘ื”ื ื—ื” ืฉืฆืคื™ืคื•ืช ื›ื“ื•"ื” ืงื‘ื•ืขื”โ€ช ,โ€ฌื”ืจืื”โ€ช/โ€ฌื™ ื›ื™โ€ช:โ€ฌโ€ฌ
โ€ซื( ืชื ื•ืขืช ื”ื›ื“ื•ืจ ื”ื™ื ืชื”"ืคโ€ช.โ€ฌโ€ฌ
โ€ซื‘( ื”ืื ืชื ื•ืขืชื• ืชื”ื™ื” ืชื”"ืค ื’ื ืขื‘ื•ืจ ืžืงืจื™ื ื‘' ื•ื’'?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_101.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซืจื•ืฆื™ื ืœื”ื›ื ื™ืก ื˜ื™ืœ ืฉืžืกืชื• โ€ช m=1000kgโ€ฌืœืžืกืœื•ืœ ืžืขื’ืœื™ ืงื‘ื•ืขื” ืกื‘ื™ื‘ ื›ื“ื•ื”" ื ื›ืš ืฉื™ืงื™ืฃ ืื•ืชื• ืื—ืช ืœโ€ช 12-โ€ฌืฉืขื•ืชโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ืจื“ื™ื•ืก ื”ืžืกืœื•ืœ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ื˜ื™ืœ ?โ€ฌ
โ€ซื’โ€ช.โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช โ€ช v0โ€ฌื™ืฉ ืœืฉื’ืจ ืืช ื”ื˜ื™ืœ ืžืคื ื™ ื›ื“ื•ื”" ื )ื‘ื”ื–ื ื—ืช ืงื™ื•ื ื”ืื˜ืžื•ืกืคื™ืจื”( ?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_102.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซื—ืœืœื™ืช ืฉืžืกืชื” โ€ช 10000โ€ฌืง" ื’ ื ืขื” ืกื‘ื™ื‘ ื›ื“ื•ื”" ืื‘ืžืกืœื•ืœ ืžืขื’ืœื™ ืฉืจื“ื™ื•ืกื• โ€ช r = 3Rโ€ฌื‘ืืฉืจ )โ€ช = Rโ€ฌืจื“ื™ื•ืก ื›ื“ื•ื”" ื(โ€ช .โ€ฌื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“ โ€ชgโ€ฌโ€ฌ
โ€ซืข"ืค ื›ื“ื•ื”" ืโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ื›ื•ื‘ื“ ื‘ืžืจื—ืง โ€ช? r = 3 Rโ€ฌโ€ฌ
โ€ซื‘โ€ช.โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื ืขื” ื”ื—ืœืœื™ืช ?โ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื™ ืชื•ืกืคืช ื”ืื ืจื’ื™ื” ืฉื™ืฉ ืœืชืช ืœื—ืœืœื™ืช ื”ื " ืœ ืข" ืž ืฉืชื™ืžืœื˜ ืžืžืงื•ืžื” ?โ€ฌ
โ€ซืœื•ื•ื™ืŸ ืฉืžืกืชื• โ€ช 400โ€ฌืง" ื’ ืกื•ื‘ื‘ ืกื‘ื™ื‘ ื›ื“ื•ื”" ื ื‘ืžืจื—ืง โ€ช Rโ€ฌืžืคื ื™ ื›ื“ื•ื”" ื )โ€ช = Rโ€ฌืจื“ื™ื•ืก ื›ื“ื•ื”" ื(โ€ช .โ€ฌื‘ืขื–ืจืช ืจืงื˜ื” ืžืฉื ื™ื ืืช ืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸโ€ฌ
โ€ซื•ืžืขื‘ื™ืจื™ื ืื•ืชื• ืœืžืกืœื•ืœ ืžืขื’ืœื™ ื—ื“ืฉ ืฉืจื“ื™ื•ืกื• โ€ช.8Rโ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ืฉื”ื•ืฉืงืขื” ืข" ื™ ื”ืจืงื˜ื” ืœืฉื™ื ื•ื™ ืžืกืœื•ืœ ื”ืœื•ื•ื™ืŸ ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื™ ืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸ ื‘ืฉื ื™ ื”ืžืกืœื•ืœื™ื ?