1. Potential Energy and Conservation of Energy
Chapter 8
Potential Energy and Conservation of Energy
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8-1 Potential Energy
Potential energy (symbol = U) is energy (Joules!)
Associated with configuration of a system of objects that exert forces on one another
A system of objects may be:
Earth and a bungee jumper
Gravitational potential energy accounts for kinetic energy increase during the fall (KE increases!)
Elastic potential energy accounts for deceleration by the bungee cord (KE decreases)
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8-1 Potential Energy
Potential energy U is energy that can be associated with the configuration of a system of objects that exert forces on one another
Configuration means that WHERE objects are will matter.
Physics determines how potential energy is calculated
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8-1 Potential Energy
Potential energy U is energy that can be associated with the configuration of a system of objects that exert forces on one another
But note: Energy (in the universe) is conserved!
Energy can be transformed from potential to kinetic…
Energy can be transformed from kinetic to potential….
Energy can be transformed into thermal
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Gravitational Potential Energy
For an object being raised or lowered:
The change in gravitational potential energy is the negative of the work done by the force of gravity
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6. Gravitational potential energy
Change in gravitational potential energy is related to work done by gravity.
Work done by Gravity:
Force ● Distance
Force = mg
Distance = Dy
Angle between: 0
Work done = POSITIVE Work = +mg Dy

7. Gravitational potential energy
Gravitational potential energy is related to configuration of objects (mass “m” and Earth)
Define potential energy of a position at height “h” relative to “0” as
mgh
NOTE – where “0” is will be YOUR choice…

8. Gravitational potential energy
Change in gravitational potential energy is related to work done by gravity.
Work done by Gravity: +mgDy
Initial Potential Energy:
mgy1
Final Potential Energy:
mgy2

9. Moving DOWN
Change in gravitational potential energy is related to work done by gravity.
Work done by Gravity: +mgDy (positive!)
Initial Potential Energy:
mgy1 (start high!)
Final PE:
mgy2 (lower!)
Difference: PEfinal – PE initial
(mgy2 – mgy1) < 0
Negative!

10. Gravitational potential energy
Change in gravitational potential energy is related to work done by gravity..
OK… what about moving UPWARDS??

11. Gravitational potential energy
Change in gravitational potential energy is related to work done by gravity..
Work done by Gravity:
- mgDy

12. Gravitational potential energy
Change in gravitational potential energy is related to work done by gravity..
Work done by Gravity (up!):
- mgDy (negative!)
Initial Potential Energy:
mgy1 (lower)
Final PE:
mgy2 (higher!)
Difference: PEfinal – PE initial
(mgy2 – mgy1) > 0
Positive!

13. Gravitational potential energy
Either moving DOWN or UP, change in gravitational potential energy is equal in magnitude and opposite in sign to work done by gravity.
Work done by Gravity up: mgDy
Difference: PEfinal – PE initial
DU = (mgy2 – mgy1) > 0
DU = positive!

14. Gravitational potential energy
Either moving DOWN or UP, change in gravitational potential energy is equal in magnitude and opposite in sign to work done by gravity.
Work done by Gravity down: mgDy
Difference: PEfinal – PE initial
DU = (mgy2 – mgy1) < 0
DU = negative!

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8-1 Potential Energy
For an object being raised or lowered:
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16. 8-1 Potential Energy Key points: The system consists of two objects
A force acts between a particle & rest of system
When configuration changes from 1 to 2, force does work W1-2, changing KE to another form
When configuration is reversed back (2 to 1), force reverses the energy transfer, doing work W2-1
Thus KE of tomato becomes potential energy, & then kinetic energy again 16

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8-1 Potential Energy
Conservative forces are forces for which W1-2 = - W is always true (no matter how the configuration changes and reversals happen!)
Examples: gravitational force, spring force
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8-1 Potential Energy
Conservative forces are forces for which W1 = -W2 is always true
Nonconservative forces are those for which it is false
Examples: kinetic friction force, drag force
Kinetic energy of a moving particle is transferred to heat by friction
Thermal energy cannot be recovered back into kinetic energy of object via friction
Therefore the friction is not conservative, thermal energy is not a potential energy
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8-1 Potential Energy
When only conservative forces act on a particle, we find many problems can be simplified:
And…
Figure 8-4
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8-1 Potential Energy
Figure 8-4
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8-1 Potential Energy
Figure 8-4
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22. The conservation of mechanical energy
The total mechanical energy of a system is the sum of its kinetic energy and potential energy.
A quantity that always has the same value is called a conserved quantity.

