AP Physics 1 Unit 6 FRQ Practice: Energy and Momentum of Rotating Systems Free Response

Free-Response Question · Unit 6

Scenario: A figure skater is spinning on frictionless ice with her arms extended. Her total rotational inertia (with arms out) is I_1 = 5.0 kg·m². She is spinning at ω_1 = 2.0 rad/s. She then pulls her arms in close to her body, reducing her rotational inertia to I_2 = 1.5 kg·m².

— AP Physics 1 style problem · Topics 6.1, 6.4

Calculate the skater's angular velocity ω_2 after she pulls her arms in. Justify why angular momentum is conserved in this situation.

✓ Model answer (earns the points)

Why angular momentum is conserved: The ice is frictionless, so there is no external torque acting on the skater about her vertical rotation axis. (Her arms exert internal forces only.) Therefore, angular momentum is conserved: L_1 = L_2.

Calculation:

I_1·ω_1 = I_2·ω_2

(5.0)(2.0) = (1.5)·ω_2

10 = 1.5·ω_2

ω_2 = 10/1.5 ≈ 6.7 rad/s

The skater's angular velocity increases from 2.0 to 6.7 rad/s — about 3.3× faster.

Why it scores: (1) Correctly identifies that conservation of L applies because no external torque acts (frictionless ice). (2) Writes L_1 = L_2, expanded as I_1ω_1 = I_2ω_2. (3) Plugs in numbers correctly. (4) Reports a final answer with units (rad/s).

Calculate the skater's rotational kinetic energy BEFORE and AFTER pulling in her arms. By how much does her rotational kinetic energy change?

✓ Model answer (earns the points)

Initial rotational KE:

K_1 = ½·I_1·ω_1² = ½(5.0)(2.0)² = ½(5.0)(4) = 10 J

Final rotational KE:

K_2 = ½·I_2·ω_2² = ½(1.5)(6.67)² = ½(1.5)(44.4) ≈ 33 J

Change in rotational KE:

ΔK = K_2 − K_1 = 33 − 10 = +23 J

The rotational kinetic energy INCREASED by about 23 J (more than tripled).

Why it scores: (1) Correctly computes K_1 using K = ½Iω². (2) Correctly computes K_2 using K = ½Iω² with the ω from Part A. (3) Reports a positive ΔK with correct sign. (4) Includes units (joules).

A student claims: "Pulling in her arms increases her rotational kinetic energy. But she did no external work, so this must violate conservation of energy." Is the student's reasoning correct? Explain in detail where the extra rotational KE comes from.

✓ Model answer (earns the points)

The student is incorrect. The rotational KE does increase, but this does NOT violate conservation of energy.

Where the energy comes from: Although no EXTERNAL work is done on the skater, INTERNAL work IS done by her muscles. As she pulls her arms in toward her body, her arms are moving in circles. The centripetal acceleration of her arms requires an inward force. Her muscles supply this force AND pull her arms inward through some distance. Force times distance = work, so her muscles do POSITIVE internal work on her arms.

That internal muscle work transfers chemical energy (stored in her body from food) into rotational kinetic energy. Total energy is still conserved — it's just that energy was converted FROM chemical (in muscles) TO mechanical (rotational KE), not added from outside.

Key insight (important): This is a crucial difference between linear momentum and energy. Internal forces always come in equal-and-opposite pairs that CANCEL for momentum, but their WORK does NOT necessarily cancel. Internal forces CAN change a system's total kinetic energy (think of a person walking: the floor does almost no work, but the walker accelerates because their muscles do internal work).

So angular momentum is conserved (no external torque), but rotational KE is not — and that's perfectly consistent with energy conservation when you account for the energy her muscles supply.

Why it scores: (1) Clearly states the student is incorrect. (2) Identifies the source of the extra energy (internal muscle work). (3) Explains the mechanism (muscles pull arms in against centripetal effects, doing positive work). (4) Notes that total energy is conserved (chemical → mechanical). (5) Bonus: distinguishes between conservation of L (no external torque) and conservation of K (no work, internal or external).

How to score points on AP Physics 1 angular momentum FRQs

Always state WHY angular momentum is conserved. Specifically: "no external torque acts." Don't just write "L is conserved" — justify it.
Write out L = Iω BEFORE plugging in numbers. Showing the symbolic equation before substituting often earns its own point.
For "increase or decrease" questions, do the math first. Calculate both quantities before concluding which is bigger.
When KE changes but L is conserved, the energy came from internal work. Be specific: which muscles? Which forces? What kind of energy was converted?
Don't confuse "no work" with "no energy change." Internal forces can do net work, even when external forces don't.
Units, units, units. Angular momentum: kg·m²/s. Rotational KE: J. Angular velocity: rad/s.
For multi-part problems, use the result of Part A in Part B. Don't recalculate from scratch.