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_103.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซืฉื ื™ ื’ื•ืคื™ื ืฉืžืกืชื โ€ช 20โ€ฌืง"ื’ ืžื•ื ื—ื™ื ื‘ื˜ืขื•ืช ื‘ื• ื–ืžื ื™ืช ื›ืœื•ื•ื™ื ื™ื ืกื‘ื™ื‘ ื›ื“ื•ื”" ื ื‘ืจื“ื™ื•ืก ืกื™ื‘ื•ื‘ ืฉืœ โ€ช 107โ€ฌืžื˜ืจื™ื ื›ืืฉืจ ื”ืžืจื—ืง ื‘ื™ื ื”ืโ€ฌ
โ€ซื”ื•ื โ€ช 2*107โ€ฌืžื˜ืจื™ืโ€ช .โ€ฌื”ืžื”ื™ืจื•ืช ื ื™ืชื ืช ืœืœื•ื™ื™ื ื™ื ื‘ื›ื™ื•ื•ื ื™ื ืžื ื•ื’ื“ื™ืโ€ช.โ€ฌโ€ฌ
โ€ซืื ื”ื”ืชื ื’ืฉื•ืช ื‘ื™ืŸ ื”ื’ื•ืคื™ื ื”ื™ื ืืœืกื˜ื™ืช โ€ช:โ€ฌโ€ฌ
โ€ซืโ€ช.โ€ฌืžื”ื• ื”ื–ืžืŸ ืขื“ ืœื”ืชื ื’ืฉื•ืช ื”ืจืืฉื•ื ื” ?โ€ฌ
โ€ซื‘โ€ช.โ€ฌืžื”ื• ื”ื–ืžืŸ ื‘ื™ืŸ ื›ืœ โ€ช 2โ€ฌื”ืชื ื’ืฉื•ื™ื•ืช ืกืžื•ื›ื•ืช ?โ€ฌ
โ€ซืื ื”ื”ืชื ื’ืฉื•ืช ื‘ื™ืŸ ื”ื’ื•ืคื™ื ื”ื™ื ืคืœืกื˜ื™ืช โ€ช:โ€ฌโ€ฌ
โ€ซื’โ€ช.โ€ฌืžื”ื™ ื”ืื ืจื’ื™ื” ื”ืื•ื‘ื“ืช ื‘ื”ืชื ื’ืฉื•ืช ?โ€ฌ
โ€ซื“โ€ช.โ€ฌืžื”ื™ ื”ืžื”ื™ืจื•ืช ื‘ื” ืคื•ื’ืขื™ื ื”ื’ื•ืคื™ื ื‘ืืจืฅ ?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_104.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซโ€ช ,โ€ฌื›ืืฉืจโ€ฌ
โ€ซื”ื•ื ืจื“ื™ื•ืก ื›ื“ื”"ืโ€ช .โ€ฌื‘ืขื–ืจืช ืจืงื˜ื”โ€ช ,โ€ฌืžืฉื ื™ื ืืช ืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸโ€ฌ
โ€ซืœื•ื•ื™ืŸ ืฉืžืกืชื• โ€ช 400โ€ฌืง"ื’ ืกื•ื‘ื‘ ืกื‘ื™ื‘ ื›ื“ื”"ื ื‘ื’ื•ื‘ื”โ€ฌ
โ€ซื•ืขื‘ื™ืจื™ื ืื•ืชื• ืœืžืกืœื•ืœ ืžืขื’ืœื™ ื‘ืจื“ื™ื•ืกโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซ?ื( ืžืฆื ืืช ื”ืื ืจื’ื™ื” ืฉื”ื•ืฉืงืขื” ืข"ื™ ื”ืจืงื˜ื” ืขืœ ืžื ืช ืœืฉื ื•ืช ืืช ืžืกืœื•ืœ ื”ืœื•ื•ื™ืŸโ€ฌ
โ€ซื‘( ืžื”ื™ ืžื”ื™ืจื•ืช ื”ืœื•ื•ื™ืŸ ื‘ืฉื ื™ ื”ืžืกืœื•ืœื™ืโ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_105.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซื˜ื™ืœ ื ื•ืจื” ืžืคื ื™ ื›ื“ื”"ื ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซ?ื( ืžื”ื•ื” ื’ื•ื‘ื” ื”ืžืงืกื™ืžืœื™ ืืœื™ื• ืžื’ื™ืข ื”ื˜ื™ืœโ€ฌ
โ€ซื‘( ืžื”ื™ ืžื”ื™ืจื•ืช ื”ื˜ื™ืœ ื‘ื’ื•ื‘ื” โ€ช 100โ€ฌืง"ืž ืžืขืœ ืคื ื™ ื›ื“ื”"ืโ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_106.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซโ€ช.โ€ฌืœื•ื•ื™ืŸ ื ืžืฆื ืžืขืœ ืื•ืชื” ื ืงื•ื“ื” ืžืขืœ ืงื• ื”ืžืฉื•ื•ื”โ€ฌ
