Potential Energy and Conservation of Energy

Question 1

Give 2 definitions for a conservative force.

Answer 1

1. A force who’s net work it does on a particle moving around any closed path, from an initial point and then back to that point, is zero.
2. A force who’s net work it does on a particle moving between two points does not depend on the path taken by the particle.

Question 2

Define potential energy.

Answer 2

Energy that is associated with the configuration (an arrangement) of a system in which a conservative force acts.

ΔU = -W = - ∫F·dr.

Question 3

Define the gravitional potential energy.

Answer 3

The potential energy associated with a system consisting of Earth and a nearby particle.

ΔU = mg(y_f - y_i)

Question 4

Define elastic potential energy.

Answer 4

The energy associated with the state of compression or extension of an elastic object.

U = ½kx², where the relaxed length is x = 0.

Question 5

Suppose you need to calculate the work done by a conservative force along a given path between two points, and the calculation is difficult or even impossible without additional information. How could you go around this?

Answer 5

You can find the work by substituting some other path between those points for which the calculation is easier and possible.

Question 6

Define E_Mech.

Answer 6

E_Mech = KE + U.

Question 7

What does the principle of conservation of energy state?

Answer 7

Isolated system (No external forces casues energy changes within the system)

If only conservatice forces do work with an isolated system, then the mechanical energy of the system cannot change.

E_mech = ΔKE + ΔU = 0.

Note: There is no consideration of the intermediate motion and no finding the work done by the forces involved.

Question 8

Derive the conservation of energy equation.

Answer 8

W = ΔKE ⇒ ΔKE + ΔU = 0.

Question 9

How are F and U related?

Answer 9

F = -∇U.

Question 10

Where is KE greatest on this graph? Where is KE = 1J? Locate the turning points.

Answer 10

Question 11

For every color of E_mech find the turning points and equilibrium states.

Answer 11

Purple:

Turning points are just after x₁ and approx. x₅. Neutral equilibrium point after x₅.

Pink:

Turning points are between x₁ and x₂; second is between x₄ and x₅. Unstable equilibrium point at x₃.

Green:

Turning points before and after x₂. Stable equilibrium at x₄.

Question 12

Answer 12

F = -∇U

A. CD, AB, BC

B. +direction of x.

Question 13

What is defined to be the work done by an external force when friction is not involved? What about when it is?

Answer 13

W_ext = ΔKE +ΔU = ΔE_mech

W_ext = ΔE_mech + ΔE_th, where ΔE_th is the change of thermal energy due to friction.

ΔE_th = f_kd.

Question 14

Derive W_ext = ΔE_mech + ΔE_th from a block being across a floor by a force F while a kinetic frictional force f_k opposes the motion. The block has velocity v_o at the start of a displacement d and velocity v at the end of the displacement.

Answer 14

F -f_k =ma.

Since F is constant then a is constant.

v² - v_o² / 2d = a

Plug and chug.

For a more general case consider the block up a ramp ⇒ potential energy should also be taken into effect.

Question 15

Answer 15

f_k(d) ⇒ All the same.