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Last Chapter: Kinetic energy and work
Chapter 8
Potential Energy and
Conservation of Energy
8-1 Potential Energy
Five sections
S-1: Potential Energy
S-2: Conservation of Mechanical Energy:
S-3: Reading a Potential Energy Curve
S-4: Work Done on a System by an External Force
S-5: Conservation of Energy
8.1 Potential energy
Potential energy is energy that can be associated with
the configuration (arrangement) of a system of objects
that exert forces on one another.
Some forms of potential energy:
1.  Gravitational Potential Energy
  Interaction: gravity
  Configuration: relative position
2.  Elastic Potential Energy
•  Interaction: EM interaction between molecules
•  Configuration: Relative position among molecules
8.2 Work and potential energy:
Start from an example
F
Change in
potential
energy
Case-(a)
Case-(b)
0+
=
+
+0
=
-
Work = Fd
F
+
Always have
opposite sign
8.2 Work and potential energy
The change ΔU in potential energy (gravitational,
elastic, etc) is defined as being equal to the negative
of the work done on the object by the force
(gravitational, elastic, etc)
Do we care about the kinetic energy in this workenergy process?
No: If only the potential energy changes = If we
keep the speed of the object unchanged
Yes: Regarding the total energy that has a variable
kinetic energy
8.2 Conservative and non-conservative forces
A popular scenario in reality
1. A system consists of two or more objects.
2. A force acts between a particle-like object in the system and the rest of the system.
3. When the system configuration changes, the force does work (call it W1) on the
object-1  Transferring energy between the kinetic energy of the object, K, and
some other type of energy of the system.
4. When the configuration change is reversed, the force reverses the energy transfer,
doing work W2 in the process.
8.2 Conservative and non-conservative forces
In a situation in which
(1)  W1= -W2 is always true
(2)  the other type of energy is a potential
energy
Then, the force is said to be a
conservative force.
Or, the work done by the force in moving
a particle between two points is
independent of the path taken:
•  gravitational force
•  spring force
•  magnetic force
•  electric force
A force that is not
conservative is called a
non-conservative force:
• 
• 
• 
The kinetic frictional force
Air/fluid drag force
non-elastic scattering force
8.3 Path Independence of Conservative Forces
The net work done by a
conservative force on a
particle moving around
any closed path is zero.
If the work done from a to b
along path 1 as Wab,1 and the
work done from b back to a
along path 2 as Wba,2. If the
force is conservative, then
the net work done during the
round trip must be zero
If the force is conservative,
Sample problem, slippery cheese
Sample problem, slippery cheese
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Sample problem, slippery cheese
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Sample problem, slippery cheese
Calculations: Let us choose the dashed path in Fig. 8-5b; it
consists of two straight segments. Along the horizontal
segment, the angle φ is a constant 90°. Even though we do
not know the displacement along that horizontal segment,
the work Wh done there is
Along the vertical segment, the displacement d is 0.80 m
and, with and both downward, the angle is a constant = 0°.
Thus, for the work Wv done along the vertical part of the
dashed path,
The total work done on the cheese by Fg as the cheese
moves from point a to point b along the dashed path is then
This is also the work done as the cheese slides along the
track from a to b.
Sample problem, slippery cheese: Let’s try this together
Sample problem, slippery cheese: Let’s try this together
2 kg
0.8 m
α
And we need this
α
to write down the calculations
March 27
8.4: Determining Potential Energy values:
For the most general case, in which the force may vary with
position, we may write the work W:
We can use it to derive “Potential Energy values”
corresponding different forces!
8.4: Determining Potential Energy values:
Gravitational Potential Energy
A particle with mass m moving vertically along a y axis (the positive
direction is upward). As the particle moves from point yi to point yf, the
gravitational force does work on it. The corresponding change in the
gravitational potential energy of the particle–Earth system is:
Reference point
The gravitational potential energy associated with a particle–Earth
system depends only on the vertical position y (or height) of the
particle relative to the reference position y =0, not on the horizontal
position. So,
8.4: Determining Potential Energy values:
Elastic Potential Energy
In a block–spring system, the block is moving on the end of a spring of
spring constant k. As the block moves from point xi to point xf , the
spring force Fx =- kx does work on the block. The corresponding change
in the elastic potential energy of the block–spring system is
Reference point
If the reference configuration is when the spring is at its relaxed length,
and the block is at xi = 0.
