Download 14.1 The Work of a Force

CHAPTER 14
Kinetics of a particle:Work and Energy
14.1 The Work of a Force
1. Definition of Work

A force F does work in a particle only when the particle
undergoes a displacement in the direction of the force.

r
Here
dr
ds
r

F
=force acts on the particle
dr = r  -r =displacement of the particle

ds = dr
 =angle between F and

dr
 positive

dU  negtive
0

du  F cos  ds
 
 F * dr
00    900

0
0
90    180 

0 
  90 .dr  0 

F S  cos 
Unit of Work (SI)
1 N*m=1 joule (J)
1 ft*lb=

F S  cos 
2. Work of a Variable Force F  F S 

F
S1


F cos
r1
S2
r2
S2
S1
 
ds
dU  Fdr
 F cos ds
2
U1 2   dU
1

r1  
s1
   Fdr   F ( s ) cos ds
r2
s2
S
3. Work of a Constant Force
moving along a straight live
U 1 2 

Fc
S1
y
z
s1
ds




r  xi  yj  zk




dr  dxi  dyj  dzk


w   wk
(position)
(displacement)
 
  Fdr 
S1
y1
S2
s2
 F cos  S 2  S1 
S2
4. Work of a Weight
S
F cos ds
 F cos  

Fc cos
W
s
1

S
s2
r1
r2
y2
r2  
U1 2
 r1 wdr




r2

 wk  dxi  dyj  dzk
r1
 


z1
wdz   w
z2
z2
z1

dz   w z 2  z1 
 positive  Z  0
  wZ  
negtive  Z  0
5. Work of a spring forces
Consider a linear spring force Fs  ks
(1) Work done on the spring
S
sd
Fs
S1
S2
Fs
Fs  ks
S1
S2
s2
S
s2
U12   Fsds   ksds
s1
s1
 12 ks22  12 ks12  12 k ( s 2  s 2 )
(2)Work of a spring done on the particle
A particle (or only) attached to a spring
F.B.D of spring and particle
spring force on the particle Fs  ks
work of a spring on the particle is
U1 2
2
 
  Fs  dr   ksds
1
  ks
1
2
2
s2
s1
  12 k ( s2  s1 )  0
2
2
14.2 Principle of Work and Energy (PWE)
1. P.W.E
The particle’s initial kinetic energy plus the work done by
all the forces acting on the particle as it moves from its
initial to its final position is equal to the particle’s final
kinetic energy.
T1  U12  T2 or 12 mv12  U12  12 mv2 2
z
2
1
Here T  2 mv  kinetic energy

P( FR )t
U12  Work done by forces.

r
2. Derivation


FR   F external force
Equation of motion of particle




FR  ma  mat ut  manun
 
 
 ( FR ) t ut  ( FR ) n un


 FR   F
t
n ( Fe ) n
y
x
Initial frame

Work done on particle P by external force FR is






FR  dr   F  dr  ma  dr


 F  dr   Fds cos   ( F cos u )ds



a  at ut  an u n


dr  dsut
d 2s
( F )t  mat  m dt


 a  dr  at ds


  F  dr  ( F ) t ds  mat ds
(1)Applying the kinetic equation at ds  Vdv to the above
 
equation yields  Fv dr   Ft ds  mvdv
(2)Integrate both sides to yield
v 1
r1  
r2
2
2
2 2
1
1
U

mv

mv

mv
  r F  dr   r mvdv  12 2 v1 2 2 2 1  T2  T1
2
1
or
T1  U12  T2
3. Remark
(1) PWE represents an integrated form of equation of
motion  F t  mat
s2
v2
dv
Ft  mat   Ft  mv ds    s Ft ds   v mvdv

at 
vdv
ds
 F ds  mvdv
1
1
2
1
U12  12 mv2  12 mv  T2  T1
t
2
(2) PWE provides a convenient substitution for  F t  mat
When solving kinetic problems involving force , velocity and
n
displacement.