โ€ซ?ื( ืžื”ื• ื–ืžืŸ ื”ืžื—ื–ื•ืจ ืฉืœ ืกื™ื‘ื•ื‘ ื”ืœื•ื•ื™ืŸโ€ฌ
โ€ซ?ื‘( ืžื”ื• ืจื“ื™ื•ืก ื”ืกื™ื‘ื•ื‘ ืฉืœื•โ€ฌ
โ€ซื’( ืžื”ื• ื’ื•ื‘ื” ื”ืœื•ื•ื™ืŸ ืžืขืœ ืคื ื™ ื›ื“ื”"ืโ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_107.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซื•ืžืกื•ืชื™ื”ืโ€ฌ
โ€ซื•โ€ฌ
โ€ซื ืžืฆืื™ื ื‘ืžืฆื‘ ื ื™ื™ื— ื›ืืฉืจ ืžืจื›ื–ื™ื”ื ื‘ืžืจื—ืงโ€ฌ
โ€ซืฉื ื™ ื›ื•ื›ื‘ื™ื ืฉืจื“ื™ื•ืกืโ€ฌ
โ€ซโ€ช.โ€ฌื–ื” ืœืขื•ืžืช ื–ื”โ€ฌ
โ€ซ?ื( ืžื”ื ื—ื•ืงื™ ื”ืฉื™ืžื•ืจ ื‘ื‘ืขื™ื” ื–ื•โ€ฌ
โ€ซ?ื‘( ืžื”ื™ ื”ืื ืจื’ื™ื” ื”ืคื•ื˜ื ืฆื™ืืœื™ืช ื”ื”ืชื—ืœืชื™ืช ืฉืœ ื”ืžืขืจื›ืชโ€ฌ
โ€ซ?ื’( ืžื”ื™ื” ืžื”ื™ืจื•ืชื• ืฉืœ ื›ืœ ืื—ื“ ืžื”ื›ื•ื›ื‘ื™ื ืจื’ืข ืœืคื ื™ ื”ื”ืชื ื’ืฉื•ืชโ€ฌ
โ€ซื“( ืžื”ื™ ืขื‘ื•ื“ืช ื›ื•ื— ื”ืจื‘ื™ื˜ืฆื™ื” ืขืœ ื”ื’ื•ืคื™ื ืขื“ ืœืจื’ืข ื”ื”ืชื ื’ืฉื•ืชโ€ฌ
โ€ซื–ื” ืžื–ื”โ€ช .โ€ฌืžืžืฆื‘ ื–ื”โ€ช ,โ€ฌื”ื ืžืชื—ื™ืœื™ื ืœื™ืคื•ืœโ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_108.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซื‘ืžืกืœื•ืœ ืžืขื’ืœื™ ื‘ืžื”ื™ืจื•ืช โ€ช .โ€ฌืœื•ื•ื™ืŸ ืฉื ื™ ื ืข ืกื‘ื™ื‘ ืื•ืชื• ื›ื•ื›ื‘ ื‘ืžื”ื™ืจื•ืชโ€ฌ
โ€ซโ€ช.โ€ฌโ€ฌ
โ€ซืœื•ื•ื™ืŸ ืฉืžืกืชื• ื ืข ืกื‘ื™ื‘ ื›ื•ื›ื‘ ืฉืžืกืชื•โ€ฌ
โ€ซ?ื( ืœืื™ื–ื” ืžื”ืœื•ื•ื™ื ื™ื ืจื“ื™ื•ืก ืกื™ื‘ื•ื‘ ื’ื“ื•ืœ ื™ื•ืชืจ ื•ืคื™ ื›ืžื”โ€ฌ
โ€ซ?ื‘( ืœืื™ื–ื” ืžื”ื ื–ืžืŸ ืžื—ื–ื•ืจ ื’ื“ื•ืœ ื™ื•ืชืจ ื•ืคื™ ื›ืžื”โ€ฌ
โ€ซื’( ืžื˜ืื•ืจื™ื˜ ืคื’ืข ื‘ืœื•ื•ื™ืŸ ื”ืจืืฉื•ืŸ ื‘ื›ื™ื•ื•ืŸ ืžืฉื™ืง ืœืชื ื•ืขืชื• ื•ื’ืจื ืœื”ื›ืคืœืช ืžื”ื™ืจื•ืชื•โ€ช .โ€ฌื”ืื ื”ืœื•ื•ื™ืŸ ื™ืชื ืชืง ืžื”ื›ื•ื›ื‘โ€ฌ
โ€ซโ€ช25_Gravity/e_25_5_109.htmlโ€ฌโ€ฌ
โ€ซื’ืจื‘ื™ื˜ืฆื™ื”โ€ฌ
โ€ซื—ืœืœื™ืช ืฉืžืกืชื” โ€ช 10000โ€ฌืง"ื’ ื ืขื” ืกื‘ื™ื‘ ื›ื“ื•ื”"ื ื‘ืžืกืœื•ืœ ืžืขื’ืœื™ ืฉืจื“ื™ื•ืกื• โ€ชr = 3Rโ€ฌโ€ฌ
โ€ซ)โ€ช = Rโ€ฌืจื“ื™ื•ืก ื›ื“ื•ื”"ื(โ€ช.โ€ฌโ€ฌ
โ€ซื ืชื•ื ื” ืชืื•ืฆืช ื”ื›ื•ื‘ื“ โ€ช gโ€ฌืข"ืค ื›ื“ื•ื”"ืโ€ช.โ€ฌโ€ฌ
โ€ซืโ€ช .โ€ฌืžื”ื™ ืชืื•ืฆืช ื”ื›ื•ื‘ื“ ื‘ืžืจื—ืง โ€ช? r = 3 Rโ€ฌโ€ฌ
โ€ซื‘โ€ช .โ€ฌื‘ืื™ื–ื• ืžื”ื™ืจื•ืช ื ืขื” ื”ื—ืœืœื™ืช ?โ€ฌ
โ€ซื’โ€ช .โ€ฌืžื”ื™ ืชื•ืกืคืช ื”ืื ืจื’ื™ื” ืฉื™ืฉ ืœืชืช ืœื—ืœืœื™ืช ื”ื "ืœ ืข"ืž ืฉืชื™ืžืœื˜ ืžืžืงื•ืžื” ?โ€ฌ
