The questions are on the moment of inertia test. Take it under real conditions before reading the mark scheme.
Quick answer: Moment of inertia is I = ∫r² dm, the sum of each mass element times the square of its distance from the rotation axis. This mark scheme gives the model answers and the marks for the 8 questions of the AP Physics C test on moment of inertia, from moment of inertia to disc formula, with the equivalent answers to accept.
Model answers with a marking guide. Give method marks for a correct set-up of the integral or the equations of motion even when the arithmetic fails, and apply error carried forward when later parts use earlier wrong values. Units are required for every final answer.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 5
Question 1 Moment of inertia (2 marks)
a. I = ∫r² dm, the sum of each mass element times the square of its perpendicular distance from the axis. b. Στ = Iα.Marking: 1 mark each.
Question 2 Rod by integration (3 marks)
a. dm = (M/L) dx. b. I = (M/L)∫₀^L x² dx = ML²/3. c. I = 0.50 × 0.36/3 = 0.060 kg·m².Marking: 1 mark for each part.
Question 3 Parallel axis (3 marks)
a. Icm = 0.50 × 0.36/12 = 0.015 kg·m². b. I = Icm + Md². c. I = 0.015 + 0.50 × 0.040 = 0.035 kg·m².Marking: 1 mark each.
Question 4 Pulley and mass (4 marks)
a. mg − T = ma; TR = Iα with I = ½MR² and α = a/R, so T = ½Ma. b. a = mg/(m + M/2) = 14.7/2.5 = 5.9 m/s². c. T = ½Ma = 1.0 × 5.89 = 5.9 N. d. α = a/R = 5.89/0.10 = 59 rad/s².Marking: 1 mark per part.
Question 5 Braking a flywheel (3 marks)
a. α = τ/I = 20/0.80 = 25 rad/s². b. t = ω/α = 50/25 = 2.0 s. c. K = ½Iω² = ½ × 0.80 × 2500 = 1.0 × 10³ J.Marking: 1 mark per part.
Question 6 Hoop and disc (2 marks)
The disc arrives first. The hoop has the larger moment of inertia (MR² against ½MR²), so a larger share of the energy is rotational, leaving less for translation and giving a smaller acceleration.Marking: 1 mark for the disc; 1 mark for the I explanation.
Question 7 Spinning skater (2 marks)
No external torque acts, so L = Iω is constant. Reducing I forces ω up. If I halves, ω doubles.Marking: 1 mark for conservation of L; 1 mark for the factor 2.
Question 8 Disc formula (1 mark)
I = ½MR².Marking: 1 mark.