Ex:
r
From P.W.E we have
T1  U12  T2
1
2

mv1  mgr sin   mv2
2
V2  2 gr sin 
1
2
T ?
2
mg
t
14.3 Principle of Wok and Energy for a
Si
System of Particles

principle of Work and Energy for
Particle i



ri 2
 ri 2   1
2
2
1
2 mi vi1    Fx  dri    f x  dri  2 mi vi 2
ri1
ri1
i
Fi
n
t
fi

Fi =resultant external force on ith particle

fi 
n


f ij
=resultant internal force on ith particle
Since work and energy are scalars both work and kinetic
energy applied to each particle of the system may be added
together algebraically.  

 
r
r

2
2
i2
i2
1
1
m
v

F

d
r

f

d
r

m
v
So that  2 i i 2   r i i   r i i  2 i i 2
i1
i1
or
T  U  T
j 1
j i

1

1 2

2
T
T
U
Here
=System’s initial kinetic energy
2 =System’s final kinetic energy
12 =Work done by all external and internal forces
1
Note: r  
i2
(1)   ri1 f i  dri  0 ,since the paths over which
corresponding particles travel will be different.
(1)
f ij
i
f ji
Si
(2)
(Non rigid Body)
Elastic..plastic…
i
j
 j
Rigid body
j
S

j
ri 2

 f i  dri  0


(2)
,if the particles are contained within
ri1
the boundary of a translating rigid body , or particles
connected by inextensible cables.
i
14.4 Power and Efficiency
1. Power
The amount of work per unit of time.
U
dv
Pav 
or
P 
t
dt


dv  F  dr  work




 
F  dr
dr
P 
 F
 F v
dt
dt
unit of power
1N  m
s
 1J
s
 1W
2. Efficiency 效率
poweroutput

powerinput
energyoutput

energyinput
14.5 Conservative Forces and Potential
Energy
1. Conservative force
The force moves the particle form one point to another point
to produce work which is independent of the path
followed by the particle.






 1 F  dr   2 F  dr   3 F  dr
 F  Conservative force
1
F
2
3
(1) Work done by weight
U  Wy
(2) Work done by the spring force on a particle
S
w
w
V  ( ks2  ks )
1
2
Spring force
S1
y
y0
S2
2
1
2
2
1
F (保守力)
2. Potential Energy 位能
P
A measure of the amount of work a conservation force will
datum
do when it moves from a given position to the datum or a
Vg  Wy w
推回基準作
reference plane.
P
 y 的功
F
datum
(1) Gravitational potential energy
y
Vg  Wy (y=positive upward) Vg  Wy
作負功
(2) Elastic Potential Energy
Ve   12 ks2
always positive.
The spring force has the capacity for always positive work on the particle.
Unstretched
S
Ve  ?
S
push
datum
pull
3. Potential Function V
V  Vg  Ve
 The work done by conservation forces(W and Fs) in
moving the particle from point ( x1 , y1 , z1 ) to point ( x2 , y2 , z2 )
Is V12  V1  V2
Ex:
(U )  Wy  W ( S  S )
1 2 w
2
(Unstretched position)
2
V1  Vg1  Ve1  WS1  ks1
1
2
datum
S1
2
V1  V2  W ( S 2  S1 )  12 ks1  12 ks2
2
 (U12 ) w  (V12 ) Fs  U12
w
2
V2  Vg 2  Ve2  WS 2  ks2
1
2
S2
v2
1
(V12 ) Fs  ( 12 ks2  12 ks1 )
l0
v1
2
2
14.6 Conservation of Energy
The principle of work and energy is rewritten as
T1  (U12 ) cons  (U12 ) noncons  T2
(U12 ) cons  V1  V2  Work done by conservative forces
(U12 ) noncons  Work done by nonconservative forces
T1  V1  (U12 ) noncons  T2  V2
If (U12 ) noncons  0 , then
T1  V1  T2  V2  Conservation of energy
The sum of the particle’s kinetic and potential energy remains
constant during the motion .
 Conservation of energy for a system of particles is
T  V  T  V
1
1
2
2
end