Sample problem: gravitational potential energy
Sample problem: gravitational potential energy
Reference point with U = 0
Sample problem: gravitational potential energy
8.5: Conservation of Mechanical Energy
Principle of Conservation of Energy
In an isolated system where only conservative forces cause energy changes, the
kinetic energy and potential energy can change, but their sum, the mechanical
energy Emec of the system, cannot change.
The mechanical energy Emec of a system is the sum of its potential energy U and the
kinetic energy K of the objects within it:
With the absence of
Potential Energy
With the absence of
Kinetic Energy
With both Potential Energy
and Kinetic Energy
8.5: Conservation of Mechanical Energy: An example
Potential energy: Reference point
for zero U
In stages (a) and (e), all the energy is
kinetic energy.
In stages (c) and (g), all the energy is
potential energy.
In stages (b), (d), ( f ), and (h), half the
energy is kinetic energy and half is
potential energy.
Note: If the swinging involved a
frictional force then Emec would not be
conserved, and eventually the
pendulum would stop.
March 30
8.5: Conservation of Mechanical Energy
Principle of Conservation of Energy
In an ISOLATED system where only conservative forces cause energy changes, the
kinetic energy and potential energy can change, but their sum, the mechanical
energy Emec of the system, cannot change.
Sample problem: water slide
Possible ways we can try
(1) Newton’s Laws: Motion, Force,
Acceleration, …
-  Will it work?
-  No: Force, track, …
(2) Energy conservation: U, K
-  Will it work?
-  Yes
•  Two forces: (a) mg does
work, (b) normal force
doesn’t do work
Sample problem: water slide
8.6: Reading a Potential Energy Curve: Potential Energy & Force
Negative slope!
A plot of U(x): the potential energy function of
a system containing a particle confined to move
along an x axis. (Since there is no friction, the
mechanical energy is conserved.)
A plot of the force F(x) acting on the
particle, derived from the potential energy
plot by taking its slope at various points.
8.6: Reading a Potential Energy Curve: Potential & kinetic energy
Escape: With K>= 0 outside the
potential well (x  infinity).
8.6: Potential Energy Curve: Equilibrium Points (assuming K=0 on the line)
Case-3: An particle at any point to the
right of x5
•  The system’s mechanical energy is
equal to its potential energy, and so it
must be stationary.
A particle at such a position is said to be in
neutral equilibrium.
Case-1: An particle at x2 or x4
•  It will be stuck there, cannot move
left or right on its own because to
do so would require a negative
kinetic energy.
•  If we push it slightly left or right, a
restoring force appears that moves
it back to x2 or x4.
A particle at such a position is said to
be in stable equilibrium.
Case-2: An particle at x3
When the particle at x3 has K =0.
•  If the particle is located exactly there,
the force on it is also zero, and the
particle remains stationary.
•  If it is displaced even slightly in either
direction, a nonzero force pushes it
farther in the same direction, and the
particle continues to move.
A particle at such a position is said to be in
unstable equilibrium.
Useful in describing processes in subatomic world
Sample problem: reading a potential energy graph
Sample problem:
reading a potential
energy graph
Sample problem:
reading a potential
energy graph
April 1
Sample problem:
reading a potential
energy graph
Principle of Conservation of Energy
In an isolated system where only conservative forces cause
energy changes, the kinetic energy and potential energy can
change, but their sum, the mechanical energy Emec. of the
system, remains as a constant.
How about a non-isolated system?
–
A system subject to external forces
8.7: Work done on a System by an External Force
Work is energy transferred to or from a system by means of an
external force acting on that system.