23. The conservation of mechanical energy
When only force of gravity does work on a system, total mechanical energy of that system is conserved.
Gravity is known as a “conservative” force

24. An example using energy conservation
0.145 kg baseball thrown straight 20m/s. How high?

25. An example using energy conservation
0.145 kg baseball thrown straight 20m/s. How high?
Use Energy Bar Graphs to track total, KE, and PE:

26. When forces other than gravity do work
Now add the launch force!
Move hand .50 m upward while accelerating the ball

27. When forces other than gravity do work
Now add the launch force!
Move hand .50 m upward while accelerating the ball

28. Work and energy along a curved path
Use same expression for gravitational PE whether path is curved or straight.

29. Energy in projectile motion
Two identical balls leave from the same height with the same speed but at different angles. Prove they have the same speed at any height h (neglecting air resistance)

30. Motion in a vertical circle with no friction
Speed at bottom of ramp of radius R = 3.00 m?

31. Motion in a vertical circle with no friction
Speed at bottom of ramp of radius R = 3.00 m?

32. Motion in a vertical circle with no friction
Normal force DOES NO WORK!

33. Motion in a vertical circle with friction
Revisit the same ramp, but this time with friction.
If his speed at bottom is 6.00 m/s, what was work by friction?

34. Moving a crate on an inclined plane with friction
Slide 12 kg crate up 2.5 m incline without friction at 5.0 m/s.
With friction, it goes only 1.6 m up the slope.
What is fk?
How fast is it moving at the bottom?

35. Moving a crate on an inclined plane with friction
Slide 12 kg crate up 2.5 m incline without friction at 5.0 m/s.
With friction, it goes only 1.6 m up the slope.
What is fk?
How fast is it moving at the bottom?

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8-1 Potential Energy
For an object being raised or lowered:
This also applies to an elastic block-spring system
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8-1 Potential Energy
Mathematically:
This result allows you to substitute a simpler path for a more complex one if only conservative forces are involved
Figure 8-5
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8-1 Potential Energy
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8-1 Potential Energy
Answer: No. The paths from a → b have different signs. One pair of paths allows the formation of a zero-work loop. The other does not.
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8-1 Potential Energy
For the general case, we calculate work as:
So we calculate potential energy as:
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8-1 Potential Energy
So we calculate potential energy as:
Using this to calculate gravitational PE, relative to a reference configuration with reference point yi = 0:
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8-1 Potential Energy
Use the same process to calculate spring PE:
With reference point xi = 0 for a relaxed spring:
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43. Elastic potential energy
A body is elastic if it returns to its original shape after being deformed.
Elastic potential energy is the energy stored in an elastic body, such as a spring.
Elastic potential energy stored in an ideal spring is Uel = 1/2 kx2.
Figure shows graph of elastic potential energy for ideal spring.

44. Motion with elastic potential energy
Glider of mass 200 g on frictionless air track, connected to spring with k = 5.00 N/m. Stretch it 10 cm, and release from rest.
=> What is velocity when x = 0.08 m?

45. Motion with elastic potential energy
Glider of mass 200 g on frictionless air track, connected to spring with k = 5.00 N/m. Stretch it 10 cm, and release from rest.
=> What is velocity when x = 0.08 m?

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8-1 Potential Energy
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8-1 Potential Energy
Answer: (3), (1), (2); a positive force does positive work, decreasing the PE; a negative force (e.g., 3) does negative work, increasing the PE
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48. Situations with both gravitational and elastic forces
When a situation involves both gravitational and elastic forces, the total potential energy is the sum of the gravitational potential energy and the elastic potential energy: U = Ugrav + Uel.

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8-2 Conservation of Mechanical Energy
The mechanical energy of a system is the sum of its potential energy U and kinetic energy K:
Work done by conservative forces increases K and decreases U by that amount, so:
Using subscripts to refer to different instants of time:
In other words:
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8-2 Conservation of Mechanical Energy
This is the principle of the conservation of mechanical energy:
This is very powerful tool:
One application:
Choose the lowest point in the system as U = 0
Then at the highest point U = max, and K = min
Eq. (8-18)
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8-2 Conservation of Mechanical Energy
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8-2 Conservation of Mechanical Energy
Answer: Since there are no nonconservative forces, all of the difference in potential energy must go to kinetic energy. Therefore all are equal in (a). Because of this fact, they are also all equal in (b).
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53. Conservative and nonconservative forces
A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative.
The work done between two points by any conservative force
a) can be expressed in terms of a potential energy function.
b) is reversible.
c) is independent of the path between the two points.
d) is zero if the starting and ending points are the same.

54. Conservative and nonconservative forces
A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative.
A force (such as friction) that is not conservative is called a non-conservative force, or a dissipative force.

55. Frictional work depends on the path
Move 40.0 kg futon 2.50 m across room; slide it along paths shown. How much work required if mk = .200

56. Conservative or nonconservative force?
Suppose force F = Cx in the y direction. What is work required in a round trip around square of length L?

57. Conservation of energy
Nonconservative forces do not store potential energy, but they do change the internal energy of a system.
The law of the conservation of energy means that energy is never created or destroyed; it only changes form.
This law can be expressed as K + U + Uint = 0.

