The questions are on the simple harmonic motion test. Take it under real conditions before reading the mark scheme.
Quick answer: Simple harmonic motion obeys d²x/dt² = −ω²x, with solution x = A cos(ωt + φ). This mark scheme gives the model answers and the marks for the 8 questions of the AP Physics C test on simple harmonic motion, from spring oscillator to verify a solution, with the equivalent answers to accept.
The model answers show where each mark is earned. Derivations are marked on the logic of each step, so a correct final expression without the steps does not earn full credit. Numerical answers need a unit.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 7
Question 1 Spring oscillator (2 marks)
ω = √(k/m) = √(100 / 0.25) = 20 rad/s. T = 2π/ω = 0.31 s.Marking: ω (1); T (1).
Question 2 Calculus on x(t) (3 marks)
a. v = −0.60 sin 10t, so vmax = 0.60 m/s.b. a = −6.0 cos 10t, so amax = 6.0 m/s².
c. x = 0.060 cos(1.0) = 0.060 × 0.540 = 0.032 m.
Marking: one mark per part.
Question 3 Amplitude and period (2 marks)
ω² = k/m depends only on the spring constant and the mass. The solution x = A cos(ωt + φ) has A as a free constant set by the start conditions. Changing A therefore leaves T = 2π/ω unchanged.Marking: ω depends on k and m (1); A set by initial conditions (1).
Question 4 Energy and speed (3 marks)
a. E = ½kA² = ½ × 250 × 0.0064 = 0.80 J.b. U = ½ × 250 × 0.0016 = 0.20 J.
c. K = 0.60 J, so v = √(2 × 0.60 / 0.40) = 1.7 m/s.
Marking: one mark per part.
Question 5 Physical pendulum (3 marks)
The torque about O is τ = −mgd sinθ. For small θ, sinθ ≈ θ, so τ = −mgd θ. Newton’s second law for rotation gives I d²θ/dt² = −mgd θ, which is d²θ/dt² = −(mgd/I)θ. This matches d²θ/dt² = −ω²θ with ω = √(mgd/I).Marking: torque expression (1); small-angle step (1); comparison with SHM equation (1).
Question 6 Swinging rod (3 marks)
I = ⅓ML² and d = L/2, so T = 2π√(I / (Mgd)) = 2π√(2L / 3g) = 2π√(1.2 / 29.43) = 2π × 0.2019 = 1.27 s.Marking: I and d (1); formula (1); answer with unit (1).
Question 7 Doubling the period (2 marks)
T ∝ √L, so doubling T needs four times the length: L = 4.0 m.Marking: T ∝ √L (1); answer (1).
Question 8 Verify a solution (2 marks)
dx/dt = Aω cos ωt, so d²x/dt² = −Aω² sin ωt = −ω²x.Marking: first derivative (1); second derivative and conclusion (1).
