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5
C H A P T E R
Force and
Motion
Standard: Understand that types of motion may be described,
measured, and predicted.
Standard: Understand that the types of force that act on an
object and the effect of that force can be described,
measured, and predicted.
What you should know:
• Changes in motion and position can be measured.
• The types of forces that act upon an object can be predicted
and measured.
• Gravity is a universal force that every mass exerts on every other mass.
• Many forces act at a distance.
• Common contact forces include friction and buoyancy.
• Simple machines can be used to change the direction or size of a force.
• An object at rest will stay at rest unless acted upon by an outside force.
• An object in motion will remain in motion unless acted upon
by an outside force.
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Motion
What is motion? How do you tell if an object is moving? If you observe an object from a constant position for a few minutes and its
position does not appear to change, you generally are satisfied that
it is not moving. However, if you observe the object for another few
minutes and its position seems to be moving straight away at a
slow but constant speed, who or what is actually moving? It may
be that you are moving away from the object, or that the object is
moving away from you, the observer. It becomes clear that when
you talk about motion, you need to have a reference point, or a
designated position in space, in order to describe it.
To measure motion, you need reference points. You can measure an object’s displacement, or the distance that it traveled from
the reference point. You can also easily measure the time it takes for
the object to travel that distance from the reference point. Using a
reference point, you can also determine the direction of movement
of the object. Motion then can be described as a displacement from
an original position to a new position, as shown in Figure 5-1.
The distance can be determined by subtraction of the original
position from the new position. This distance is the change in position. It tells you how far the object has moved. If you know the
time that has elapsed for the object to move from the original location to the new one, you can divide the distance by the time and
get the average speed. That is,
vd
t
where v average speed, d distance traveled, and t time. Note
that speed is the rate of movement over a distance and indicates
magnitude (how big something is). It does not indicate direction
and is, therefore, a scalar quantity.
You can also use the change in distance divided by the change
in time to determine the velocity, as in the formula
Velocity change in distance
change in time
Object's path
(actual distance)
Initial
position
Displacement vector S
Final position
Figure 5-1. Displacement is the distance an object has traveled from a fixed
reference point.
Chapter 5: Force and Motion
111
Velocity is the change in distance per unit of time. Velocity is a vector quantity, which also includes the direction of motion. (A vector quantity is a physical quantity or measurement that specifies
both direction and magnitude. Magnitude means how much or to
what extent.) All motion is relative to the frame of reference. Sometimes our frame of reference is moving as well. Even now you are
moving through space as Earth orbits around the Sun, which is
moving around our galaxy. In addition, Earth is rotating on its axis.
There is no absolute frame of reference in describing motion.
Skill Activities
Multiple Choice
Use the graph below to answer questions 1–4.
Distance (meters)
The graph represents the movement of a person with respect to a single observation point
(O), which is also the point of origin.
D
25
20
C
E
15
B
10
F
A
5
0
(Origin)
4
8
12
16
Time (minutes)
G
20
1. At Point A, the person is
a. standing still
b. slowly walking toward the point of origin
c. slowly walking away from the point of origin
d. returning to the point of origin
2. At Point B, the person is
a. standing still
b. returning to the point of origin
c. slowly walking toward the point of origin
d. rapidly walking away from the point of origin
3. At Point C, the person is
a. changing direction
b. returning to the point of origin
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X
24
c. slowly walking toward the point of origin
d. rapidly walking away from the point of origin
4. At Point F, the person is
a. standing still
b. returning to the point of origin
c. slowly walking away from the point of origin
d. rapidly walking away from the point of origin
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INQUIRE
EXPLAIN
Short Response
5. Distinguish between displacement and distance.
6. How do you measure speed?
7. What is the formula for velocity?
Gridded Response
8. What is the average speed of an object that moves 10 kilometers in 2 hours? Show
your work and enter your answer on a copy of the grid below. Use your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
9. If a person travels at 100 meters/second, how far will he move in 40 seconds? Show
your work and enter your answer on a copy of the grid below. Use your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
Chapter 5: Force and Motion
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10. How long will it take for an airplane traveling at 400 kilometers / hour to reach a city
that is 880 kilometers away? Show your work and enter your answer on a copy of the
grid below. Use your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
11. If a car moves 3.0 kilometers in 30 minutes, what is its speed? Show your work and
enter your answer on a copy of the grid below. Use your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
12. If an object moves 10 meters in 5 seconds, what is its velocity? Show your work and
enter your answer on a copy of the grid below. Use your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
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Types of Motion and Forces
The study of motion and forces is called mechanics. Force, meaning a push or pull on an object, may cause motion or alter motion
(see Figure 5-2a and b).
(a)
(b)
Figure 5-2. Forces of pulling (a) and pushing (b) are used in opening and
closing doors.
4.54
newtons
of force
N
oz
0
2
4
6
8
10
12
14
16
18
0
8
16
1 pound
of force
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Gravity, for example, is a force we deal with every day. It is an attractive force or a pull that Earth and an object have for each other.
The force of gravity gives an object its weight. When we weigh an
object on a spring scale, we are measuring the pull between the object and Earth. (See Figure 5-3.)
32
40
Newton’s Laws of Motion
48
56
64
1 pound
Figure 5-3. When we
weigh an object on a spring
scale, we are measuring
the pull between the object
and Earth.
First Law of Motion
Sir Isaac Newton best described the most basic property that objects have with respect to force and motion. Newton’s first law of
motion states: An object at rest tends to remain at rest. This property is called inertia. The more mass an object has, or the more it
weighs, the more inertia it has. Inertia is not just a resistance to motion, but it also describes how objects resist changes in movement
once put into motion. An object in motion tends to remain in motion traveling at a uniform speed in a straight line unless acted
upon by an unbalanced force. You feel inertia when a car or airplane starts and stops quickly. (See Figure 5-4 on the next page.)
