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Ch. 11 Energy I:
Work and kinetic energy
Ch.11 Energy I:
Work and kinetic energy
11-1 Work and energy
Example: If a person pulls an object uphill. After
some time, he becomes tired and stops.
We can analyze the forces exerted in this
problem based on Newton’s Laws, but those laws
can not explain: why the man’s ability to exert a
force to move forward becomes used up.
For this analysis, we must introduce the new
concepts of “Work and Energy”.
Ch.11 Energy I:
Work and kinetic energy
Notes: 1) The “physics concept of work” is
different from the “work in daily life”;
2) The “energy” of a system is a measure
of its capacity to do work.
Ch.11 Energy I:
Work and kinetic energy
11-2 Work done by a constant force
11-3 Power
1.Definition of ‘Work’

The work W done by a constant force F that

moves a body through a displacement s in the
directions of the force as the product of the
magnitudes of the force and the displacement:
 
W  Fs (Here s //F )
(11-1)
Ch.11 Energy I:
Work and kinetic energy
Example: In Fig11-5, a block is
sliding down a plane.


The normal force N does zero

work; the friction force f does
negative work, the gravitational

force m g does positive work which
is
mgs cos   mgh

N
v
f
s

mg

h
Fig 11-5
or
mgs cos   s (mg cos  )
Ch.11 Energy I:
Work and kinetic energy
v
s

mg

h
2. Work as a dot product

The work done by a force F can be written as
 
(11-2)
W  F s



(1) If F  s , the work done by the F is zero.
(2) Unlike mass and volume, work is not an intrinsic
property of a body. It is related to the external force.
(3) Unit of work:
Newton-meter (Joule)
(4) The value of the work depends on the
inertial reference frame of the observer.
Ch.11 Energy I:
Work and kinetic energy
3. Definition of power:
The rate at which work is done.
If a certain force performs work Won a body in a
time t , the average power due to the force is
W
(11-7)
Pav 
t
The instantaneous power P is
dW
(11-8)
P
dt
If the power is constant in time, then P  Pav .
Ch.11 Energy I:
Work and kinetic energy

If the body moves a displacement d s in a time dt,



dW F  d s  d s  
P

 F
 F v
dt
dt
dt
Unit of power:
(11-10)
joule/second (Watt)
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Ch.11 Energy I:
Work and kinetic energy
11-4
11-5
Work done by a variable force
1.One-dimensional
situation
The smooth curve in
Fig 11-12 shows an
arbitrary force F(x) that
acts on a body that
moves from x i to x f .
Fig 11-12
F
F2
Fx (x)
F1
xi
x
xf
x
Ch.11 Energy I:
Work and kinetic energy
We divide the total displacement into a number N
of small intervals of equal width x . This interval so
small that the F(x) is approximately constant. Then
in the interval x1 to x1 +dx , the work W1  F1x
and similar W2  F2x ……The total work is
W  W1  W2  ...  F1x  F2x  ...
or
N
W   Fn x
(11-12)
n 1
Ch.11 Energy I:
Work and kinetic energy
To make a better approximation, we let x go to
zero and the number of intervals N go to infinity.
Hence the exact result is
or
W  lim
x  0
N
N
 F x
n 1
W  lim  Fn x  
x  0
n 1
(11-13)
n
xf
xi
F ( x)dx
(11-14)
Numerically, this quantity is exactly equal to the
area between the force curve and the x axis
between limits x i and x f .
Ch.11 Energy I:
Work and kinetic energy
Example: Work done by the
spring force
Fig 11-13
In Fig 11-13, the spring is in the
Relaxed length
relaxed state, that is no force
applied, and the body is
located at x =0.
o
Ws   Fs dx  
xf
xi
x
1
2
2
kxdx   k ( x f  xi )
2
Fs  kx
Only depend on initial
and final positions
Ch.11 Energy I:
Work and kinetic energy
2.Two-dimensional situation
Fig 11-16 shows a particle moves
along a curve from i to f . The
element of work



y
F
dW  F  d s

ds
The total work done is
f 

W   F d s  
i
i
f
(11-19)
F cos  ds
f
or W  i ( Fx i  Fy j )(dxi  dy j )
f
  ( Fx dx  Fy dy )
i

(11-20)


f
i
o
x
Fig 11-16
Ch.11 Energy I:
Work and kinetic energy
Sample problem 11-5
A small object of mass m is
suspended from a string of length L.
The object is pulled sideways by a

force F that is always horizontal, until
the string finally makes an angle  m
with the vertical. The displacement is
accomplished at a small constant
speed. Find the work done by all the
forces that act on the object.
Fig 11-17
y
x

m
T ds

F
m
mg
Ch.11 Energy I:
Work and kinetic energy
11-6 Kinetic energy and work-energy theorem



Relationship between
F, a, v
Work and Energy
1. Definition of kinetic energy K:
1 2
K  mv
2
for a body of mass m moving with speed v.
2. The work-energy theorem:
WnetF
1
1
2
2
 mv f  mvi
2
2
(11-24)
“The net work done by the forces acting on a body is equal
to the change in the kinetic energy of the body.”
Ch.11 Energy I:
Work and kinetic energy
3. General proof of the work-energy theorem

