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 AP Physics C lab on moment of inertia gives the aim, the equipment, the safety points, a step-by-step method, sample results and 7 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
A rod swinging about a pivot has period T = 2π√[(Icm + Md²)/(Mgd)]. Rearranged, T²d is linear in d², with a gradient 4π²/g. Measuring T for several pivot positions tests the parallel axis theorem and gives g and the radius of gyration without needing to know the mass.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 5
Aim
Check the parallel axis theorem by timing a metre rule that swings as a pendulum about pivots at different distances d from its centre of mass, and use the graph to find g and the radius of gyration.
Equipment
- Wooden metre rule with small holes drilled at 5, 10, 20, 30 and 40 cm (teacher prepared)
- Pin or small bolt as pivot, in a clamp on a stand
- Phone stopwatch
- Second metre rule
- Card marker for the equilibrium position
Safety
Use a short, blunt bolt rather than a sharp pin, and clamp the stand to the bench or weigh it down. Stay clear of the swinging end, and wear goggles in case the rule slips off the pivot.
Method
- Hang the rule from the pivot at the 40 cm hole, so d = 0.10 m from the centre at 50 cm.
- Pull the lower end aside by a small angle (under 10°) and release it without twisting.
- Time 20 complete oscillations from the equilibrium mark and record t₂₀.
- Repeat with holes at 30, 20, 10 and 5 cm (d = 0.20, 0.30, 0.40 and 0.45 m).
- Calculate T = t₂₀/20, then T²d and d² for each position.
Results
| d (m) | t₂₀ (s) | T (s) | T²d (s²·m) | d² (m²) |
|---|---|---|---|---|
| 0.10 | 38.8 | 1.94 | 0.376 | 0.0100 |
| 0.20 | 31.5 | 1.58 | 0.496 | 0.0400 |
| 0.30 | 30.6 | 1.53 | 0.702 | 0.0900 |
| 0.40 | 31.2 | 1.56 | 0.973 | 0.1600 |
| 0.45 | 32.0 | 1.60 | 1.152 | 0.2025 |
Analysis questions
Question 1 Period equation
Show that T²d = (4π²/g)(k² + d²), where Icm = Mk².Question 2 Plot the graph
Plot T²d against d² and find the gradient and intercept from the sample data.Question 3 Find g
Use the gradient to calculate g.Question 4 Radius of gyration
Use the intercept and gradient to find k, and compare with L/√12 for a rod of length 1.00 m.Question 5 Why small angles
Explain why the amplitude must be small.Question 6 Uncertainty
Timing 20 oscillations gives t₂₀ with an uncertainty of ±0.2 s. Calculate the percentage uncertainty in T.Question 7 Evaluation
State two sources of error and one improvement.