To make an object change its motion or travel in a circular path
requires a force. An object in orbit around Earth is attracted toward Earth by Earth’s gravitational pull (force). If you could switch
off gravity, the orbiting object would move off in a straight line at
a constant speed. A ball spinning about your head on a string travels in a circular path as long as the string is connected to the ball
Chapter 5: Force and Motion
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You feel pulled back
You feel pushed forward
Car starts
Car stops
Figure 5-4. You feel inertia when a car starts and stops quickly.
and is being pulled by a force toward the center of the circle. If you
cut the string, the ball will fly off in a straight line.
An object traveling in a straight line at a constant speed requires
a force acting on it to make it change speed or direction. If the force
is applied to the right, then the object will move to the right, following a curved path. If the force is applied in the opposite direction, to the left, then the object would veer to the left. A force acting
in the opposite direction of motion slows the velocity of the object.
If the force is acting in the same direction as the motion of the object, its velocity increases. An object traveling in a straight line at
a constant speed must have a force acting on it to make it slow
down. You are accustomed to objects slowing their motion because
of friction (discussed in more detail on page 124 of this chapter).
Friction is a common force that slows moving objects until they
stop. (See Figure 5-5.)
Second Law of Motion
Newton’s second law tells us about the relationship among force,
mass, and the change in motion. That is, it states that the net force
on an object is the product of its acceleration and mass. We can describe this relationship with the equation
F ma
where F is the force in newtons, m is the mass in kilograms, and a
is the acceleration in meters/second2. Acceleration (a) is the
rate of change of velocity. The force (F ) required to accelerate
an object is directly proportional to its mass (m). (See Figure
Push direction
Air friction (resistance)
Gravity
Ball
B
Figure 5-5. Air friction causes the ball to slow down in its movement.
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Preparing for FCAT Science: Grade 8
(a)
(b)
Figure 5-6. A greater force is required to push and accelerate movement of
the pickup truck because it has a greater mass.
5-6.) The greater the force, the greater the acceleration. Using the
same amount of force, lighter masses accelerate more than larger
masses. (We discuss gravitational acceleration further on page 126
of this chapter.)
This relationship also provides some units that we can use for
force. Mass is measured in kilograms, and acceleration is measured
in meters per second per second. Force is measured in kilogram
meters per second per second or kg.m/sec2. The derived unit of
force is called a newton or N. It is the force required to accelerate
a 1.0 kilogram mass by 1.0 meter per second per second.
Acceleration means that an object is constantly changing its
speed or direction. Positive acceleration occurs when the speed or
velocity of an object keeps increasing. Negative acceleration (deceleration) occurs when the speed or velocity of an object continues to decrease. If the speed of an object is constant, then there is
no acceleration. To find the acceleration of an object, divide the
change in velocity by the change in time it takes to make the
change in velocity. The change in velocity is found by subtracting
the initial or starting velocity from the ending or final velocity.
Change in velocity (V) final velocity starting velocity
or
Change in velocity (V) Vf Vi
Acceleration is
a
Vf Vi
t f ti
or the rate in change of velocity. The units for acceleration are
miles per hour per second, kilometers per hour per second, or meters per second per second. (We discuss acceleration due to gravity later in this chapter.)
The forces we have been talking about are called unbalanced
forces. By unbalanced forces we mean forces that can produce moChapter 5: Force and Motion
117
Figure 5-7. The force
exerted by the table is
equal to gravity’s pull on
the rock. The two forces
cancel each other out and
no motion is produced.
tion or changes in motion. Another term that is used is net force,
which is the sum of all the forces acting on an object. Most of the
time there is actually more than one force acting on an object.
These forces are in equilibrium with each other, that is, two equal
but opposite forces produce no motion or change in motion. They
cancel each other out. For example, if an object is sitting on a table,
gravity is trying to pull it toward the ground. The table is exerting an
upward force on the object equal to the pull of gravity, but in the opposite direction. The object stays put, as shown in Figure 5-7.
Student teams in a tug-of-war pull a rope in opposite directions.
As long as their pulls are equal, there is no net movement. Balanced
forces produce no net change in motion, as shown in Figure 5-8a. If
one team pulls harder than the other, then a net motion occurs along
the rope. The forces have become unbalanced. (See Figure 5-8b.)
Since forces can be described by vectors (an arrow indicating
size and direction), so can the resulting motions. Suppose we have
a small boat that we can use to cross a river, and it can travel at 4.0
kilometers/hour as long as there is no current. However, there is a
current in the river the day we want to cross it. If the boat drifts
with the current, it will travel at 3.0 kilometers/hour without our
rowing. Question: If our rower turns the boat into the current of
the river and rows upstream, how fast will we go? Answer: Rowing
produces a speed of 4.0 kilometers/hour upstream, and the current
flowing in the opposite direction pushes against the boat at 3.0 kilometers/hour. The net change, or our speed upstream, will be 1.0
kilometer/hour. What happens when the boat travels downstream?
The river pushes the boat at 3.0 kilometers/hour, and the rower
adds 4.0 kilometers/hour in the same direction. Therefore, the boat
travels at 7.0 kilometers/hour downstream. The speed vectors are
added together to get the resulting velocity.
Force
Force
(a)
Balanced forces
Force
Force
(b)
Unbalanced forces
Figure 5-8. Tug-of-war. (a) Balanced forces cancel each other out. (b)
Unbalanced forces produce a net movement along the rope.
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Preparing for FCAT Science: Grade 8
Boat going upriver
4.0 km/hr
3.0 km/hr
1.0 km/hr
resultant
(No current)
boat speed
4.0 km/hr
Boat going downriver
3.0 km/hr
4.0 km/hr
7.0 km/hr
resultant
Boat going across river
3.0 km/hr
River current
3.0 km/hr
4.0 km/hr
Resultant
5.0 km/hr
Figure 5-9. The boat travels out from shore at a speed of 4.0 kilometers/
hour when there is no current. The boat is pushed downstream when there
is a current of 3.0 kilometers/hour. If the boat is going up the river, the
resultant vector is 1.0 kilometer/hour. If the boat is going down the river, the
resultant vector is 7.0 kilometers/hour. If the boat is going across the river,
the vector diagram is a right triangle with a resultant of 5.0 kilometers/hour.
