Quick answer: Simple harmonic motion obeys d²x/dt² = −ω²x, with solution x = A cos(ωt + φ). This AP Physics C lab on simple harmonic motion 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 metre stick can swing from any hole drilled along its length, and the swing time changes with the hole chosen. This lab measures the period for several pivot distances and tests the physical pendulum equation, then uses a straight-line graph to extract g and the radius of gyration of the stick.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 7
Aim
To measure the period of a metre stick swinging about different pivot holes and use the physical pendulum equation to find g and the radius of gyration of the stick.
Equipment
- A metre stick drilled with small holes at 10, 20, 30, 40 and 45 cm from the centre
- A clamp and a horizontal steel pin or nail
- A stopwatch or phone timer
- A protractor to check the starting angle
Safety
Clamp the pin firmly to the bench edge and keep faces and hands clear of the swinging stick. Wear closed shoes and stand to the side.
Method
- Hang the stick from the pin through the hole at d = 0.10 m.
- Pull it aside by about 8° (less than 10°) and release it without pushing.
- Time 10 complete swings and divide by ten to get T.
- Repeat for each hole.
- For each hole calculate T²d and d². Theory predicts T²d = (4π²/g)(k² + d²), where k² = L²/12 is the square of the radius of gyration.
Results
Sample measurements for a stick of length L = 1.00 m:
| d (m) | Time for 10 swings (s) | T (s) | T²d (s² m) | d² (m²) |
|---|---|---|---|---|
| 0.10 | 19.4 | 1.94 | 0.376 | 0.0100 |
| 0.20 | 15.8 | 1.58 | 0.499 | 0.0400 |
| 0.30 | 15.3 | 1.53 | 0.702 | 0.0900 |
| 0.40 | 15.7 | 1.57 | 0.986 | 0.1600 |
| 0.45 | 16.0 | 1.60 | 1.152 | 0.2025 |
Analysis questions
Question 1 Check the table
Check the values of T and T²d for d = 0.30 m and d = 0.45 m.Question 2 Straight-line graph
Explain why a graph of T²d against d² should be a straight line, and state its gradient and intercept in terms of g and k.Question 3 Find g
Take the outer two rows of the table, work out the slope of T²d against d², then deduce a value for the gravitational field strength.Question 4 Radius of gyration
Use the intercept to find k² and compare it with L²/12.Question 5 Timing uncertainty
The reaction-time error in starting and stopping the timer is about ±0.2 s. Calculate the uncertainty in T and its percentage for d = 0.20 m.Question 6 Shape of the data
Explain why T is larger at d = 0.10 m than at d = 0.30 m, and why it rises again towards d = 0.45 m.Question 7 Amplitude check
Explain why the starting angle must be below about 10°, and estimate the error in T for 40°.
