1. Chapter 7
Potential Energy

2. Potential Energy
Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other
The forces are internal to the system

3. Types of Potential Energy
There are many forms of potential energy, including:
Gravitational
Electromagnetic
Chemical
Nuclear
One form of energy in a system can be converted into another

4. System Example This system consists of Earth and a book
Do work on the system by lifting the book through Dy
The work done is mgyb - mgya

5. Potential Energy
The energy storage mechanism is called potential energy
A potential energy can only be associated with specific types of forces
Potential energy is always associated with a system of two or more interacting objects

6. Gravitational Potential Energy
Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface
Assume the object is in equilibrium and moving at constant velocity
The work done on the object is done by and the upward displacement is

7. Gravitational Potential Energy, cont
The quantity mgy is identified as the gravitational potential energy, Ug
Ug = mgy
Units are joules (J)

8. Gravitational Potential Energy, final
The gravitational potential energy depends only on the vertical height of the object above Earth’s surface
In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero
The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

9. Conservation of Mechanical Energy
The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system
Emech = K + Ug
The statement of Conservation of Mechanical Energy for an isolated system is Kf + Ugf = Ki+ Ugi
An isolated system is one for which there are no energy transfers across the boundary

10. Conservation of Mechanical Energy, example
Look at the work done by the book as it falls from some height to a lower height
Won book = DKbook
Also, W = mgyb – mgya
So, DK = -DUg

11. Conservative Forces
A conservative force is a force between members of a system that causes no transformation of mechanical energy within the system
The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle
The work done by a conservative force on a particle moving through any closed path is zero

12. Nonconservative Forces
A nonconservative force does not satisfy the conditions of conservative forces
Nonconservative forces acting in a system cause a change in the mechanical energy of the system

13. Nonconservative Force, Example
Friction is an example of a nonconservative force
The work done depends on the path
The red path will take more work than the blue path

14. Elastic Potential Energy
Elastic Potential Energy is associated with a spring, Us = 1/2 k x2
The work done by an external applied force on a spring-block system is
W = 1/2 kxf2 – 1/2 kxi2
The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

15. Elastic Potential Energy, cont
This expression is the elastic potential energy: Us = 1/2 kx2
The elastic potential energy can be thought of as the energy stored in the deformed spring
The stored potential energy can be converted into kinetic energy

16. Elastic Potential Energy, final
The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0)
The energy is stored in the spring only when the spring is stretched or compressed
The elastic potential energy is a maximum when the spring has reached its maximum extension or compression
The elastic potential energy is always positive
x2 will always be positive

17. Conservation of Energy, Extended
Including all the types of energy discussed so far, Conservation of Energy can be expressed as
DK + DU + DEint = DE system = 0 or
K + U + E int = constant
K would include all objects
U would be all types of potential energy

18. Problem Solving Strategy – Conservation of Mechanical Energy
Conceptualize: Define the isolated system and the initial and final configuration of the system
The system may include two or more interacting particles
The system may also include springs or other structures in which elastic potential energy can be stored
Also include all components of the system that exert forces on each other

19. Problem-Solving Strategy, 2
Categorize: Determine if any energy transfers across the boundary
If it does, use the nonisolated system model, DE system = ST
If not, use the isolated system model, DEsystem = 0
Determine if any nonconservative forces are involved

20. Problem-Solving Strategy, 3
Analyze: Identify the configuration for zero potential energy
Include both gravitational and elastic potential energies
If more than one force is acting within the system, write an expression for the potential energy associated with each force

21. Problem-Solving Strategy, 4
If friction or air resistance is present, mechanical energy of the system is not conserved
Use energy with non-conservative forces instead
The difference between initial and final energies equals the energy transformed to or from internal energy by the nonconservative forces

22. Problem-Solving Strategy, 5
If the mechanical energy of the system is conserved, write the total energy as
Ei = Ki + Ui for the initial configuration
Ef = Kf + Uf for the final configuration
Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

23. Mechanical Energy and Nonconservative Forces
In general, if friction is acting in a system:
DEmech = DK + DU = -ƒkd
DU is the change in all forms of potential energy
If friction is zero, this equation becomes the same as Conservation of Mechanical Energy

24. Nonconservative Forces, Example 1 (Slide)
DEmech = DK + DU
DEmech =(Kf – Ki) +
(Uf – Ui)
DEmech = (Kf + Uf) –
(Ki + Ui)
DEmech = 1/2 mvf2 – mgh = -ƒkd

25. Nonconservative Forces, Example 2 (Spring-Mass)
Without friction, the energy continues to be transformed between kinetic and elastic potential energies and the total energy remains the same
If friction is present, the energy decreases
DEmech = -ƒkd

26. Conservative Forces and Potential Energy
Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system
The work done by such a force, F, is
DU is negative when F and x are in the same direction

27. Conservative Forces and Potential Energy
The conservative force is related to the potential energy function through
The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system
This can be extended to three dimensions

28. Conservative Forces and Potential Energy – Check
Look at the case of an object located some distance y above some reference point:
This is the expression for the vertical component of the gravitational force

29. Nonisolated System in Steady State
A system can be nonisolated with 0 = DT
This occurs if the rate at which energy is entering the system is equal to the rate at which it is leaving
There can be multiple competing transfers

30. Nonisolated System in Steady State, House Example

31. Potential Energy for Gravitational Forces
Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation:
The potential energy then is

32. Potential Energy for Gravitational Forces, Final
The result for the earth-object system can be extended to any two objects:
For three or more particles, this becomes

33. Electric Potential Energy
Coulomb’s Law gives the electrostatic force between two particles
This gives an electric potential energy function of

34. Energy Diagrams and Stable Equilibrium
The x = 0 position is one of stable equilibrium
Configurations of stable equilibrium correspond to those for which U(x) is a minimum
x=xmax and x=-xmax are called the turning points

35. Energy Diagrams and Unstable Equilibrium
Fx = 0 at x = 0, so the particle is in equilibrium
For any other value of x, the particle moves away from the equilibrium position
This is an example of unstable equilibrium
Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

36. Neutral Equilibrium
Neutral equilibrium occurs in a configuration when U is constant over some region
A small displacement from a position in this region will produce neither restoring nor disrupting forces

37. Potential Energy in Fuels
Fuel represents a storage mechanism for potential energy
Chemical reactions release energy that is used to operate the automobile