What do you think happens when the rower tries to row directly
across the river at 90 degrees to the current? As the boat moves out
from shore at 4.0 kilometers/hour, it is pushed downstream at 3
kilometers/hour at the same time. We can draw the vector diagram
by adding the tail of the current vector to the tip of the boat speed
vector at 90 degrees downstream. The resulting vector starts from
the shore and goes straight to the tip of the current vector, forming
a right triangle. We can measure the resulting vector with a ruler if
the vectors are drawn to scale. We can also use geometry. Since this
is a right triangle, its hypotenuse will be 5 kilometers/hour (3, 4, 5
right triangle), as shown in Figure 5-9.
Third Law of Motion
Newton’s third law states that for every action there is an equal
but opposite reaction. The air rushing out of a deflating balloon
propels the balloon in the opposite direction, as shown in Figure 510a on the next page. This same principle is used in all rocket motors to propel spacecraft. (See Figure 5-10b on the next page.) The
impulse, which is a force applied for a very short period of time,
acts on the balloon as it is deflating and is equal to and opposite the
impulse from the exhaust. The impulse is the product of the force
(F) times the change in time (tf ti ).
Chapter 5: Force and Motion
119
Reaction
(a)
(b)
Action
Rocket moves
upward
Hot gases are forced out
Figure 5-10. (a) Air rushing out of a balloon causes it to travel in the
opposite direction. (b) Rockets operate on the principle of Newton’s third
law: For every action there is an equal but opposite reaction.
Forces may act by touching the object, or they may act on the
object over a distance, as gravity does. Statics is the study of
forces on stationary objects. In a static equilibrium state, although
there may be several forces acting on a stationary object, the forces
have to be balanced against each other, since there is no net force.
Dynamics is the study of the relationships between forces and
motion. Newton’s laws summarize these interactions.
READ
INQUIRE
EXPLAIN
Skill Activities
Short Response
13. State Newton’s first law of motion.
14. State Newton’s second law of motion.
15. State Newton’s third law of motion.
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Preparing for FCAT Science: Grade 8
16. Explain why a larger mass is harder to move than a smaller mass.
17. You throw a ball across a field.
a. Describe its path, direction of movement, and speed. (Draw a diagram to illustrate
your answer.)
b. Describe the effects of friction and gravity on the ball’s movement.
18. You throw a ball straight up into the air.
a. What force is acting on the ball as it moves upward?
b. What happens to the speed of the ball as it is moving upward? Why?
c. What happens to the speed of the ball as it is moving downward? Why?
19. What happens to the motion of an object if no forces are acting upon it? Why?
20. Draw a diagram of a block of wood with four forces acting on it in equilibrium.
Gridded Response
21. A book weighs 9.8 newtons. What is the force on the book exerted by the bookshelf?
Answer on your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
22. If a moving object is accelerating at 100 meters/second 2 when a force of 10 newtons
is applied to it, what is its mass? Answer on your own paper.
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
Chapter 5: Force and Motion
121
More on Acceleration and Friction
As we stated earlier, acceleration is defined as the rate of change
of velocity. Any change in velocity is acceleration. Falling objects
accelerate as they fall toward the ground, going faster and faster.
This is expressed in the formula
v vi
a f
t f ti
where a acceleration, vf vi change in velocity, and tf ti =
change in time. The values for gravitational acceleration are
a 32 feet/second/second in the English measurement, and a 9.8
meters/second/second in the metric system.
There are also several other forms of the acceleration equation.
For example, an object dropped from a very high tower is moving at
58.8 meters/second after 6 seconds. What is its acceleration? Solution: If you substitute the measurements into the formula, you get
a (58.8 meters/second)/(6.0 seconds)
9.8 meters/second/second
This agrees with the commonly accepted value for the acceleration
of falling objects where there is little or no friction. Larger objects
may accelerate more slowly due to increasing air friction slowing
them down. If an object falls far enough it will stop accelerating,
and its speed or terminal velocity remains the same. The drag,
which is the resistance of an object to movement through a fluid,
is equal to the gravitational acceleration, so the object will continue to fall but not increase its velocity. An object’s shape or its
“streamlining” helps determine its fluid friction, or drag, so that
more streamlined objects experience less drag and have faster terminal velocities. (See Figure 5-11.)
To calculate the final velocity of an object from a resting position, use the formula
Vf at
Problem: If an object is accelerated 4.0 meters/second/second
for 10 seconds, what will its final velocity be?
Solution: Substituting these numbers in the above expression
we get
Vf (4.0 meters/second/second) (10 seconds)
Multiplying, we get 40 meters/second for the final velocity. This assumes that friction or drag is very small with little influence on the
velocity.
If an object is already moving, the final velocity can be calculated by
Vf Vs at
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Preparing for FCAT Science: Grade 8
Rock
Sphere
Ground
Figure 5-11. Streamlining allows the sphere to experience less friction and
fall faster than the rock with the irregular shape.
where Vs is the starting velocity and the rest of the formula is what
we have already used as the final velocity.
The distance an object travels from its initial acceleration time
is calculated by
d 1 at 2
2
READ
INQUIRE
EXPLAIN
Skill Activities
Short Response
Use the acceleration table below to answer questions 23a–c.
Acceleration
Time (sec)
Velocity (m/sec)
Change in Velocity (m/sec)
Acceleration (m/sec)
0
0
0
0.0
5
10
10
5.0
10
40
30
6.0
20
60
30
30
90
40
3.0
39
3.0
23. a. What is the acceleration between 10 and 20 seconds?
b. What is the velocity change between 20 and 30 seconds?
c. What is the velocity at the end of 40 seconds?
Chapter 5: Force and Motion
123
Friction
The force that acts against motion is friction. When one surface
moves over another, frictional force always acts to slow and stop
the motion. The amount of friction depends upon several factors:
the materials the surfaces are made from, the texture or smoothness of the surfaces, and the amount of force pressing the surfaces
together.
Friction shows up in several different forms. The first is static
or starting friction. It is the difference in the force required to
start an object moving compared to the force required to keep it
moving. When you drag a mass up an inclined plane, you can measure the pull using a spring scale. It takes 24 newtons to get the motion started (starting friction) and only 20 newtons to keep it sliding
(sliding friction).