โ€ซโ€ช25_Gravity/e_25_8_001.htmlโ€ฌโ€ฌ
โ€ซโ€ช7. A typical neutron star may have a mass equal to that of the Sun but a radius of only 10.0 km.โ€ฌโ€ฌ
โ€ซ?โ€ช(a) What is the gravitational acceleration at the surface of such a starโ€ฌโ€ฌ
โ€ซ?โ€ช(b) How fast would an object be moving if it fell from rest through a distance of 1.20 m on such a starโ€ฌโ€ฌ
โ€ซโ€ช25_Gravity/e_25_8_002.htmlโ€ฌโ€ฌ
โ€ซโ€ช10. Two concentric shells of uniform density having masses M1 and M2 are situated as shown in Fig. 14-29.โ€ฌโ€ฌ
โ€ซโ€ชFind the force on a particle of mass m when the particle is located atโ€ฌโ€ฌ
โ€ซโ€ช(a) r = a,โ€ฌโ€ฌ
โ€ซโ€ช(b) r = b, andโ€ฌโ€ฌ
โ€ซโ€ช(c) r = c. The distance r is measured from the center of the shells.โ€ฌโ€ฌ
25_Gravity/e_25_8_003.html
14. Show that the velocity of escape from the Sun at the Earth's distance from the Sun is
the Earth in its orbit, assumed to be a circle.
(This is a specific case of a general result for circular orbits: Vesc = 2VOrb.)
2 times the speed of
25_Gravity/e_25_8_004.html
15. A rocket is accelerated to a speed of v = 2(gRE)1/2 near the Earth's surface and then coasts (moves)
upward.
(a) Show that it will escape from the Earth.
(b) Show that very far from the Earth its speed is v = (2gRE)1/2.
25_Gravity/e_25_8_005.html
16. The Sun, mass 2.0 X 1030 kg, is revolving about the center of the Milky Way galaxy, which is 2.2 X 1020
m away.
It completes one revolution every 2.5 X l08 years. Estimate the number of stars in the Milky Way.
(Hint: Assume for simplicity that the stars are distributed with spherical symmetry about the galactic center and
that our Sun is essentially at the galactic edge.)
25_Gravity/e_25_8_006.html
19. Two neutron stars are separated by a center-to-center distance of 93.4 km. They each have a mass of 1.56 X
l030 kg and a radius of 12.6 km.
They are initially at rest with respect to one another.
(a) How fast are they moving when their separation has decreased to one-half of its initial value?
(b) How fast are they moving just before they collide? Ignore relativistic effects.
