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Chapter 9 Work and Energy
第九章 功與能
What is energy?
The term energy is so broad that a clear definition is
difficult to write.
Energy is a scalar quantity associated with the state
(or condition) of one or more objects.
Energy can be transformed from one type to another
and transferred from one object to another, but the
total amount is always the same (energy is conserved).
No exception to this principle of energy conservation
has ever been found.
Kinetic energy
Before collision:
1-D collision:
After collision:
m1v1  m1v1  m2v2
Momentum conservation:
After collision in the rest frame of particle 2:
 v1  v1  v2
Thus:
v1 
m1  m2
v1
m1  m2
v2 
2m1
v1
m1  m2
Verify: m1v12  m2v22  m1v12 so the kinetic energy is conserved.
1 2
K .E.  mv
2
Why the prefactor should be ½ ?
What is work?
Energy can be transformed or transferred by force.
But force alone might not change the energy of an
object, e.g. centripetal force on an object in uniform
circular motion.
Work W is energy transferred to or from an object by
means of a force acting on the object.
Energy transferred to the object is positive work, and
energy transferred from the object is negative work.
Work by a constant force
1 2
K .E.  mv
2
Why the prefactor should be ½ ?
d
d 1 2
( K .E.)  ( mv )  mva  Fv
dt
dt 2

i
f
t
d ( K .E.)   Fvdt  Fx
0
( K .E.)  Fx  work
Work done in curvilinear motion
 
dW  F  dr

 
 
dv 
1  
dW  F  dr  m  (v dt )  mv  dv  d ( mv  v )
dt
2
  1 2 1 2
W   dW   F  dr  mv f  mvi
2
2
path
path
Work done by a variable force

 
rf 

W   F  rn   F  dr
n
ri
For 1D along x axis: W 

xf
xi
Fx dx
Work-energy theorem
1 2 1 2
W  mv f  mvi
2
2
or
W  K
The unit of energy:
1 J  1 kg m s  1 N m
1 cal  4.186 J
2 -2
Power
 
dW F  dr  
P

 F v
dt
dt
The unit of power:
1 W 1J s  1 N m s
-1
1 horsepower  745.7 W
1 kilowatt hour  3.6 106 J
-1
Tables of energy and power
Work done by gravity



F  mg  mgj
 
W  mg  s



 (mgj )  (xi  yj )
W  mg ( y f  yi )
For arbitrary path:
yf
 
W   dW   mg  ds  (mg )  dy  mg ( y f  yi )
yi
Work done by a spring
Fspring  kx
xf
Wspring    kxdx
1
2
2
Wspring   k ( x f  xi )
2
xi
Work done by friction
Wby A on B  Wby B on A
W f   fk s
Note that the work done
by friction depends on
the total path length, not
just the initial and final
coordinates.
fk
Example 0
A crate of mass m is dropped onto a conveyor belt that moves at a
constant speed v (see the figure below). The coefficient of kinetic
friction is k. (a) What is the work done by friction? (b) How far
does the crate move before reaching its final speed? (c) When the
crate reaches its final speed, how far has the belt moved?
Potential energy
Potential energy
Dutch Scientist Christian Huygens
showed that speed v reached of a
fall from a height h is given by
v2  h
Gottfried W. Leibnitz thought a
falling body acquires a “force”
that can carry it back to its original
level.
His “force” is what we call energy.
Potential energy
Potential energy
Work-energy theorem leads to the
principle of conservation of
mechanical energy.
Wext  U  U f  U i
The zero of the potential energy
can be chosen arbitrarily.
If the potential energy of the
initial configuration is chosen to
be zero, i.e. Ui = 0, then the
potential energy of a system is
the external work that needed to
take the system to its final state.
Conservative forces
If the work done by a force to an
object depends on only its initial
and final positions and not the
path taken, then it is called a
conservative force, otherwise it is
a nonconservative force.
Work done by gravity:
Wg  mg ( y f  yi )
Work done by a spring:
1
Wsp   k ( x 2f  xi2 )
2
Work done by friction:
Depending on the path.
Conservative forces
For a conservative force:
1)
( 2)
Wa(

W
b
a b
Wa(2)b  Wb(2)a
1)
( 2)
Wa(

W
b
ba  0
The work done by a conservative force around any closed loop
is zero.
Potential energy and conservative forces
Since the work done by a conservative force to an object
depends only on its initial and final positions, we can write:


WC  (U (rf )  U (ri ))
The reason for the minus sign will become clear later. If we
assume K = 0 when the object is moved, then it requires an
external force to counter balance the conservative force FC.
It seems more natural to have:
WEXT


 WC  U (rf )  U (ri )
Potential energy and conservative forces
For a fixed initial position r0, we can rewrite WC as a
function of the final position r.



