1. Potential Energy
~March 1, 2006

2. Coming Up Exam on Friday Next Week Material through today.
Look for current WA
Next Week
Important Topic: Conservation of Momentum and collisions between objects.

3. Potential Energy
Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other
This can be used only with conservative forces
Conservative forces are NOT Republicans
When conservative forces act within an isolated system, the kinetic energy gained (or lost) by the system as its members change their relative positions is balanced by an equal loss (or gain) in potential energy.
This is Conservation of Mechanical Energy

4. 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
Nuclear heat easy
Heat Nuclear probably impossible!
Conversion from one type to another type of energy is not always reversible.

5. Systems with Multiple Particles
We can extend our definition of a system to include multiple objects
The force can be internal to the system
The kinetic energy of the system is the algebraic sum of the kinetic energies of the individual objects
Sometimes, the kinetic energy of one of the objects may be negligible

6. System Example This system consists of Earth and a book
Do work on the system by lifting the book through Dy
The work done by you is mg(yb – ya)
At the top it is at rest.
The amount of work that you did is called the potential energy of the system with respect to the ground.
The PE’s initial value is mgya
The FINAL value is mgyb
The difference is the work done.

7. Let’s drop the book from yb and see what it is doing at ya.
Multiply by m and divide by 2:
The decrease in Potential Energy equals the increase in Kinetic Energy

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

9. 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 Fapp and the upward displacement is

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

11. Energy Problems Draw a diagram of the situation. ESTABLISH AN ORIGIN.
THE POTENTIAL ENERGY OF A PARTICAL OF MASS M IS ALWAYS MEASURED WITH RESPECT TO THIS ORIGIN.
The potential energy of a particle is defined as being ZERO when it is at the origin.
At some height above the origin, the value of the PE is mgh.

12. 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

13. 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 + Uf = Ki+ Ui
An isolated system is one for which there are no energy transfers across the boundary

14. Let’s look at the more general case.
y=h m
W=0 d d D D
W=mgD
W=mgD
We do work to
move mass to
y=h. W=mgh F
y=0
(origin) m m
Add'em Up - Same result ... mgh

15. ANY PATH
Can be broken up into a series of very small vertical moves and horizontal moves.
The horizontal moves require no work.
The force is at right angles to the motion. Dot product is zero.
The vertical moves are

16. in raising it a distance h is "mgh" no matter what the path!
The work done on a mass
in raising it a distance h
is "mgh" no matter what the path!

17. 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

18. Elastic Potential Energy
Elastic Potential Energy is associated with a spring
The force the spring exerts (on a block, for example) is Fs = - kx
The work done by an external applied force on a spring-block system is
W = ½ kxf2 – ½ kxi2
The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

19. Elastic Potential Energy, cont
This expression is the elastic potential energy: Us = ½ 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

20. 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

21. Problem Solving Strategy – Conservation of Mechanical Energy
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

22. Problem-Solving Strategy, 2
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

23. Problem-Solving Strategy, 3
If friction or air resistance is present, mechanical energy of the system is not conserved
Use energy with non-conservative forces instead

24. Problem-Solving Strategy, 4
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

25. Conservation of Energy, Example 1 (Drop a Ball)
Initial conditions:
Ei = Ki + Ui = mgh
The ball is dropped, so Ki = 0
The configuration for zero potential energy is the ground
Conservation rules applied at some point y above the ground gives
½ mvf2 + mgy = mgh

26. Conservation of Energy, Example 2 (Pendulum)
As the pendulum swings, there is a continuous change between potential and kinetic energies
At A, the energy is potential
At B, all of the potential energy at A is transformed into kinetic energy
Let zero potential energy be at B
At C, the kinetic energy has been transformed back into potential energy

27. Conservation of Energy, Example 3 (Spring Gun)
Choose point A as the initial point and C as the final point
EA = EC
KA + UgA + UsA = KA + UgA + UsA
½ kx2 = mgh

28. Conservative Forces
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
A closed path is one in which the beginning and ending points are the same

29. Conservative Forces, cont
Examples of conservative forces:
Gravity
Spring force
We can associate a potential energy for a system with any conservative force acting between members of the system
This can be done only for conservative forces
In general: WC = - DU

30. 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

31. 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

32. Nonconservative Forces, cont
The work done against friction is greater along the red path than along the blue path
Because the work done depends on the path, friction is a nonconservative force

33. Problem Solving Strategies – Nonconservative Forces
Define the isolated system and the initial and final configuration of the system
Identify the configuration for zero potential energy
These are the same as for Conservation of Energy
The difference between the final and initial energies is the change in mechanical energy due to friction

34. Nonconservative Forces, Example 1 (Slide)
DEmech = DK + DU
DEmech =(Kf – Ki) +
(Uf – Ui)
DEmech = (Kf + Uf) –
(Ki + Ui)
DEmech = ½ mvf2 – mgh = -ƒkd

35. 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

36. Nonconservative Forces, Example 3 (Connected Blocks)
The system consists of the two blocks, the spring, and Earth
Gravitational and potential energies are involved
The kinetic energy is zero if our initial and final configurations are at rest

37. Connected Blocks, cont
Block 2 undergoes a change in gravitational potential energy
The spring undergoes a change in elastic potential energy
The coefficient of kinetic energy can be measured

38. 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

39. Conservative Forces and Potential Energy
The conservative force is related to the potential energy function through
The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x

40. Conservative Forces and Potential Energy – Check
Look at the case of a deformed spring
This is Hooke’s Law

41. Energy Diagrams and Equilibrium
Motion in a system can be observed in terms of a graph of its position and energy
In a spring-mass system example, the block oscillates between the turning points, x = ±xmax
The block will always accelerate back toward x = 0

42. 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

43. 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

44. 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 either restoring or disrupting forces

45. Potential Energy in Molecules
There is potential energy associated with the force between two neutral atoms in a molecule which can be modeled by the Lennard-Jones function

46. Potential Energy Curve of a Molecule
Find the minimum of the function (take the derivative and set it equal to 0) to find the separation for stable equilibrium
The graph of the Lennard-Jones function shows the most likely separation between the atoms in the molecule (at minimum energy)

47. Force Acting in a Molecule
The force is repulsive (positive) at small separations
The force is zero at the point of stable equilibrium
The force is attractive (negative) when the separation increases
At great distances, the force approaches zero