Sliding friction, or kinetic friction, is the force opposing the
motion of sliding surfaces. Smooth surfaces reduce sliding friction,
and rough surfaces increase it. Hard surfaces reduce friction, while
soft surfaces increase it. Surfaces made of the same material tend
to have greater friction than those of different materials. Materials
placed between surfaces like oil, Teflon, or graphite decrease sliding friction and are called lubricants.
Rolling friction is produced when one object rolls over another like wheels on the road. The amount of friction is dependent
on the force between the surfaces, the area of contact, and the ma-
To start motion
24N
Mass
(a) Static friction
Sliding at a constant speed
20N
Mass
(b) Kinetic friction
10N
Mass
(c) Rolling friction
Figure 5-12. (a) Starting friction: A spring scale drags a mass little or no
distance as motion starts. (b) Kinetic friction: A spring scale drags the same
mass half the distance. (c) Rolling friction: A spring scale pulls the same
block with rollers under it, causing it to slide the same distance.
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Preparing for FCAT Science: Grade 8
terials in contact. If car tires had no friction with the road it would
not be possible to start, stop, or turn a car. When northern roads
ice, we find out the results of “frictionless roads.”
Fluid friction occurs when objects move through the air or
other fluids, for example, putting your hand out the window of a
fast moving car and having it pushed backward by the air. As we
have previously said, fluid friction is also called drag. The amount
of fluid friction depends upon the frontal surface area of the object, how fast the object is moving, and the viscosity of the fluid.
(See Figure 5-12a-c.) (Viscosity is the resistance of a fluid to
flow. For example, pancake syrup has greater viscosity than water.
The syrup flows more slowly, and objects move through the water
faster.)
Skill Activities
Short Response
24.
25.
26.
27.
Explain what is meant by static or starting friction.
What are three factors that influence sliding friction?
What force keeps underwater vehicles from traveling as fast as vehicles in the air?
What force is great enough in hurricane winds to flatten a concrete wall?
Gridded Response
28. The graph below shows the velocity of a car driving through a city neighborhood.
What is the driver’s acceleration rate at 3 minutes? At 6 minutes? (Hint: Remember
v vi
.)
a f
t f ti
30
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
Velocity (miles/hour)
READ
INQUIRE
EXPLAIN
25
20
15
10
5
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
0
1
2
3
6
4
5
Time (minutes)
7
8
9
Chapter 5: Force and Motion
10
125
Other Forces
Many forces act over distances rather than by “contact.” The most
common force is gravity. Gravity is an attractive force between all
objects in the universe. Although gravity is not a very strong force,
it can act over vast distances on all the objects around it. Newton
realized that any object that had mass had a gravitational pull. All
the objects in the universe have a pull for all the others. Newton
summarized his Universal Law of Gravitation with the following
equation:
f
Gm1m2
r2
Force (N)
which states that the gravitational force between two objects, f, is
equal to the universal gravitational constant, G, times the masses
of the two objects, m1 and m2, divided by the square of the distance
between them, r. The gravitational force is directly proportional to
the masses m1 and m2, and inversely proportional to the square of
the distance between the objects. In other words, larger masses
have greater pulls while smaller masses have lesser pulls. Earth
has a pull for a baseball. The baseball also has a pull for Earth. It is
the overwhelming mass of Earth that determines the weight of the
baseball, and the pull of the baseball for Earth is almost nothing in
comparison. Increasing the amount of mass of a body increases its
attractive force.
Like many other forces that act at a distance, the gravitational
force follows the inverse square law, that is, the distance between
two objects is doubled, the gravitational pull between them decreases by one-fourth; tripling the distance between two objects
cuts the force to one-ninth of the original force. Figure 5-13 shows
this relationship. Gravitation acts only as an attractive force, a pull
but not a push.
Electrostatic force also follows the inverse square law with
respect to distance but, in a different manner from gravitation, it
can be attractive or repulsive because unlike electrical charges attract each other and like charges repel each other. (See Figure 5-14.)
(More discussion of static electricity and electrostatic force is in the
next section.) Larger charges produce greater forces. These forces
decrease rapidly as electrical charges move apart and increase as
they move closer together.
0
Distance (m)
Figure 5-13. Inverse square
relationship. Gravitational
force is inversely proportional to the square of the
distance between objects.
126
Electrical Forces
Electrical charges can produce electrical fields. What is an electrical charge? There are two types of electrical charges. The first type
is static charge, which does not move. The second type of electrical charge is an electrical current or moving charge.
Preparing for FCAT Science: Grade 8
–
–
–
–
–
–
–
–
–
(a)
–
+
–
–
–
–
–
–
–
–
–
–
–
–
–
–
+
+
+
+
–
–
–
–
–
–
(b)
Figure 5-14. (a) The negatively charged balloons repel each other. (b) The
negatively charged balloon is attracted to the positively charged rod.
What makes these electrical charges? Electrons, the outermost
parts of atoms, create charge. Electrons have only a negative
charge, yet we often talk about positive charge. Protons are positively charged parts of atoms, but they do not create static electrical charge or electrical currents. This is because they cannot move.
They are firmly attached to each other in the nucleus of the atom
and are shielded from the outside by the electrons. (Remember
also our discussion of protons and electrons in Chapter 3.) The
electrical charge is the presence or absence of electrons. An accumulation of electrons produces a negative charge. The greater
the number of electrons, the greater the negative charge. A positive
charge is produced when there is a lack of electrons or fewer electrons there than somewhere else.
If we rub a glass rod with a silk cloth, the cloth removes electrons from the rod, leaving the silk negatively charged and the glass
positively charged. The best static charges are generated during the
winter when the humidity is low. (This is because the particles of
water vapor attract the electrons.) You can build up a large static
charge in your body by walking on a carpet with your sneakers. You
discharge this large static charge when you touch a metal door handle. Ouch!
Electrical charges exert a force on each other, with like charges
repelling each other and opposite charges attracting each other.