For 1 D case: Fnet represents the net force acting on
the body.
Fnet
dv x
dv x dx
dv x
 max  m
m
 mvx
dt
dx dt
dx

The work done by Fnet is
Wnet
dvx
  Fnet dx   mvx
dx
dx
vxf
1
1
2
2
  mvx dvx  mvxf  mvxi
2
2
vxi
It is also true for the case in two or three
dimensional cases
Ch.11 Energy I:
Work and kinetic energy
Please relate a) point to
conservation of momentum
4.Notes of work-energy theorem:
WnetF
1
1
2
2
 mv f  mvi
2
2
a). In different inertial reference frames?
The work-energy theorem survives in different inertial
reference frames.
But the values of the work and kinetic energy in their
respective reference frames may be different.
b). Limitation of the theorem
It applies only to single mass points.
Ch.11 Energy I:
Work and kinetic energy
Ch.11 Energy I:
Work and kinetic energy
11-7 Work and kinetic energy in
rotational motion
1.Work in rotation
Fig11-19 shows an
arbitrary rigid body to
which an external agent

applies a force F at
point p, a distance r
from the rotational axis.
Fig 11-19

y
F
ds 
O
d

P
r
x
Ch.11 Energy I:
Work and kinetic energy
As the body rotates through a small angle d
about the axis, point p moves through a
distance ds  rd . The component of the force in
the direction of motion of p is F sin  ,and so the
work dw done by the force is
dW  F sin  ds
 ( F sin  )( rd )
 ( rF sin  ) d
  z d



  r F
Ch.11 Energy I:
Work and kinetic energy
So for a rotation from angle  i to  f angle, the
work in the rotation is
f
(11-25)
W   d

z
i
The instantaneous power expended in rotation
motion is
dW
d
(11-27)
P
z
  z wz
dt
dt
Ch.11 Energy I:
Work and kinetic energy
2. Rotational kinetic energy
Fig 11-20
Fig 11-20 shows a rigid body rotating
about a fixed axis with angular speed
y
 . We can consider the body as a
m1 ω
collection of N particles m1, m2 ……
moving with tangential speed v1 , v2
…… If rn indicates the distance of
particle m n from the axis, then v n  rn
and its kinetic energy is
1
1
2
2
mn v n  mn rn  2 . The total kinetic
2
m2
r1
r2
O
2
energy of the entire rotating body is
Ch.11 Energy I:
Work and kinetic energy
x
1
1
2 2
2
K  m1 r1   m2 r2  2       
2
2
1
2
 ( mn rn ) 2
(11-28)
2
m r
n n
then
2
 I is the rotational inertia of the body,
1 2
K  I
2
(11-29)
3. The rotational form of the work-energy theorem
W  K
which can be obtained similarly as for single particles.
Ch.11 Energy I:
Work and kinetic energy
Table 11-1
Translational quantity Rotational quantity
Work
W   Fx dx
W    z d
Power
P  Fx v x
P   zz
Kinetic energy K  1 mv 2
1 2
K  I
2
Work-energy theorem
W  K
W  K
2
Ch.11 Energy I:
Work and kinetic energy
Sample problem 11-10
A space probe coasting (航线) in a region of
negligible gravity is rotating with an angular speed
of 2.4rev/s about an axis that points in its direction
of motion.
The spacecraft is in the form of a thin spherical
shell of radius 1.7m and mass 245kg.
It is necessary to reduce the rotational speed to
1.8rev/s by firing tangential thrusters (推进器) along
the equator of the probe.
What constant force must the thruster exert if the
change of angular speed is to be accomplished as
the probe rotates through 3.0 revolution?
Ch.11 Energy I:
Work and kinetic energy
Solution: For a thin spherical shell
2
2
2
I  MR  (245kg)  (1.7m) 2  472kg  m 2
3
3
The change in rotational kinetic energy is
1
1
2
2
K  I  f  I i
2
2
 2.67  104 J
 in –z direction

F
Z

v

z
The rotational work is W      RF  K
z
K
 2.67  10 7 J
then F 

 833N
 R  (1.7m)  [( 2 )  (3.0rev)]
Ch.11 Energy I:
Work and kinetic energy
11-8 Kinetic energy in collision
We reconsider a collision between two bodies that
move
along the x axis with the analysis of kinetic energy.
1. Elastic collision: the total kinetic energy before
collision equals
after the
k i the
k f total kinetic energy
(11-30)
collision.
Ch.11 Energy I:
Work and kinetic energy
Fig (6-17)
Two-body collisions in cm frame

1.
2.
m1
P1i
'
P2i
'
m2


P1f

'
P2f
initial
'
elastic
3.
inelastic
4.
Completely
inelastic
5.
explosive
Final
Ch.11 Energy I:
Work and kinetic energy
2. Inelastic collision: the total final kinetic energy is
less
than the total initial kinetic energy.
(If you drop a tennis ball on a hard surface, it
does not quite bounce to its original height.)
3. Completely inelastic collision: two bodies stick
together. This type of collision loses the maximum
amount of kinetic energy, consistent with the
conservation
ofenergy
momentum.
4.
Explosive or
releasing collision:
“The total final kinetic energy is greater than the
total initial kinetic energy.” Often occur in nuclear
reactions.
Ch.11 Energy I:
Work and kinetic energy