Simple harmonic motion: spring and pendulum

Practical method

Aim

Determine g from the period of a simple pendulum and investigate a mass–spring system, as required for simple harmonic motion.

Variables

Independent variable: pendulum length l
Dependent variable: period T
Control variables: small angle of swing (below 10°); the same bob.

Apparatus

• Pendulum bob on a string
• Clamp, stand and split cork
• Metre rule
• Stopwatch
• Fiducial marker
• Protractor

Method

1. Clamp the string between two pieces of cork; measure l from the point of suspension to the centre of the bob.
2. Displace the bob through a small angle (under 10°) and release it.
3. Time 20 oscillations from the fiducial marker at the lowest point; repeat and average.
4. Repeat for lengths from 0.200 m to 1.000 m.
5. Plot T² against l. From T = 2π\(\sqrt{\dfrac{l}{g}}\), the gradient is \(\dfrac{4\pi ^{2}}{g}\).

Sample results

Simple pendulum, amplitude under 10°.

Analysis

T² is proportional to l with gradient 4.02 s² m⁻¹, so g = \(\dfrac{4\pi ^{2}}{4.02}\) = 9.81 m s⁻². Period does not depend on the bob's mass. The same analysis applied to a mass–spring system (T² against m) gives k.

Uncertainty

Length is measured to ±1 mm (0.5% at 0.200 m); the largest uncertainty is timing. If l is measured to the top of the bob rather than its centre, the line has an intercept but the gradient, and so g, is unchanged.

Errors and improvements

Risk assessment

Common mistakes

• Measuring l to the top of the bob only.
• Using large angles.
• Plotting T against l.

Worked analysis

Gradient = \(\dfrac{4.024}{1.000}\) = 4.02 s² m⁻¹. g = \(\dfrac{4\pi ^{2}}{\text{gradient}}\) = \(\dfrac{39.48}{4.024}\) = 9.81 m s⁻².

Common misconception

A heavier bob does not change the period of a simple pendulum; only the length and g matter for small swings.