These charges can be moved by using contact or induction charging,
as shown above in Figure 5-14. Induction charging occurs when a
nearby charged object rearranges the electrons on a neutral object
without touching it. Using the charged glass rod and silk cloth, we
can cause the electrons on the silk cloth to move by simply moving
the charged glass rod near the silk cloth. Contact charging occurs
when electrons are transferred. For example, when a wire from an
Chapter 5: Force and Motion
127
electron energy source such as a battery touches something else,
electrons are transferred.
An electrical field is an electrical force acting at a distance.
Electrical fields surround charges (affect other nearby charges)
and produce a push or a pull that is stronger when the electrical
fields are closer and grow weaker the farther away they are. The
greater the size of the charge of the electrical field, the greater its
force will be. Unlike electrical charges (, ) will attract each
other, and like charges (, )(, ) will repel each other.
Static electricity often accumulates on substances that electrons can’t easily move in or on. These substances are called insulators. Glass, rubber, and plastic are examples of insulators. They
are nonconductors because they do not have free-moving electrons. A substance that allows electrons to flow over and through
it easily is called a conductor. Metals are conductors because their
electrons are not tightly bound to their nuclei and therefore flow
freely.
Current electricity is the flow of electrons from one place to
another. The pathway that electrons follow is called an electric
circuit. An electric circuit consists of a source, a conductor, a load
(a device such as a motor that uses the electricity), and a switch.
Electrical current flows through the conductors of an electric circuit. Electrons are supplied by a source. One type of source is a
battery. In the battery, a chemical reaction takes place that pushes
the electrons out through a wire into the circuit. The terminal
where the electrons are pushed out is called the negative electrode. These electrons ultimately return to the battery at the positive electrode.
The number of electrons flowing through the circuit is the electrical current. The push or potential driving these electrons
through the circuit is the voltage. Voltage is measured in units
called volts by an instrument called a voltmeter. The voltage is
related to the chemical reaction used in the battery. The resistance
to the flow of electrons through the circuit is the resistance or
electrical friction. These three factors are related to each other
where
V IR
This relationship is called Ohm’s law. V is the voltage, I is the current measured in amperes, and R is the resistance of the circuit
measured in ohms. Figure 5-15 shows a series circuit that has a single pathway for the electrons to flow through.
Series or Parallel Circuits
Electric circuits (source, conductors, controls, and loads) can be
connected so that the flow of electrons from the negative electrode
of the source has only one path. This is called a series circuit. If
we place a voltmeter anywhere in the circuit and close the switch,
it will read the source voltage. If we place a dry cell battery (a bat128
Preparing for FCAT Science: Grade 8
Figure 5-15. A series circuit with a single pathway for the electrons to flow
through. Note that any gap or break in the pathway will interrupt the flow of
electrons.
tery where the conducting material is a solid substance) in the circuit and wire it in series (, ), the voltage adds together. For
example, if the source voltage is 1.5 V, 1.5 V 1.5 V gives a reading
of 3.0 V on the meter. Question: What would happen to the reading
if we added a third battery in series? Answer: Our meter would
read 4.5 V. What happens if we wire the batteries together in parallel as shown in the diagram below? (See Figure 5-16.)
e–
Switch
e–
e–
e–
1 ohm
Dry cells
e–
e–
Figure 5-16. In a parallel circuit, electrons can flow through more than one
path. Here, electrons can continue to flow through the other path even
though a break in one pathway has occurred.
We get a voltage reading of 1.5 V. In parallel circuits, the voltage
is not additive. Adding a third or fourth battery would still provide
the same 1.5 V reading. Your parallel circuit would last much longer
than the series circuit because the energy driving the load is divided among more sources.
What happens when there is an additional lightbulb added in series? What we see is that the voltage through the circuit does not
change. If one bulb burned out, the series circuit would not operate, since the circuit is broken. If the lamps were wired into the circuit in parallel, there would be an additional path for the electricity
to flow through. If one bulb burned out, the other would still work.
Chapter 5: Force and Motion
129
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Skill Activities
Short Response
29. What is the advantage of being able to hook batteries in series?
30. What is the advantage of being able to connect batteries in parallel?
Use the diagram below, which represents a DC circuit with two batteries wired in parallel and a third battery wired in series to the other two, to answer questions 31 and 32.
–
+
–
+
–
+
Lamp
Switch
(a) Series Circuit
+
+
+
+
+
+
–
Lamp –
Lamp
–
–
–
–
Switch
Switch
(b) Parallel Circuit
(c) Parallel and Series Circuit
31. What would the voltage reading be for this circuit?
32. Name some devices that can serve as a load in an electric circuit.
Magnetic Fields
Magnetic fields also produce attractive or repulsive forces. We can
easily see a magnetic field around a magnet by sprinkling iron filings around the magnet on a sheet of paper. A bar magnet shows
magnetic lines of force coming into and going out of the poles of
the magnet. (A pole is the area of a magnet where the magnetic effect is strongest.) The lines of force curve back to connect to the
pole at the opposite end of the magnet. Figure 5-17a illustrates this
pattern. If we bring two magnets close together with their opposite
poles toward each other, because they are attracting each other,
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Preparing for FCAT Science: Grade 8
N
S
N
(a)
S
(b)
S
S
(c)
Figure 5-17. (a) Magnetic lines of force coming into and going out of the
poles of a bar magnet. (b) When opposite poles of two bar magnets are
brought close together, the lines of force flow into each other. (c) When the
same poles of two bar magnets are brought close together, the magnetic
lines of force bend away from each other and are repelled or deflected away
from the poles.
the lines of force flow into each other. (See Figure 5-17b.) If two of
the same poles are brought near each other, we can see that the
magnetic lines of force bend away from each other and are deflected away from the poles. (See Figure 5-17c.)
Magnetic fields are different from electrical and gravitational
forces because they do not have to follow the inverse square law.
This is because magnetic lines of force do not spread out into a
sphere. They are controlled by the shape of the magnet and can be
concentrated into localized areas.
Electrical, magnetic, and gravitational fields are vector quantities, which means, as we’ve said earlier, that they have direction
and a strength, or magnitude. Depending upon their direction and
strength, these vector quantities can either reinforce or cancel each
other. The fields themselves show these same variations, particularly when two are interacting with each other.
