Ch.11 Energy I:Work and kinetic energy
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
1.Definition of ‘Work’
The work W done by a constant force that moves a body through a displacement in the directions of the force as the product of the magnitudes of the force and the displacement:
(11-1)
F
s
(Here )FsW F//s
11-3 Power
Ch.11 Energy I:Work and kinetic energy
The normal force does zero
work; the friction force does
negative work, the gravitational
force does positive work which is
or
N
N
mghmgs cos
)cos(cos mgsmgs
gm
gm
f
shv
Fig 11-5
f
Example: In Fig11-5, a block is sliding down a plane.
Ch.11 Energy I:Work and kinetic energy
2. Work as a dot product
The work done by a force can be written as
(11-2)
(1) If , the work done by the 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.
F
W F s
sF
F
gm
shv
Ch.11 Energy I:Work and kinetic energy
If a certain force performs work on a body in a time , the average power due to the force is
(11-7)
The instantaneous power P is
(11-8)
If the power is constant in time, then .
av
WP
t
dWP
dt
avPP
3. Definition of power: The rate at which work is done.
Wt
Ch.11 Energy I:Work and kinetic energy
(11-10)
Unit of power: joule/second (Watt)
dW F d s d sP F F v
dt dt dt
If the body moves a displacement in a time dt,
sd
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Ch.11 Energy I:Work and kinetic energy
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 to .
Fig 11-12
ix
ixfx
fxx
x
F
1F
2F )(xFx
11-511-4
Ch.11 Energy I:Work and kinetic energy
We divide the total displacement into a number N
of small intervals of equal width . This interval so
small that the F(x) is approximately constant. Then
in the interval to +dx , the work
and similar ……The total work is
or
(11-12)
1 1W F x
x
2 2W F x 1x1x
N
nn xFW
1
...... 2121 xFxFWWW
Ch.11 Energy I:Work and kinetic energy
To make a better approximation, we let go to zero and the number of intervals N go to infinity. Hence the exact result is
or
x
01
limN
nx
n
W F x
01
lim ( )f
i
N x
n xxn
W F x F x dx
(11-13)
(11-14)
ix fx
Numerically, this quantity is exactly equal to the area between the force curve and the x axis between limits and .
Ch.11 Energy I:Work and kinetic energy
Example: Work done by the spring force
In Fig 11-13, the spring is in the relaxed state, that is no force applied, and the body is located at x =0. o x
Relaxed length
Fig 11-13
2 21( )
2
f
i
x
s s f ixW F dx kxdx k x x
kxFs Only depend on initial and final positions
Ch.11 Energy I:Work and kinetic energy
Fig 11-16 shows a particle moves along a curve from to f . The element of work
The total work done is (11-19)
i
f
F
sd
x
y
o
i
dW F d s
cosf f
i iW F d s F ds
( )( )
( )
f
x yi
f
x yi
W F i F j dxi dy j
F dx F dy
Fig 11-16
2.Two-dimensional situation
(11-20)
or
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 that is always horizontal, until
the string finally makes an angle
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.
y
mgm
F
m
dsT
Fig 11-17
F
mx
Ch.11 Energy I:Work and kinetic energy
11-6 Kinetic energy and work-energy theorem
for a body of mass m moving with speed v.
21
2K mv
,F
,a
v Relationship between
Work and Energy1. Definition of kinetic energy K:
2. The work-energy theorem:
2 21 1
2 2netF f iW mv mv (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: represents the net force acting on the body.
netF
dx
dvmv
dt
dx
dx
dvm
dt
dvmmaF x
xxx
xnet
The work done by isnetF
xnet net x
dvW F dx mv dx
dx
It is also true for the case in two or three dimensional cases
2 21 1
2 2
xf
xi
v
x x xf xi
v
mv dv mv mv
Ch.11 Energy I:Work and kinetic energy
4.Notes of work-energy theorem:
2 21 1
2 2netF f iW mv mv
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.
Please relate a) point to conservation of momentum
a). In different inertial reference frames?
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 at
point p, a distance r
from the rotational axis.
Fig 11-19
x
y
Pr
F
F
d
O
ds
Ch.11 Energy I:Work and kinetic energy
As the body rotates through a small angle
about the axis, point p moves through a
distance . The component of the force in
the direction of motion of p is ,and so the
work dw done by the force is
rdds
sin
( sin )( )
( sin )
dW F ds
F rd
rF d
d
sinF
zd
Fr
Ch.11 Energy I:Work and kinetic energy
So for a rotation from angle to angle, the work in the rotation is
(11-25)
The instantaneous power expended in rotation
motion is
i f
f
izW d
z z z
dW dP w
dt dt
(11-27)
Ch.11 Energy I:Work and kinetic energy
2. Rotational kinetic energy
Fig 11-20 shows a rigid body rotating about a fixed axis with angular speed
. We can consider the body as a collection of N particles , ……
moving with tangential speed , …… If indicates the distance of particle from the axis, then and its kinetic energy is
. The total kinetic energy of the entire rotating body is
1m 2m
2v1v
nm nn rv
222
2
1
2
1 nnnn rmvm x
y
O
nr
1m
2m
2r1r
ω
Fig 11-20
Ch.11 Energy I:Work and kinetic energy
is the rotational inertia of the body,
then
22
2222
2211
)(2
12
1
2
1
nnrm
rmrmK
Irm nn 2
2
2
1 IK
(11-28)
(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
Power
Kinetic energy
Work-energy theorem W K
2
2
1 IK
W K
zzP
2
2
1mvK
xxvFP
xW F dx zW d
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
F
v
Solution: For a thin spherical shell
The change in rotational kinetic energy is
222 472)7.1()245(3
2
3
2mkgmkgMRI
2 2
4
1 1
2 2
2.67 10
f iK I I
J
Z
Nrevm
J
R
KF 833
)]0.3()2[()7.1(
1067.2 7
zW RF K
in –z directionz
The rotational work is
then
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 the total kinetic energy after the
collision.
fi kk (11-30)
Ch.11 Energy I:Work and kinetic energy
Fig (6-17) Two-body collisions in cm frame
1minitial
elastic
inelastic
Completely inelastic
explosive
1.
3.4.5.
2m'
1iP'
2iP
Final
2.
'1fP
'
2fP
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 of momentum.
4. Explosive or energy releasing collision: “The total final kinetic energy is greater than the total initial kinetic energy.” Often occur in nuclear reactions.
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