Chapter 7 OutlinePotential Energy and Energy Conservation

• Gravitational potential energy

• Conservation of mechanical energy

• Elastic potential energy

• Springs

• Conservative and non-conservative forces

• Conservation of energy

• Force and potential energy

• Energy diagrams

Potential Energy

• Kinetic energy depends on motion.

• Potential energy depends on position.

• Energy can be converted between these forms.

• Total energy will remain constant.

• Gravitational potential energy, , relative to some reference is equal to the work done against gravity to lift an object from the reference point to some height.

Gravitational Potential Energy

• Can we discuss an exact value for gravitational potential energy?

• We calculated the change in from the gravitational force times the distance.

• We cannot define the gravitational potential energy without first defining the zero point.

• Only the change in gravitational potential energy is relevant. We are free to set at any point we want.

• Only the change in height matters; the path is irrelevant.

Conservation of Mechanical Energy

• If we only have gravitational force, the sum of gravitational potential energy and kinetic energy must be constant.

• This is conservation of mechanical energy.

Potential Energy Example

Elastic Potential Energy

• Last chapter, we discussed the work needed to compress (or stretch) a spring.

• Since we were doing work on the spring, where was the energy going?

• It was stored as elastic potential energy.

• The term elastic implies that the energy stored in the spring can be converted completely into kinetic energy.

• The work done on the spring is equal in magnitude but opposite in sign to the change in the energy stored in the spring.

Work Done by Other Forces

• What about work done by other forces, such as friction?

• Total energy is always conserved!

• Where does the other energy go?

• Frictional forces are non-conservative.

• Energy is still conserved, but some of it is converted into heat.

Potential Energy Example

Conservative vs. Non-conservative Forces

• When we throw a ball in the air, its kinetic energy is “stored” as gravitational potential energy as it approaches its maximum height, and converted back to kinetic energy as it comes down.

• Because this back and forth conversion between kinetic and potential energy is possible, we call gravity a conservative force.

• Mechanical energy is conserved:

• Consider instead a box sliding to a stop because of friction.

• The kinetic energy is converted to heat by the frictional force.

• This cannot be reversed, so it is a non-conservative force.

Properties of Conservative Forces

• The work done by any conservative force has these four properties.

1. Can be expressed as difference of potential energy.

2. Is reversible.

3. Is path-independent.

4. Closed loop work is zero.

Conservative or Non-conservative Example?

Law of Conservation of Energy

• While the total mechanical energy can vary due to work done by non-conservative forces, energy is never created or destroyed.

• Consider a car skidding to a stop.

• The kinetic energy is converted to heat, increasing the temperature of the tires and pavement.

• This increases their internal energy.

• Since the work done by friction is negative and the change in internal energy is positive, .

• The law of conservation of energy is:

• This is always true.

Force and Potential Energy (1D)

• We have found expressions for the potential energy associated with gravity and springs from the forces.

• What if we know the potential energy function and want to find the corresponding force? (Consider 1D first.)

• The work done by a conservative force equals the negative of the change in potential energy.

, or

• Since , an infinitesimal bit of work, is , so,

• Solving for the force,

Force and Potential Energy (3D)

• The potential energy will in general depend on all three spatial dimensions.

• We can repeat the analysis for and , but we need to introduce a new mathematical notation.

• To find the total vector force, we look at each direction independently.

• When we move only in the direction, and remain constant.

• We take the derivative of with respect to while treating and as constants. This is called the partial derivative.

• Repeating this for and ,

Force and Potential Energy (3D)

• Combing these in vector form,

• We can write this more succinctly using the “del” operator.

• The force is the negative gradient of the potential.

Energy Diagram

• We can glean a lot of information by looking at graph of the potential energy.

Energy Diagram Example

Chapter 7 SummaryPotential Energy and Energy Conservation

• Gravitational potential energy:

• Conservation of mechanical energy

• Elastic potential energy:

• Conservative forces

• Potential energy, reversible, path-independent, zero closed loop

• Conservation of energy:

• Force and potential energy:

• Energy diagrams

• Stable minima and unstable maxima