Electrical force and magnetic force are also both aspects of a
single force. Electrical currents, or moving electrons, flow through
a wire and create a magnetic field. The magnetic field wraps around
the wire in concentric layers, as shown in Figure 5-18 on page 132.
Magnets also have an effect on electrical currents. A conductor
passing through a magnetic field experiences a force when a current
Chapter 5: Force and Motion
131
N
S
N
S
Battery
S
N
N
S
Electron
flow
Figure 5-18. A straight, current-carrying wire conductor is surrounded by the
concentric layers of a magnetic field.
passes through it. (See Figure 5-19.) This is the basis for electric motors and generators. An electromagnet is created by wrapping a
wire with a flowing current of electricity around an iron rod. The
number of turns of the wire around the rod determine the strength of
the magnetic field.
Electrical currents also influence other currents by the magnetic fields they produce. If two conductors have a current moving
through them in the same direction, they tend to attract each other.
Conductors with currents moving through them in opposite directions repel each other. Currents in wires produce magnetic fields,
and wires moving in a magnetic field produce a current. This is
strong evidence that electrical and magnetic phenomena are the result of the same basic principle. Moving charges create forces that
influence other moving charges. (See Figures 5-20a and b.)
There are many types of magnets. Permanent magnets have
their crystals aligned so that most of them are in the same direction, creating a larger magnetic field. Electromagnets offer many
advantages over permanent magnets. They can be switched on and
off, their strength and polarity regulated by the amount and direction of the electrical current. Most magnets today are electromagnets. Superconducting magnets and pulse magnets can produce
magnetic fields many times stronger than that of Earth.
Turning
magnetic field
Electron
flow
Figure 5-19. You can visualize the direction of a magnetic field around a
straight conductor. If you grasp a straight, current-carrying wire with your left
hand, your thumb will point in the direction of current flow. The direction of
the magnetic field moving around the wire is shown by the curvature of your
fingers around the wire. This is also known as left-hand rule #1.
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Preparing for FCAT Science: Grade 8
Magnetic field
S
N
S
N
Electron
flow
(a)
Figure 5-20a. Left-hand rule #2 shows the direction of the magnetic field
through a coil. Your fingers wrap around the coil and show the direction of
current flow through the coil. Your thumb points in the direction of the
current flow.
Figure 5-20b. Left-hand rule #3. You can locate a magnetic force on a wire
conductor in a magnetic field: Your thumb points in the direction of the
current flow. Your forefinger points to the direction of the magnetic field (B)
between the two poles N and S surrounding the wire. Your middle finger
points to the direction of the magnetic force exerted on the wire (F).
READ
INQUIRE
EXPLAIN
Skill Activities
Short Response
33. State Newton’s gravitational law.
34. How are pairs of charged particles affected by distance and magnitude?
35. What is an electrical charge?
36. How do you predict whether two charges are likely to attract or repel each other?
Chapter 5: Force and Motion
133
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EXPLAIN
Extended Response
Force (N)
37. The graph below shows the relationship between gravitational force and the distance
between masses.
0
Distance (m)
a. If you double the distance between two planets, what happens to the gravitational
force between them?
b. If you double the mass of one of the planets, what happens to the gravitational
force between them?
Simple Machines
We use machines every day in our daily lives. Machines decrease
the effort we need to exert to do work. They do this by changing the
direction of force, multiplying force, or multiplying speed to make
work easier. Simple machines include the lever, inclined plane, pulley, wheel-and-axle, screw, and wedge.
Simple machines make work easier because they require less
force to move an object than it would take by just using our unaided muscles. Work is the force applied times the distance over
which the force is applied, or
Work (W ) Force (N ) Distance (m)
Even though the force used to do the work is less when using a simple machine, the force (or effort) has to be applied over a greater
distance. This means that you must do more work to exert less effort. (Resistance is the force that the effort has to overcome.)
Machines that allow you to move things a farther distance and at
a faster rate require a larger force (or effort) over a shorter time
period.
The advantage that machines give us in using less effort to move
an object a greater distance is called mechanical advantage. Mechanical advantage can also be used to change speed. The mechan134
Preparing for FCAT Science: Grade 8
Figure 5-21. The ideal mechanical advantage (IMA) can be calculated based
on the proportions of an inclined plane.
ical advantage can also be thought of as the numerical advantage of
using a machine. For example, if a machine has a mechanical advantage of 3, it multiplies the effort needed by 3. The mechanical
advantage of a simple machine can be calculated using the formula
MA Fout
Fin
where mechanical advantage is equal to the ratio of force out to
force in.
The actual mechanical advantage (AMA) of a simple machine is obtained by measuring the forces required to do the work.
Ideal mechanical advantage (IMA) can be calculated based on
the proportions of the simple machine. Ideal mechanical advantage
does not include the friction of the simple machine, so that the
IMA will always be more than the AMA. (See Figure 5-21.) Note
that the calculation of each AMA depends on the machine used.
Each machine has its own formula for AMA. This is because AMA
takes friction into account, whereas IMA is what the MA of the machine would be in the absence of friction.
Table 5-1 provides several formulas you will find useful in problem solving.
Table 5-1. Reference Formulas
DE
or L
DR
H
AMA R
E
Work output R DR
IMA Work input E DE
Machine efficiency AMA 100
IMA
Chapter 5: Force and Motion
135
The Inclined Plane
The inclined plane allows objects to be lifted to a new position with
less effort, but the amount of work done will be more than just lifting
the object. A wheelchair ramp is an example of an inclined plane.
Problem: If you lift a 10-newton object to a 2.0-meter-high platform,
how much work do you do?
Solution:
W F D or Work Force Distance
Substituting F 10 newtons and D 2.0 meters,
W 10 newtons 2.0 meters
20 newton-meters work
If you use an inclined plane with an IMA of 3 (3.0 meters distance/1.0
meter rise), this will cut the force required by about one-third, but
will require moving the object three times as far. This means that F 3.5 newtons (because the force of 10 newtons has been cut by about
one-third) and D 6.0 meters (because the object moved a distance
three times as far, which is now 6.0 meters). Therefore
W 3.5 newtons 6.0 meters
21 newton-meters of work
Where did the extra work come from? Lifting the object has almost
no friction, but rolling or sliding up an inclined ramp does add
friction.