8.7: Work done on a System by an External Force
1. When FRICTION is NOT involved:
Attention:
The conservative
gravitational force: internal
force: ΔEmec=ΔK+ΔU=0
8.7: Work done on a System by an External Force
2. When FRICTION is involved:
Net external force
except frictions
Next slide
To include
conservative
force
Fdx ! fk dx = ma " dx
d
d
d
# F dx ! # f
0
k
0
dx =
# ma " dx
0
d
dv
Fd = fk d + # m " dx =
dt
0
d
#
0
vd
dx
fk dx + # m " dv
dt
v0
vd
Fd = fk d + # mv " dv
v0
1
1
1
Fd = fk d + (mvd2 ! mv02 ) = fk d + ( mvd2 ! mv02 )
2
2
2
Fd = fk d + $K
(8-34)
!Fdx ! fk dx = !ma " dx
!Fd = !#K + fk d
Fd = #K ! fk d
W = #Emec ! #Eth
(8-34)
Work by the external force
!Emec = W + !Eth
Mechanical energy
“Work/Energy” source of the changes in energies
Sample problem: change thermal energy
F
m
m
d
How could this
happen?
Sample problem: change in thermal energy
How all types of energies vary in the process?
• 
• 
• 
• 
• 
• 
Force F // d
Work done by F to the crate-floor system: +
Change in the kinetic energy of the crate: Change in the potential energy of the crate: 0
What is missing?
Thermal energy changes due to friction!
Sample problem: change in thermal energy
(8-34)
8.8: Conservation of Energy: with the presence of non-conservative
external forces & internal structure
Law of Conservation of Energy
The total energy E of a system can change only by amounts of
energy that are transferred to or from the system.
Emec: any change in the mechanical energy of the system
Eth: any change in the thermal energy of the system
Eint: any change in any other type of internal energy of the
system
No energy or material
exchanges with environment!
The total energy E of an isolated system cannot change.
Sample problem: energy, friction, spring, and tamales
Sample problem: energy, friction, spring, and tamales
System: The package–spring–floor–wall system
includes all these forces and energy transfers in
one isolated system. From conservation of
energy,
Forces:
(1)  The normal force on the package from the floor 
does no work on the package.
(2)  The gravitational force on the package  does no
work.
(3)  As the spring is compressed, a spring force 
does work on the package.
(4)  The spring force also pushes against a rigid wall
 does no work.
(5)  There is friction between the package and the
floor  increases their thermal energies.
April 6
8.8: Conservation of Energy: Attention !!
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.
Change in potential energy
8.8: Conservation of Energy: Attention!!
Should not treat the skater as a simple point.
Instead,
the internal “structure” needs to be taken into account.
What actually did the work are internal force(s)!
8.8: Conservation of Energy: Power
In general, power P is the rate at which energy is transferred by a force
from one type to another. If an amount of energy E is transferred in an
amount of time t, the average power due to the force is
Pavg
!E
=
!t
and the instantaneous power due to the force is
dE
P=
dt
The most important conclusions and relations
Conservative Forces
 
Potential Energy
Net work on a particle over a
closed path is 0
 
Energy associated with the
configuration of a system and a
conservative force
Eq. (8-6)
Gravitational Potential
Energy
 
Energy associated with Earth +
a nearby particle
Eq. (8-9)
Elastic Potential Energy
 
Energy associated with
compression or extension of a
spring
Eq. (8-11)
The most important conclusions and relations
Potential Energy Curves
Mechanical Energy
Eq. (8-22)
Eq. (8-12)
 
For only conservative forces
within an isolated system,
mechanical energy is
conserved
 
 
Work Done on a System by
an External Force
 
Without/with friction:
Eq. (8-26)
Eq. (8-33)
At turning points a particle
reverses direction
At equilibrium, slope of U(x) is 0
Conservation of Energy
 
The total energy can change
only by amounts transferred in
or out of the system
Eq. (8-35)
The most important conclusions and relations
Power
 
 
The rate at which a force
transfers energy
Average power:
Eq. (8-40)
 
Instantaneous power:
Eq. (8-41)
8.7.2. A rubber ball is dropped from rest from a height h. The ball bounces off the
floor and reaches a height of 2h/3. How can we use the principle of the
conservation of mechanical energy to interpret this observation?
