Quick answer: Simple harmonic motion is repeated oscillation where the acceleration is proportional to the displacement from equilibrium and points towards it, a = −ω²x. This Physics 11-12 lab on simple harmonic motion gives the aim, the equipment, the safety points, a step-by-step method, sample results and 8 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
How does the oscillation period of a hanging mass change as the mass increases? Use a simple spring, a stopwatch and a graph of T² against m to test the relationship and to measure the stiffness of the spring.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: IB C.1
Aim
To find how the period of a mass on a spring depends on the mass, and to determine the spring stiffness k from a graph of T² against m.
Equipment
- Coil spring (stiffness about 20 N/m), clamp stand, boss and clamp
- Slotted masses (0.100 kg to 0.500 kg) and hanger
- Stopwatch (0.1 s resolution), metre ruler
- Small paper flag or marker as a fiducial mark
Safety
Clamp the stand to the bench or weigh down its base so it cannot tip. Keep feet clear of falling masses and wear eye protection in case the spring unhooks.
Method
- Hang the spring from the clamp and add the hanger plus a 0.100 kg mass.
- Place the fiducial marker level with the bottom of the mass at rest.
- Pull the mass down about 2 cm and release; check that it moves straight up and down.
- Start the stopwatch as the mass passes the marker moving upwards and count 20 complete oscillations.
- Record the time t for 20 oscillations, then repeat for masses up to 0.500 kg.
- Calculate T = t ÷ 20 and T² for each mass.
Results
Sample measurements (t is the time for 20 oscillations):
| m / kg | t for 20 oscillations / s | T / s | T² / s² |
|---|---|---|---|
| 0.100 | 9.3 | 0.465 | 0.216 |
| 0.200 | 12.9 | 0.645 | 0.416 |
| 0.300 | 15.6 | 0.780 | 0.608 |
| 0.400 | 18.0 | 0.900 | 0.810 |
| 0.500 | 20.1 | 1.005 | 1.010 |
Analysis questions
Question 1 Calculate the period
Explain why the time for 20 oscillations is recorded, and check the entry for m = 0.300 kg by calculating T and T².Question 2 Uncertainty in the timing
Take the timing error of the 20-oscillation measurement as ±0.2 s. For m = 0.200 kg, find the percentage uncertainty in T and then in T².Question 3 Plot and gradient
Plot T² (y-axis) against m (x-axis), draw a best-fit line and calculate its gradient using the first and last points.Question 4 Spring stiffness
From T² = (4π²/k) m + c, find k from the gradient.Question 5 The intercept
The line does not pass through the origin. Calculate the intercept and explain what it shows.Question 6 Testing proportionality
A student claims that T is proportional to m. Use the data to show that this is wrong and state the true relationship.Question 7 Amplitude check
Describe how to test that the period does not depend on amplitude, and state the expected result.Question 8 Improvements
Suggest two improvements to reduce random and systematic errors.