How well a machine reduces effort can be expressed as efficiency. One way you can calculate efficiency is by comparing the
IMA with the AMA. If you use the formula for work efficiency from
above:
DEe
DR
6.0 meters
2.0 meters
3
IMA and
10 newtons
3.5 newtons
2.86
AMA machine efficiency is equal to AMA 100 or
IMA
2.86
3.0
95.2%
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You can also calculate efficiency by dividing the work output by
the work input:
Work output
100
Work input
20 newton-meters
100
21 newton-meters
95.2%
Efficiency Note that efficiency is less than 100 percent. This is because the
actual mechanical advantage is less than the ideal mechanical advantage. One important thing to remember about simple machine
problems is to draw the diagrams and fill in all the available
information.
Other Simple Machines
The Inclined Plane—Wedge and Screw
A more dramatic example of changing the direction of a force is the
wedge, which is a double inclined plane. Its primary use is to
change the direction of force to a right angle from the direction the
wedge moves. Driving a wedge into the end of a log will split the
log. Driving a wedge between a heavy object and its base will lift
the object. (See Figure 5-22a.)
A screw is an inclined plane wrapped around a cylinder or a
wedge. Rotation of the screw pulls it down into the wood. (See Figure 5-22b.) Knives, axes, and nails are variations of the inclined
plane.
Lever
A lever can be a bar or plank balanced on a fulcrum or pivot point.
The lever is one of the most useful simple machines because the
A screw is an
inclined plane
wrapped around
a pole.
(a)
(b)
Figure 5-22. (a) Nails, knives, and axes are examples of wedges and are
variations of the inclined plane. (b) A screw is also an inclined plane. It is an
inclined plane wrapped around a cylinder or wedge. Rotation of the screw
pulls it down into the wood.
Chapter 5: Force and Motion
137
fulcrum can be constructed so that the friction is minimal. This
makes the lever highly efficient.
A typical lever consists of an effort arm and a resistance arm
with a fulcrum or pivot point located somewhere between the effort arm and a resistance arm. The resistance and effort arms are
both parts of the lever. The resistance is the object being moved by
the resistance arm, and the force applied to move it is the effort.
The effort is applied to the fulcrum by the resistance arm.
Levers are classified into three basic types according to the location of the fulcrum, the effort, and the load (resistance). Firstclass levers have the fulcrum between the effort and the load,
multiplying the force, motion, or direction of the force. In first-class
levers, the effort and the load move in opposite directions, like a
seesaw or pry bar.
Second-class levers have the fulcrum at the opposite end of
the effort, and the load is between the fulcrum and the effort. The
load moves in the same direction as the effort. A wheelbarrow uses
two of these levers.
A third-class lever has the fulcrum at one end and the effort
next to it, with the load at the far end. Examples are a fishing rod
and tweezers. This type of lever usually multiplies motion. Figure
5-23a–c show each type of lever and give examples.
Figure 5-23. (a) A first-class lever. (b) A second-class lever. (c) A third-class
lever.
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Preparing for FCAT Science: Grade 8
Most of the formulas used for inclined planes can also be applied to the lever. In the lever, all distances are measured from the
fulcrum (DE length of the effort arm; DR length of the resistance or load arm). The resistance (R) and the effort (E ) are forces
measured in newtons. Since the amount of friction in levers is very
small, you can assume that the IMA and AMA are equal. IMA is calculated by the formulas
DE
,
DR
L
IMA E , or
LR
IMA R
E
IMA The work principle is work input work output, or, in terms of the
lever:
R DR E DE
A wheel-and-axle is a simple machine that is a modified lever.
The larger wheel is attached to a smaller wheel at the center, which
is the axle. Since they are attached, when you turn one, you turn
the other. Turning the outer wheel moves it a greater distance than
the inner wheel or axle. Forces applied to the outer wheel are multiplied by the axle. Turning the axle instead makes the wheel turn
farther, multiplying speed.
Figure 5-24. A single fixed
pulley.
Pulleys
Pulleys are also a modified form of levers used to amplify force
or change the direction of a force. The wheels of a pulley have a
groove down the middle through which rope or a chain is run. The
rope is pulled to exert the effort force.
There are two basic types of pulley systems. The single fixed
pulley does not move, but the rope running through it allows the
direction of the force to change. (See Figure 5-24.)
Pulling down on the rope on a flagpole moves the flag up. It
does not multiply force or reduce the amount of pull required to lift
the flag. A movable pulley is attached to the load or resistance and
moves with it. This system reduces the amount of effort needed to
move the resistance. It has a mechanical advantage of 2, because
the rope will have to be pulled twice as far to lift the load or resistance half the distance. In other words, it would require a little
more than 5.0 newtons of force to lift a 10-newton load 1.0 meter,
but you would have to pull the rope 2.0 meters.
Attaching multiple movable pulleys to each other is used to
make a system called a block-and-tackle. It may have a mechanical advantage of up to 6 or more and is used for moving heavy
loads. Adding more pulleys increases the amount of friction, limiting its usefulness. (See Figure 5-25 on the next page.)
Chapter 5: Force and Motion
139
Figure 5-25. Multiple pulleys.
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Skill Activities
Short Response
38. Using the diagrams of the tools in the illustration below, classify each as to the type
of lever used.
Tweezers
Crowbar
Shears
Broom
Can
opener
Paper cutter
Question 39 refers to the figure below.
39. Assume that the horizontal line is a 6-meter board, balanced as shown in part (a). In
part (b), which diagram best represents the board if a 20-newton weight is placed at
position B and another at position D?
A
B
C
D
(a)
140
(b)
Preparing for FCAT Science: Grade 8
(1)
(3)
(2)
(4) None of these
Fluid Forces
Fluids or materials that can flow like liquids and gases, can produce pressure when their molecules collide with the surfaces of
their container or objects in them. Air molecules run into you from
all directions and exert a force all over the area of your body. Pressure is the amount of force caused by all the small collisions striking the surface area of your body. The formula for pressure is the
force/area over which it acts:
Pressure Force
Area
or
P F
A
Pressure can be measured in atmospheres, pounds per square
inch (psi), or newtons per square meter. One newton per meter
squared is equal to 1 pascal. One atmosphere is 101,300 pascals, or
14.7 pounds per square inch.