58. A system having two potential energies and friction
Gravity, a spring, and friction all act on the elevator.
2000 kg elevator with broken cables moving at 4.00 m/s
Contacts spring at bottom, compressing it 2.00 m.
Safety clamp applies constant 17,000 N friction force as it falls.

59. A system having two potential energies and friction
What is the spring constant k for the spring so it stops in 2.00 meters?

60. Force and potential energy in one dimension
In one dimension, a conservative force can be obtained from its potential energy function using
Fx(x) = –dU(x)/dx

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8-3 Reading a Potential Energy Curve
For one dimension, force and potential energy are related (by work) as:
Therefore we can find the force F(x) from a plot of the potential energy U(x), by taking the derivative (slope)
If we write the mechanical energy out:
We see how K(x) varies with U(x):
Eq. (8-22)
Eq. (8-23)
Eq. (8-24)
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62. Force and potential energy in one dimension
In one dimension, a conservative force can be obtained from its potential energy function using
Fx(x) = –dU(x)/dx

63. Force and potential energy in two dimensions
In two dimension, the components of a conservative force can be obtained from its potential energy function using
Fx = –U/dx and Fy = –U/dy

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8-3 Reading a Potential Energy Curve
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65. Energy diagrams
An energy diagram is a graph that shows both the potential-energy function U(x) and the total mechanical energy E.

66. Force and a graph of its potential-energy function

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8-3 Reading a Potential Energy Curve
To find K(x) at any place, take the total mechanical energy (constant) and subtract U(x)
Places where K = 0 are turning points
There, the particle changes direction (K cannot be negative)
At equilibrium points, the slope of U(x) is 0
A particle in neutral equilibrium is stationary, with potential energy only, and net force = 0
If displaced to one side slightly, it would remain in its new position
Example: a marble on a flat tabletop
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8-3 Reading a Potential Energy Curve
A particle in unstable equilibrium is stationary, with potential energy only, and net force = 0
If displaced slightly to one direction, it will feel a force propelling it in that direction
Example: a marble balanced on a bowling ball
A particle in stable equilibrium is stationary, with potential energy only, and net force = 0
If displaced to one side slightly, it will feel a force returning it to its original position
Example: a marble placed at the bottom of a bowl
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8-3 Reading a Potential Energy Curve
Plot (a) shows the potential U(x)
Plot (b) shows the force F(x)
If we draw a horizontal line, (c) or (f) for example, we can see the range of possible positions
x < x1 is forbidden for the Emec in (c): the particle does not have the energy to reach those points
Figure 8-9
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8-3 Reading a Potential Energy Curve
Answer: (a) CD, AB, BC (b) to the right
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8-4 Work Done on a System by an External Force
We can extend work on an object to work on a system:
For a system of more than 1 particle, work can change both K and U, or other forms of energy of the system
For a frictionless system:
Eq. (8-25)
Eq. (8-26)
Figure 8-12
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8-4 Work Done on a System by an External Force
For a system with friction:
The thermal energy comes from the forming and breaking of the welds between the sliding surfaces
Eq. (8-31)
Eq. (8-33)
Figure 8-13
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8-4 Work Done on a System by an External Force
Answer: All trials result in equal thermal energy change. The value of fk is the same in all cases, since μk has only 1 value.
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8-5 Conservation of Energy
Energy transferred between systems can always be accounted for
The law of conservation of energy concerns
The total energy E of a system
Which includes mechanical, thermal, and other internal energy
Considering only energy transfer through work:
Eq. (8-35)
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8-5 Conservation of Energy
An isolated system is one for which there can be no external energy transfer
Energy transfers may happen internal to the system
We can write:
Or, for two instants of time:
Eq. (8-36)
Eq. (8-37)
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8-5 Conservation of Energy
External forces can act on a system without doing work:
The skater pushes herself away from the wall
She turns internal chemical energy in her muscles into kinetic energy
Her K change is caused by the force from the wall, but the wall does not provide her any energy
Figure 8-15
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8-5 Conservation of Energy
We can expand the definition of power
In general, power is the rate at which energy is transferred by a force from one type to another
If energy ΔE is transferred in time Δt, the average power is:
And the instantaneous power is:
Eq. (8-40)
Eq. (8-41)
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8 Summary
Conservative Forces
Net work on a particle over a closed path is 0
Potential Energy
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
Elastic Potential Energy
Energy associated with compression or extension of a spring
Eq. (8-9)
Eq. (8-11)
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8 Summary
Mechanical Energy
For only conservative forces within an isolated system, mechanical energy is conserved
Potential Energy Curves
At turning points a particle reverses direction
At equilibrium, slope of U(x) is 0
Eq. (8-12)
Eq. (8-22)
Work Done on a System by an External Force
Without/with friction:
Conservation of Energy
The total energy can change only by amounts transferred in or out of the system
Eq. (8-26)
Eq. (8-33)
Eq. (8-35)
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8 Summary
Power
The rate at which a force transfers energy
Average power:
Instantaneous power:
Eq. (8-40)
Eq. (8-41)
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