a) During the collision with the floor, the floor did not push hard enough on the ball
for it to reach its original height.
b) Some of the ball s potential energy was lost in accelerating it toward the floor.
c) The force of the earth s gravity on the ball prevented it from returning to its
original height.
d) Work was done on the ball by the gravitational force that reduced the ball s
kinetic energy.
e) Work was done on the ball by non-conservative forces that resulted in the ball
having less total mechanical energy after the bounce.
8.7.2. A rubber ball is dropped from rest from a height h. The ball bounces off the
floor and reaches a height of 2h/3. How can we use the principle of the
conservation of mechanical energy to interpret this observation?
a) During the collision with the floor, the floor did not push hard enough on the ball
for it to reach its original height.
b) Some of the ball s potential energy was lost in accelerating it toward the floor.
c) The force of the earth s gravity on the ball prevented it from returning to its
original height.
d) Work was done on the ball by the gravitational force that reduced the ball s
kinetic energy.
e) Work was done on the ball by non-conservative forces that resulted in the ball
having less total mechanical energy after the bounce.
8.5.5. You are investigating the safety of a playground slide. You are
interested in finding out what the maximum speed will be of
children sliding on it when the conditions make it very slippery
(assume frictionless). The height of the slide is 2.5 m. What is
that maximum speed of a child if she starts from rest at the top?
a) 1.9 m/s
b) 2.5 m/s
c) 4.9 m/s
d) 7.0 m/s
e) 9.8 m/s
8.5.5. You are investigating the safety of a playground slide. You are
interested in finding out what the maximum speed will be of
children sliding on it when the conditions make it very slippery
(assume frictionless). The height of the slide is 2.5 m. What is
that maximum speed of a child if she starts from rest at the top?
a) 1.9 m/s
b) 2.5 m/s
c) 4.9 m/s
d) 7.0 m/s
e) 9.8 m/s
-  Maximum speed … frictionless
-  Initial velocity = 0
1 2
mgh = mv
2
v = 2gh = 2 ! 9.8 ! 2.5 = 7.0m / s
8.5.6. A quarter is dropped from rest from the fifth floor of a very tall
building. The speed of the quarter is v just before striking the
ground. From what floor would the quarter have to be dropped
from rest for the speed just before striking the ground to be
approximately 2v? Ignore all air resistance effects to determine
your answer.
a) 14
b) 25
c) 20
d) 7
e) 10
8.5.6. A quarter is dropped from rest from the fifth floor of a very tall
building. The speed of the quarter is v just before striking the
ground. From what floor would the quarter have to be dropped
from rest for the speed just before striking the ground to be
approximately 2v? Ignore all air resistance effects to determine
your answer.
-  Initial velocity = 0
-  One floor … H
a) 14
b) 25
c) 20
d) 7
e) 10
1 2
mv
(1)
2
1
mg(nH ) = m(2v)2
(2)
2
(2) :
(1)
n
=4
5
n = 20
mg(5H ) =
8.5.7. Two identical balls are thrown from the same height from the roof of a
building. One ball is thrown upward with an initial speed v. The second ball is
thrown downward with the same initial speed v. When the balls reach the
ground, how do the kinetic energies of the two balls compare? Ignore any air
resistance effects.
a) The kinetic energies of the two balls will be the same.
b) The first ball will have twice the kinetic energy as the second ball.
c) The first ball will have one half the kinetic energy as the second ball.
d) The first ball will have four times the kinetic energy as the second ball.
e) The first ball will have three times the kinetic energy as the second ball.