An object surrounded by a fluid like air or water has pressure
acting on it from all directions. The deeper you go into a fluid, the
greater the pressure. In water, increasing the depth increases the
pressure by 1 atmosphere for every 10 meters of depth. At a depth
of 100 meters, the water pressure is 10 atmospheres more than at
the surface.
A buoyant force is also experienced in fluids like air and water.
Buoyant force is the upward force exerted by a fluid on a submerged object. This buoyant force is what pushes a floating object
toward lower pressure in the air or water. The buoyant force in the
water is equal to the mass of the water displaced when the object
floats. In other words, the buoyant force is equal to the weight of
the object. This same buoyant force supports your body when you
float in water, and it also supports a balloon floating in air.
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Skill Activity
Short Response
40. A ship weighing 454,000 kilograms sits out at sea. How much is the buoyant force?
Chapter 5: Force and Motion
141
Review Questions
Multiple Choice.
Choose the BEST answer.
1. Which of the following statements is
true?
a) All motion is relative to a frame of
reference.
b) Our frame of reference must always
remain stationary.
c) Motion can be described by many
absolute frames of reference.
d) The frame of reference is insignificant when objects are moving very fast.
Base your answers to Questions 2 and 3 on
the following information. You are traveling
south to Miami and have traveled a distance
of 34 kilometers in 17 minutes.
2. What is your velocity in kilometers/
minute?
a) 0.5
c) 58
b) 2.0
d) 92
3. What is your velocity in kilometers/
hour?
a) 34
c) 102
b) 68
d) 120
4. A child swings a ball on a string high in
the air. The ball travels in a circle. According to Newton’s first law of motion,
what will happen to the ball if the string
is cut?
a) The ball will immediately fall to the
ground.
b) The ball will fly off in a straight line.
c) The ball’s path will curve to the right.
d) The ball’s path will curve to the left.
142
Preparing for FCAT Science: Grade 8
5. A rocket leaves the Kennedy Space Center. Its launch demonstrates Newton’s
law that
a) an object at rest tends to remain at
rest
b) the greater the mass, the greater the
force
c) the greater the force, the greater the
acceleration
d) for every action, there is an equal but
opposite reaction
6. According to Newton’s second law of
motion,
a) an object with small mass cannot be
moved by a large force
b) lighter masses show no change in acceleration when subjected to a large
force
c) lighter masses accelerate more than
larger masses when subjected to the
same force
d) heavier masses accelerate more than
larger masses when subjected to the
same force
7. If an object has an acceleration of 2.0
meters/second2 and a mass of 40 kilograms, what amount of force will move
it?
a) 0.5 newtons
b) 20 newtons
c) 40 newtons
d) 80 newtons
8. Lubricating the axles of a car
a) decreases friction
b) increases friction
c) increases resistance
d) reduces work output
9. Why does a spacecraft continue to move
after its engines shut down?
a) drag
b) inertia
c) friction
d) resistance
10. Which of the following is an example of
static friction?
a) a tire rolling over asphalt
b) a propeller moving through water
c) a sled sliding over snow
d) the beginning movement of a block
down an inclined plane
11. Which of the following statements is
true?
a) Frictional force speeds up motion.
b) Frictional force has no effect on
motion.
c) Frictional force slows down and ultimately stops motion.
d) Frictional force does not depend on
the nature of the surfaces in contact.
12. If the distance between two charged objects is doubled, the electrical force between the objects
a) doubles
b) remains the same
c) becomes one-fourth as strong
d) becomes one-half as strong
13. Applying the left-hand rule #1
a) allows measurement of the intensity
of magnetic fields
b) gives an indication of the size of a
magnet in the vicinity of a current
c) identifies the direction of magnetic
field lines inside a coil-shaped current
conductor
d) gives an indication of the magnetic
field lines around a straight-line current
conductor
Gridded Response
Make a copy of the grids provided or use
your own paper; do not write in this
book. Be sure to show your work and enter
your answer in the answer grids.
14. Forces of 3.0 newtons and 4.0 newtons
are pulling away from each other at an
angle of 90° (see figure below). What is
the magnitude of the resultant vector
force?
4.0 N
0 0 0 0 0
1 1 1 1 1
90°
2 2 2 2 2
3 3 3 3 3
3.0 N
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
Vector diagram
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
15. The figure below shows two opposing
vectors. Vector 1 has a force of 2.0 newtons and vector 2 has a force of 6.0
newtons. What is the magnitude and direction of the net force?
(2.0 N)
(6.0 N)
Vector
1
Vector
2
West
East
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
8 8 8 8 8
9 9 9 9 9
Chapter 5: Force and Motion
143
16. If a dish falls off a table, and it takes 3
seconds for the dish to hit the floor, how
fast is the dish moving when it hits the
floor? (Hint: The acceleration of gravity
is 9.8 meters/second 2.)
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
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Extended Response
21. a. On your own paper, show the position
of a moving object from a specific reference point by graphing the data from the
table below. Remember to use the x-axis
for time and the y-axis for the distance
from the origin.
b. On your own paper, use the data from
the previous graph to plot velocity versus
time.
5 5 5 5 5
6 6 6 6 6
7 7 7 7 7
Position Data
8 8 8 8 8
Time (minutes)
Distance (ft) from Origin
9 9 9 9 9
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EXPLAIN
Short Response
17. Describe one way that the velocity of an
object can be changed.
18. How do magnetic forces differ from
gravitational forces?
19. Conductors are substances that allow
electrons to flow through them. When
electricity flows through a conductor,
what can form around it?
20. How do conductors differ from insulators?
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Preparing for FCAT Science: Grade 8
0
0.0
1
2.5
2
5.0
3
7.5
4
10
5
10
6
10
7
10
8
17.5
9
25
10
17.5
11
10
12
5
13
5
14
4
15
3
16
2
17
1
18
0