The method and the results are on the moment of inertia lab. Do the analysis yourself first, then check the answers below.
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. These answers go with the AP Physics C lab on moment of inertia: the graph, the calculations, the uncertainties and the conclusion explained, for all 7 analysis questions.
Model answers based on the sample data. A best-fit line through the five points gives a gradient close to 4.0 s²/m and an intercept near 0.34 s²·m. Your own data may vary slightly, but g should come out within a few per cent of 9.8 m/s².
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 5
I = Mk² + Md², and for small swings T = 2π√(I/(Mgd)). Squaring gives T² = 4π²(k² + d²)/(gd), so T²d = (4π²/g)(k² + d²).
Question 2 Plot the graph
The points lie on a straight line. Gradient ≈ (1.152 − 0.376)/(0.2025 − 0.0100) = 4.0 s²/m and intercept ≈ 0.34 s²·m.
Question 3 Find g
Gradient = 4π²/g, so g = 39.48/4.03 = 9.8 m/s².
Question 4 Radius of gyration
Intercept/gradient = k² = 0.34/4.03 = 0.084 m², so k = 0.29 m. For a rod, L/√12 = 1.00/3.464 = 0.289 m, which agrees.
Question 5 Why small angles
The equation uses sin θ ≈ θ, so the torque is proportional to the angle and the motion is simple harmonic. At large angles the period depends on amplitude.
Question 6 Uncertainty
Percentage uncertainty = 0.2/31.5 = 0.6 %, the same as in T, because dividing by 20 is exact. The uncertainty in T²d is about 1.3 %.
Question 7 Evaluation
The rule may twist or the pivot may rub; amplitude may be above 10°. Using a smoother pivot and timing 30 oscillations reduces both. Smoother pivot, longer timing.
