Energy of Harmonic Oscillators

Part A = A

Part B = A

Part C = moving toward equilibrium.

Part D = C

Part E = C

Part F = D

Part G = 3/8kA^{2}

Solutions Below:

Part A

A

B

C

D

The maximum PE is when the spring is fully compressed. D might look like the right answer, but actually A is when the spring is at amplitude. Even though it’s stretched, that still counts as compression:

A

Part B

A

B

C

D

When PE is maximized, KE is minimized, so the correct answer is the same as from Part A:

A

Part C

A. at the equilibrium position.

B. at the amplitude displacement.

C. moving to the right.

D. moving to the left.

E. moving away from equilibrium.

F. moving toward equilibrium.

KE is maximized when PE is minimized. PE is greatest at amplitude and KE is greatest at equilibrium. So choice F is correct:

F. moving toward equilibrium.

Part D

A

B

C

D

KE is greatest at equilibrium:

C

Part E

A

B

C

D

Minimum PE is when KE is greatest, i.e. at equilibrium:

C

Part F

When U = KE, U = 1/2U_{max} (this is just a fact, which we won’t bother solving for here). So:

U = 1/2(U)

1/2kx^{2} = 1/2(1/2kA^{2})

x^{2} = 1/2A^{2}

x = sqrt(1/2A^{2})

x = Asqrt(2)/2

Note- if it isn’t obvious where the sqrt(2)/2 came from, use 2/4 for the fraction above instead of 1/2:

x = sqrt((2/4)A^{2})

x = Asqrt(2)/2

The diagram doesn’t give this as an answer, but it does give -Asqrt(2)/2, which is equivalent:

D

Part G

Express your answer in terms of k and A.

Since total energy = KE + PE and we only have enough information to find PE, we can work backwards by first finding the maximum PE (the PE when the spring is fully compressed) and then subtracting the PE at point B.

Maximum PE:

PE_{max} = 1/2kA^{2}

PE at Point B:

PE_{B} = 1/2k(A/2)^{2}

PE_{B} = 1/2k(A^{2}/4)

PE_{B} = 1/8k(A^{2})

Now find the difference between the PEs, and that difference must be KE since there is no friction:

KE_{B} = PE_{max} – PE_{B}

KE_{B} = 1/2kA^{2} – 1/8k(A^{2})

KE_{B} = 3/8kA^{2}

AMAZING!!! BEST THING YETTT

Thank you so much!

This was extremely helpful!

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