25_Gravity/e_25_8_007.html
20. Two particles of mass m and M are initially at rest an infinite distance apart.
Show that at any instant their relative velocity of approach attributable to gravitational attraction is (2G(M +
m)/d)1/2 ,
where d is their separation at that instant.
25_Gravity/e_25_8_008.html
24. Determine the mass of the Earth from the period T and the radius r of the Moon's orbit about the Earth: T =
27.3 days and r = 3.82 X 105 km.
25_Gravity/e_25_8_009.html
26. Spy satellites have been placed in the geosynchronous orbit above the Earth's equator.
What is the greatest latitude L from which the satellites are visible from the Earth's surface? See Fig. 14-30.
25_Gravity/e_25_8_010.html
28. Use conservation of energy and an expression for the total energy to show that the speed v of an object in an
elliptical orbit
satisfies the relation v2 = GM(2/r-1/a). Here r is the distance of the orbiting body from the central body of mass
M.
25_Gravity/e_25_8_011.html
29. A comet moving in an orbit of eccentricity 0.880 has a speed of 3.72 km/s when it is most distant from the
Sun. Find its speed when it is closest to the Sun.
25_Gravity/e_25_8_012.html
32. As shown in Fig. 14-31, two bodies (of masses m and M) in¬teracting through their mutual gravitational
force will orbit with the same angular speed
about their center of mass C.
(a) Show that in this case Kepler's law of periods becomes T2 = 4ฯ€2r3(1+R/r)2/(GM).
(b) Evaluate the correction factor (1 + R/r)2 for the motion of the Earth and the Sun and also for the motion of
the
Earth and the Moon, in each case ignoring the gravitational effect of the other bodies in the solar system.
25_Gravity/e_25_8_013.html
1. Two point-like objects, each with mass m, are connected by a massless rope of length l.
The objects are suspended vertically near the surface of Earth, so that one object is hanging below the other.
Then the objects are released. Show that the tension in the rope is T = GMml/R3 where M is the mass of the
Earth and R is its radius.
25_Gravity/e_25_8_014.html
8. The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the
equator barely provides the centripetal force needed for the rotation. (Why?)
(a) Show then that the corresponding shortest period T of rotation is given by T 2 = 3 /(G ) where is the
density of the planet, assumed to be homogeneous.
(b) Evaluate the rotation period T assuming a density of 3.0 g/cm3, typical of many planets, satellites, and
asteroids.
No such object is found to be spinning with a period shorter than found by this analysis.
25_Gravity/e_25_8_015.html
11. The following problem is from the 1946 "Olympic" examination of Moscow State University (see Fig. 1436):
A spherical hollow is made in a lead sphere of radius R, such that its surface touches the outside surface of the
lead sphere and passes through its center.
The mass of the sphere before hollowing was M. With what force, according to the law of universal gravitation,
will the hollowed lead
sphere attract a small sphere of mass m, which lies at a distance d from the center of the lead sphere on the
straight line connecting the centers of the spheres and of the hollow?
25_Gravity/e_25_8_016.html
14. Use the model of the Earth shown in Fig. 14-37 to examine the variation of g with depth in the interior of
the Earth.
(a) Find g at the core-mantle interface. How does g vary from this interface to the center of the Earth?
(b) Show that g has a local minimum within the mantle; find the distance from the Earth's center where this
occurs and the associated value of g.
(c) Make a sketch showing the variation of g within the Earth.
25_Gravity/e_25_8_017.html
20. A rocket burns out at an altitude h above the Earth's surface. Its speed vo at burnout exceeds the escape
speed vesc appropriate to the burnout altitude.
Show that the speed v of the rocket very far from the Earth is given by v = (vo2 - vesc2)1/2.
25_Gravity/e_25_8_018.html
25. Consider two satellites A and B of equal mass m, moving in the same circular orbit of radius r around the
Earth but in opposite senses of revolution and therefore on a collision course (see Fig.14-42).
(a) In terms of G, ME, m, and r, find the total mechanical energy of the two-saiellite-plus-Earth system before
collision.
(b) If the collision is completely inelastic so that wreckage remains as one piece of tangled material, find the
total mechanical energy immediately after collision.
(c) Describe the subsequent motion of the wreckage.
25_Gravity/e_25_8_019.html
26. The Sun's center is at one focus of the Earth's orbit. How far is it from the other focus?
Express your answer in terms of the radius of the Sun RS = 6.96 X 108 m.
The eccentricity of the Earth's orbit is 0.0167 and the semi-major axis is 1.50 X 1011 m.
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