The method and the results are on the rotational inertia lab. Do the analysis yourself first, then check the answers below.
Quick answer: Rotational inertia I measures how strongly an object resists changes in its rotation and depends on both mass and how far that mass is from the axis. These answers go with the AP Physics 1 lab on rotational inertia: the graph, the calculations, the uncertainties and the conclusion explained, for all 8 analysis questions.
The model answers use the sample data in the table. Your own wheel will have a different rotational inertia, but the increase caused by the clamps should be close to 2mr², and the discussion of friction and uncertainty should be similar.
Level: AP Physics 1 (algebra-based) · Curriculum: AP P1 Unit 5
Question 1 Acceleration of the mass
a = 2d/t² = 2 × 1.00 ÷ (4.06)² = 2.00 ÷ 16.48 = 0.121 m/s².
Question 2 Tension and angular acceleration
α = a/r = 0.1213 ÷ 0.050 = 2.43 rad/s². T = m(g − a) = 0.500 × (9.81 − 0.121) = 4.84 N. This is slightly less than the weight mg = 4.905 N, because the mass accelerates downward.
Question 3 Rotational inertia of the wheel
τ = Tr = 4.844 × 0.050 = 0.2422 N m. I = τ/α = 0.2422 ÷ 2.427 = 0.100 kg m².
Question 4 Rotational inertia with the clamps
a = 2.00 ÷ (4.37)² = 0.1047 m/s²; α = 2.095 rad/s²; T = 0.500 × (9.81 − 0.1047) = 4.853 N; τ = 0.2426 N m. I = 0.2426 ÷ 2.095 = 0.116 kg m².
Question 5 Testing the prediction
Predicted increase = 2mr² = 2 × 0.200 × (0.20)² = 0.016 kg m². Measured increase = 0.1158 − 0.0998 = 0.016 kg m². The two agree to within the precision of the data, supporting I = Σmr².
Question 6 Size of the correction
(4.905 − 4.844) ÷ 4.905 = 0.012, a difference of 1.2 %. For a rough estimate this is small, but because it enters I directly in a subtraction, it is better to keep the correction and use T = m(g − a).
Question 7 Uncertainty
Range = 4.10 − 4.03 = 0.07 s, so uncertainty ≈ 0.035 s, which is 0.035 ÷ 4.06 = 0.9 %. Since a ∝ 1/t², the uncertainty in a is about 1.7 %.
Question 8 Improvements
Friction in the axle slows the wheel, so the calculated I is too large; use a ball-bearing axle or measure friction by finding the mass that makes the wheel turn at a steady rate and allow for it. The string stretching or slipping on the spool changes the effective r; use a thin non-stretching thread and wind it neatly. Each reduces systematic error.