8.5.7. Two identical balls are thrown from the same height from the roof of a
building. One ball is thrown upward with an initial speed v. The second ball is
thrown downward with the same initial speed v. When the balls reach the
ground, how do the kinetic energies of the two balls compare? Ignore any air
resistance effects.
a) The kinetic energies of the two balls will be the same.
b) The first ball will have twice the kinetic energy as the second ball.
c) The first ball will have one half the kinetic energy as the second ball.
d) The first ball will have four times the kinetic energy as the second ball.
e) The first ball will have three times the kinetic energy as the second ball.
8.7.1. A car is being driven along a country road on a dark and rainy night at
a speed of 20 m/s. The section of road is horizontal and straight. The
driver sees that a tree has fallen and covered the road ahead. Panicking,
the driver locks the brakes at a distance of 20 m from the tree. If the
coefficient of friction between the wheels and road is 0.8, determine the
outcome.
a) The car stops 5.5 m before the tree.
b) The car stops just before reaching the tree.
c) As the car crashes into the tree, its speed is 18 m/s.
d) As the car crashes into the tree, its speed is 9.3 m/s.
e) This problem cannot be solved without knowing the mass of the car.
initial energy
1 2
mvi
2
8.7.1. A car is being driven along a country road
a darkby
and friction
rainy night
workondone
mgµatk D
a speed of 20 m/s. The section of road is horizontal and straight. The
1 the
1 2ahead. Panicking,
driver sees that a tree has fallen and covered
mvi2 ! road
mv f = mgµ k D
2 20 m from
2
the driver locks the brakes at a distance of
the tree. If the
2
2
! v f road
= 2gis
µk D
coefficient of friction between the wheelsvi and
0.8, determine the
outcome.
v 2f = vi2 ! 2gµ k D = 400 ! 313.6 = 86.4
a) The car stops 5.5 m before the tree.
v f = 9.29m / s
b) The car stops just before reaching the tree.
c) As the car crashes into the tree, its speed is 18 m/s.
d) As the car crashes into the tree, its speed is 9.3 m/s.
e) This problem cannot be solved without knowing the mass of the car.
8.7.3. The Jensens decided to spend their family vacation white water rafting.
During one segment of their trip down a horizontal section of the river,
the raft (total mass = 544 kg) has an initial speed of 6.75 m/s. The raft
then drops a vertical distance of 14.0 m, ending with a final speed of
15.2 m/s. How much work was done on the raft by non-conservative
forces?
a) -12 100 J
b) -18 200 J
c) -24 200 J
d) -36 300 J
e) -48 400 J
8.7.3. The Jensens decided to spend their family vacation white water rafting.
During one segment of their trip down a horizontal section of the river,
the raft (total mass = 544 kg) has an initial speed of 6.75 m/s. The raft
then drops a vertical distance of 14.0 m, ending with a final speed of
15.2 m/s. How much work was done on the raft by non-conservative
forces?
a) -12 100 J
b) -18 200 J
c) -24 200 J
d) -36 300 J
e) -48 400 J
!K + !U + W = 0
1
1
"W = mgH " ( mv 2f " mvi2 )
2
2
1
1
"W = (544)(9.8)(14.0) "[ (544)(15.2)2 " (544)(6.75)2 ]
2
2
"W = 74636.8 "[62842.88 "12393.0]
W = "24186.92J
8.8.1. A dam blocks the passage of a river and generates electricity.
Approximately, 57 000 kg of water fall each second through a
height of 19 m. If one half of the gravitational potential energy of
the water were converted to electrical energy, how much power
would be generated?
a) 2.7 × 106 W
b) 5.3 × 106 W
c) 1.1 × 107 W
d) 1.3 × 108 W
e) 2.7 × 108 W
8.8.1. A dam blocks the passage of a river and generates electricity.
Approximately, 57 000 kg of water fall each second through a
height of 19 m. If one half of the gravitational potential energy of
the water were converted to electrical energy, how much power
would be generated?
a) 2.7 × 106 W
b) 5.3 × 106 W
c) 1.1 × 107 W
d) 1.3 × 108 W
e) 2.7 × 108 W
!E 0.5" mgH
P=
=
!t
1s
P = (0.5)(57000)(9.8)(19)
P = 5,306, 700Watts
8.8.2. If the amount of energy needed to operate a 100 W light bulb for
one minute were used to launch a 2-kg projectile, what maximum
height could the projectile reach, ignoring any resistive effects?