WC (r )  U (r )  U (r0 )
 
dU  dWC  FC  dr
 
U B  U A   FC  dr
B
A
A conservative force can be associated with a scalar
potential energy function, whereas a nonconservative
force cannot.
Gravitational potential energy
Gravitational potential energy function (near the Earth surface):
U g  mgy  const
The arbitrary constant allows us to choose the zero potential
energy at any convenient position.
Conservation of mechanical energy
If a particle is only subject to a conservative force, then we
can use the potential energy with the work-energy theorem,
W = K, to obtain:
K  U  0
K f  Ki  U f  U i  0
 K f  U f  Ki  U i
We define the mechanical energy E as E  K  U
The principle of the conservation of mechanical energy:
E  E f  Ei  0
Applications of the energy conservation
1 2
E  mv  mgy  contant
2
Example 1
You throw a 0.145-kg baseball up in
the air, giving it an initial upward
velocity of magnitude 20.0 m/s. Find
how high it goes, ignoring air
resistance.
v12  2 gy1  v22  2 gy2
Work-energy theorem including
potential energy
Wother  WC  K 2  K1
WC  (U 2  U1 )
Wother  ( K 2  U 2 )  ( K1  U1 )
Wother  E2  E1
Example 2
In example 1, suppose your hand
moves up while you are throwing
the ball, which leaves your hand
with an upward velocity of 20.0
m/s. Again, ignore the air
resistance. (a) Assuming that
your hand exerts a constant
upward force on the ball, find the
magnitude of that force. (b) Find
the speed of the ball at a point
15.0 m above the point where it
leaves your hand.
Motion along a curved path


 

W  Ftotal  s  (w  Fother )  s


Fother  s  0
 
W  w  s  mg( y2  y1 )
We can use the same expression
for gravitational potential energy
whether the body’s path is curved
or straight.

s
Example 3
A batter hits two identical baseballs with the same initial speed and
height but different initial angles. Prove that at a given height h,
both balls have the same speed if air resistance can be ignored.
Example 4
Your cousin Throckmorton skateboards down a curved playground
ramp. If we treat Throcky and his skateboard as a particle, he moves
through a quarter-circle with radius R = 3.0 m. He starts from rest
and there is no friction. (a) Find his speed at the bottom of the ramp.
(b) Find the normal force that acts on him at the bottom of the curve.
1 2
2
E  mv  mgy  contant  v  2 gR  v  7.67 m/s
2
2
mv
N
 mg  3mg  735 N
R
Example 5
Assume the same initial condition as in Example 4 except that the
ramp is not frictionless. If Throcky’s speed at the bottom is only
6.00 m/s, then what was the work done by the frictional force acting
on him?
W friction  WC  K 2  K1
Example 6
We want to load a 12-kg crate into a truck by sliding it up a ramp 2.5
m long, inclined at 30°. A worker, giving no thought to friction,
calculates that he can get the crate up the ramp by giving it an initial
speed of 5.0 m/s at the bottom and letting it go. But friction is not
negligible; crate slides 1.6 m up the ramp, stops, and slides back
down. (a) Assume that the friction force acting on the crate is
constant, find it magnitude. (b) How fast the crate moving when it
reaches the bottom of the ramp?
Elastic potential energy
Work done by the spring:
Wel  
x2
x1
1
(kx)dx   k ( x22  x12 )
2
Spring potential energy function:
1 2
U el  kx
2
1 2 1 2
E  mv  kx  contant
2
2
Example 7
A glider with mass m = 0.200 kg sits on a frictionless horizontal
air track, connected to a spring with force constant k = 5.00 N/m.
You pull on the glider, stretching the spring 0.100 m, and then
release it with no initial velocity. The glider begins to move back
toward its equilibrium position (x = 0). What is its x-velocity
when x = 0.080 m?
Example 8
For the system of Example 7, suppose the glider is initially at rest
at x = 0, with the spring unstretched. You then apply a constant
force F in the +x-direction with magnitude 0.610 N to the glider.
What is the glider’s velocity when it has moved to x = 0.100 m?
Problem solving guide for energy
conservation
1. In general there may be more than one type of potential energy
and more than one particle kinetic energy involved in the problem.
E  K  U g  U sp
2. Care must be taken with the signs and the zero of the potential,
which should not be changed in one problem once it is fixed.
Example 9
In a worst case design scenario,
a 2000-kg elevator with broken
cables falling at 4.00 m/s when
it first contacts a cushioning
spring at the bottom of the shaft.
The spring is supposed to stop
the elevator, compressing 2.00
m as it does so. During the
motion a safety clamp applies a
constant 17,000-N frictional
force to the elevator. As a design
consultant, you are asked to
determine what force constant
of spring should be?.
Nonconservative forces
A force that is not conservative is called a nonconservative force.
The work done by a nonconservative force:
1. cannot be expressed as the difference of a potential-energy
function.
2. is not reversible.
3. is path-dependent.
4. is not zero when ending point is the same as the starting point.
Some nonconservative forces, such as kinetic friction or fluid
resistance, always cause the mechanical energy to be dissipated.
They are called dissipative forces. There are nonconservative
forces that increase mechanical energy, e.g. exploding
firecracker
Example 10
You are rearranging your furniture and wish to move a 40.0-kg
futon 2.50 m across the room. However, the straight-line path is
block by a heavy coffee table that you don’t want to move. Instead,
you slide the futon in a dogleg path over the floor; the doglegs are
2.00 m and 1.50 m long. Compared to the straight-line path, how
much work must you do to push the futon in the dogleg path? The
coefficient of kinetic friction is 0.200.
Example 11
In a certain region of space the force on an electron is F = Cxj,
where C is a positive constant. The electron moves in a counterclockwise direction around a square loop in the xy-plane. The
corners of the square are (x, y) = (0, 0), (L, 0), (L, L), (0, L).
Calculate the work done on the electron by the force F during one
complete trip around the square. Is this force conservative or not?
Mechanical energy and nonconservative
forces
The work-energy theorem:
Wnet  WC  WNC  K
WC  U
WNC  K  U
The law of conservation of energy:
WNC  U int
K  U  U int  0
External force and
internal energy transfer
An external force can change the kinetic
energy or potential energy of an object
without doing work on the object—that is,
without transferring energy to the object.
Instead, the force is responsible for
transfers of energy from one type to
another inside the object.
Example 12
A 2.0 kg package of tamale slides along a floor with speed v1
 4.0 m/s. It then runs into and compresses a spring, until the
package momentarily stops. Its path to the initially relaxed
spring is frictionless, but as it compresses the spring, a kinetic
frictional force from the floor, of magnitude 15 N, acts on the
package. If k  10 000 N/m, by what distance d is the spring
compressed when the package stops?
Example 13
A block of mass m is attached to a spring and moves on a rough
inclines as in the following figure. Initially, the block is at rest
with the spring unextended. A force F acting at an angle α to the
incline pulls the block. Write the modified form of the workenergy theorem.
Example 14
A block of mass m = 0.2 kg is held against, but not attached to a
spring (k = 50 N/m) which is compressed by 20 cm, as shown
below. When released, the block slides 50 cm up the rough
incline before coming to a rest. Find: (a) the force of friction; (b)
the speed of the block as it leaves the spring.
Conservative forces and potential
energy functions
A conservative force can be derived from a scalar potential
energy function.
Example 15
An electrically charged particle is held at rest at the point x = 0,
while a second particle with equal charge is free to move along
the positive x-axis. The potential energy of the system is U(x) =
C/x, where C is a positive constant that depends on the magnitude
of the charges. Derive an expression for the x-component of force
acting on the movable particle, as a function of its position.
Conservative forces and potential
energy functions
Force and potential energy function in three dimensions:
 
