A pendulum bob has potential energy U0 when held taut in a horizontal position. The bob falls until it is 30$\degree$away from the horizontal position, when it has potential energy UA. It continues to fall until the string is vertical, when it has potential energy UB. Compare its potential energies at O, A, and B.
A)U0 = UA = UB.
B)UA- UB = 2U0.
C)UA - UB = U0 - UA.
D)U0 = UB = 2UA.
E)U0 - UA = 2(UA - U

Question

A pendulum bob has potential energy U0 when held taut in a horizontal position. The bob falls until it is 30$\degree$away from the horizontal position, when it has potential energy UA. It continues to fall until the string is vertical, when it has potential energy UB. Compare its potential energies at O, A, and B. 
A)U0 = UA = UB. 
B)UA- UB = 2U0. 
C)UA - UB = U0 - UA. 
D)U0 = UB = 2UA. 
E)U0 - UA = 2(UA - U

Quizplus · Accepted Answer

The potential energy of a pendulum bob depends on its height above its equilibrium position, which is determined by the angle it makes with the vertical. At position O (the horizontal position), the bob is at its maximum height above its equilibrium position, so it has maximum potential energy U0. At position A (30 degrees from the horizontal), the bob is not as high above its equilibrium position, so it has less potential energy UA. At position B (the bottom of the swing), the bob is at its minimum height and has minimum potential energy UB. The potential energy is proportional to the height above the equilibrium position, so we have:

UA - UB = U0 / 2
UA - U0 / 2 = UB - U0 / 2
UA - U0 / 2 = (UB - U0) / 2

Multiplying all terms by 2 gives:

2(UA - U0/2) = UB - U0
UA - U0 = (UB - U0) / 2

Therefore, the potential energy at position B is half the difference between the potential energies at positions O and A, giving us choice C as the correct answer.

A Pendulum Bob Has Potential Energy U0 When Held Taut °\degree°

A Pendulum Bob Has Potential Energy U0 When Held Taut $\degree$