1. Review Chap. 7 Potential Energy and Energy Conservation
PHYS 218 sec
Review
Chap. 7
Potential Energy and Energy Conservation

2. Gravitational potential energy
Energy associated with the position of bodies in a system
A measure of the potential or possibility for work to be done
The work done on the object to change its position is stored in the object in the form of an energy
Gravitational potential energy y2 y1
When the body moves up, the work done by the gravitational force is negative and the potential energy increases.

3. Potential energy
The potential energy is a relative quantity. You have to specify the reference point when you define the potential energy. For example, the gravitational potential energy is mgy and the point where y = 0 should be specified, which is the reference point when you define the potential energy.
What is physically meaningful is the change of the potential energy. The absolute value does not have physical meaning. Note that the work done by a force is equal to the negative of the potential energy change.

4. Conservation of mechanical energy
So, K + U is conserved
This defines the total mechanical energy of the system.
Conservation of mechanical energy

5. Height of a baseball from energy conservation
Ex 7.1
Height of a baseball from energy conservation
Energy conservation is very useful to obtain speed or position, in particular, when it its very difficult to use Newton’s laws of motion.

6. When forces other than gravity do work
Therefore, for example, if there is friction force, the total mechanical energy is not conserved. Instead, we have

7. Work and energy in throwing a baseball
Ex 7.2
Work and energy in throwing a baseball
From y = 0 to y1
State 3
State 2
From y = y1 to y2
State 1
Two solutions; moving up and moving down
Here we use different choice for the y-axis from the textbook. But the final answers are the same as it should be.

8. Gravitational potential energy for motion along a curved path
Only Dy contributes
The total work becomes
The total work done by the gravitational force depends only on the difference in height

9. Calculating speed along a vertical circle
Ex 7.4
Calculating speed along a vertical circle
Point 1
Speed at the bottom of the ramp
What we need is speed not velocity, so v2 is positive
Point 2
Normal force at the bottom of the curve

10. Vertical circle with friction: same as Ex 7
Vertical circle with friction: same as Ex 7.4 but there is friction force. What is the work done by the friction force?
Ex 7.5

11. Inclined plane with friction
Ex 7.6
Inclined plane with friction
Motion of a crate: Point 1 (speed v1) g Point 2 (speed v2 = 0) g Point 3 (speed v3)
Point 2
equal to Point 1
Magnitude of a constant friction force
Point 1, 3
Consider the motion from Point 1 to Point 2

12. Consider the motion from Point 1 to Point 3
Speed at Point 3
Consider the motion from Point 1 to Point 3
Be careful:
Speed is always positive while velocity can be negative
What we want is speed not velocity, so v3 is positive

13. Elastic potential energy
Energy stored in an ideal spring.
This is a potential energy that is not gravitational in origin. Although this is not one of the fundamental forces in nature, its potential energy can be defined.

14. Gravitational potential energy plus Elastic potential energy

15. Motion with elastic potential energy
Ex 7.7
Motion with elastic potential energy m
Point 1 m
Point 2
What is v2?

16. Motion with elastic potential energy and work done by other forces: similar to Ex 7.7
The object is moving to the right. So the negative value, -0.6 m/s, is not the solution although it is a mathematical solution.

17. Motion with gravitational, elastic, and friction forces
Ex 7.9
Motion with gravitational, elastic, and friction forces
m=2000 kg
Point 1
Choose Point 1 as y = 0
Point 2

18. Conservative and nonconservative forces
Allows two-way conversion between kinetic and potential energies
Conservative force
Properties of the work done by a conservative force
The potential energy function can be defined.
It is reversible.
It is independent of the path of the body; depends only on the starting & ending points
When the starting & ending points are the same the total work is zero. f I II i

19. Nonconservative force
Does not allow two-way conversion between kinetic and potential energies
The work done by a nonconservative force cannot be represented by a potential energy.
Under the influence of some nonconservative force, the body looses its energy. So this is also called a dissipative force.
Under the influence of some nonconservative force, the body gets its energy.
Potential energy cannot be defined for nonconservative forces!

20. Conservative or nonconservative?
Ex 7.11
Conservative or nonconservative?
Leg 3
Leg 4
Leg 2
Leg 1

21. Force and potential energy
integrate
Force
Potential energy
differentiate
For 3-dim case
For 1-dim case
partial derivative

22. Energy diagram: a graph of energy vs position
Energy diagrams
Energy diagram: a graph of energy vs position
It contains the shape of the potential energy and the total energy is a straight horizontal line as it is a constant once a body is given an energy.
This allows to know the motion of the body even if the functional form for the potential energy is not known.
Slope of the tangent line is positive, so the force is negative.
The body is moving always toward the minimal potential energy point.
Slope of the tangent line is negative, so the force is positive.

23. maximum of the potential: unstable equilibrium point
Turning point
Minimum of the potential: stable equilibrium point