dU   F  dr  ( Fx dx  Fy dy  Fz dz )
U
U
U
dU 
dx 
dy 
dz
x
y
z

U ˆ U ˆ U
 ˆ  ˆ 
ˆ
ˆ
F  (i
j
k
)  (i
 j  k )U
x
y
z
x
y
z
 ˆ  ˆ 
ˆ
  (i
 j k )
x
y
z

F  U
Example 16
A puck slides on a level, frictionless air-hockey table. The
coordinates of the puck are x and y. It is acted on by a
conservative force described by the potential-energy function
U(x, y) = ½ k(x2 + y2). Derive an expression for the force acting
on the puck, and find an expression for the magnitude of the
force as a function of position.
Energy diagrams
Note that x = 0 is a stable
equilibrium point.
Potential energy curve
Note that the points x1 and x3
are stable equilibrium points;
while the points x2 and x4 are
unstable equilibrium points.
Energy diagrams
K  E U
A particle is in a bound state
when E < 0.
Gravitational potential energy
GmM
F   2 rˆ
r
Along a particular radial direction
r̂
:
r2
r2
r1
r1
U (r2 rˆ)  U (r1rˆ)    F  dr  
GmM GmM
U (r2 )  U (r1 )  

r2
r1
GmM
U (r )  
r
GmM
dr
2
r
Generalized conservation of energy
Mechanical energy can be dissipated into heat through
nonconservative forces, e.g. frictional forces.
However, we can include the thermal energy generated in
the heating process, and it is possible to convert some of
the thermal energy back into mechanical energy.
Around 1845, several scientists proposed the principle of
the conservation of energy:
Energy can change its form, but it can neither be created
nor destroyed.
Summary
The work done by a force: WAB  
B
A
The work-energy theorem: W  K
The instantaneous mechanical power:
P  F  v  dE / dt
 
F  dr
Summary
Conservative force: WA(1) B  WA(2)
B
Potential energy:
U  WC
A conservative force can be derived from
a scalar potential energy function:
U
Fx  
x
U
Fy  
y
U
F 
 U
r
U
Fz  
z
Conservation of mechanical energy:
E  K  U  WNC
Gravitational potential energy: U  
GmM
r
Further reading & homework
• The Feynman lectures on Physics, Vol. 1,
Chapter 4, 13, and 14.
• Questions: 9.1, 9.4, 9.6, 9.11, 9.12, 9.15
• Problems: 9.1, 9.5, 9.9, 9.11, 9.14, 9.19,
9.21, 9.27