a) 20 m
b) 50 m
c) 100 m
d) 200 m
e) 300 m
8.8.2. If the amount of energy needed to operate a 100 W light bulb for
one minute were used to launch a 2-kg projectile, what maximum
height could the projectile reach, ignoring any resistive effects?
a) 20 m
b) 50 m
c) 100 m
d) 200 m
e) 300 m
E = conserved
(P)(1min) = mgH
(100J / s)(60s) = (2)(9.8)H
H = (100J / s)(60s) /[(2)(9.8)]
H = 306.12m
8.8.3. A 65-kg hiker eats a 250 C-snack. Assuming the body converts
this snack with an efficiency of 25%, what change of altitude
could this hiker achieve by hiking up the side of a mountain before
completely using the energy in the snack? One food calorie (C) is
equal to 4186 joules.
a) 270 m
b) 410 m
c) 650 m
d) 880 m
e) 1600 m
8.8.3. A 65-kg hiker eats a 250 C-snack. Assuming the body converts
this snack with an efficiency of 25%, what change of altitude
could this hiker achieve by hiking up the side of a mountain before
completely using the energy in the snack? One food calorie (C) is
equal to 4186 joules.
a) 270 m
b) 410 m
c) 650 m
d) 880 m
e) 1600 m
E = conserved
E food = mgH
(25%)(250)(4186J) = (65)(9.8)H
H = (25%)(250)(4186J) /[(65)(9.8)]
H = 410.71m
Five sections
8-1 Potential
Energy + Major learning objectives
S-1: Potential Energy
1. Distinguish a conservative force from a non-conservative force.
2. For a particle moving between two points, identify that the
work done by a conservative force: It does not depend on which
path the particle takes.
3. Calculate the gravitational potential energy of a particle (or,
more properly, a particle-Earth system).
4. Calculate the elastic potential energy of a block-spring system.
8-2 Conservation of Mechanical Energy
S-2: Conservation of Mechanical Energy:
5. After first clearly defining which objects form a system, identify that
the mechanical energy of the system is the sum of the kinetic energies
and potential energies of those objects.
6. For an isolated system in which only conservative forces act, apply
the conservation of mechanical energy to relate the initial potential
and kinetic energies to the potential and kinetic energies at a later
instant.
8-2 Conservation of Mechanical Energy
S-3: Reading a Potential Energy Curve
7. Given a particle's potential energy as a function of position x, determine the
force on the particle.
8. Given a graph of potential energy versus x, determine the force on a particle.
9. On a graph of potential energy versus x, superimpose a line for a particle's
mechanical energy and determine kinetic energy for any given value of x.
10. If a particle moves along an x axis, use a potential-energy graph for that
axis and the conservation of mechanical energy to relate the energy values
at one position to those at another position.
11. On a potential-energy graph, identify any turning points and any regions
where the particle is not allowed because of energy requirements.
12. Explain neutral equilibrium, stable equilibrium, and unstable equilibrium.
8-2 Conservation of Mechanical Energy
S-4: Work Done on a System by an External Force
13. When work is done on a system by an external force with no friction
involved, determine the changes in kinetic energy and potential energy.
14. When work is done on a system by an external force with friction involved,
relate that work to the changes in kinetic energy, potential energy, and
thermal energy.
8-2 Conservation of Mechanical Energy
S-5: Conservation of Energy
15. For an isolated system (no net external force), apply the conservation of
energy to relate the initial total energy (energies of all kinds) to the total
energy at a later instant.
16. For a non-isolated system, relate the work done on the system by a net
external force to the changes in the various types of energies within the
system.
17. Apply the relationship between average power, the associated energy
transfer, and the time interval in which that transfer is made.
18. Given an energy transfer as a function of time (either as an equation or
graph), determine the instantaneous power (the transfer at any given
instant).