Simple harmonic motion: mass-spring and pendulum

Investigating simple harmonic motion using both a mass on a spring and a simple pendulum.

Theory: SHM systems: pendulums and springs

What you are trying to do

Investigate simple harmonic motion with the two standard oscillators: a simple pendulum, and a mass on a spring.

Apparatus

String and a small dense bob; a clamp stand tall enough for lengths up to a metre or more
A spring, slotted masses and hanger
Stopwatch, metre rule, and a fiducial marker (a pin or card at the equilibrium position)

Variables

Independent: the pendulum's length l; the oscillating mass m on the spring
Dependent: the period T
Control: small amplitudes throughout; the same spring; the same bob

Method

FIG. 1The pendulum's cycle. Time it as it passes the centre, where it moves fastest and the crossing is sharpest to judge.

Place the fiducial marker at the equilibrium position, start the oscillation with a small displacement, and time twenty complete cycles from a centre crossing. Divide by twenty, and repeat each timing.
For the pendulum, vary l over at least five values, measuring to the centre of the bob. Keep the swing under about ten degrees.

FIG. 2The mass-spring oscillator: the same mathematics with stiffness in place of gravity.

For the spring, vary the load over five values, displacing vertically by a small amount each time.

Analysis

Pendulum: T = 2π√(l/g), so T² against l is a straight line through the origin with gradient 4π²/g; g comes from the gradient.
Spring: T = 2π√(m/k), so T² against m gives gradient 4π²/k, and k follows.
In both cases the squared plot is the whole trick: it linearises the square root and lets every reading vote on the answer.

A worked set of readings: the pendulum

Each T is the mean of two timings of twenty oscillations:

l / m	T / s	T² / s²
0.400	1.269	1.610
0.600	1.554	2.415
0.800	1.794	3.219
1.000	2.006	4.024
1.200	2.198	4.829

The gradient of T² against l is 4.024 s² m⁻¹, and g = 4π²/gradient = 9.81 m s⁻².

A worked set of readings: the spring

Same timing routine, five loads:

m / kg	T / s	T² / s²
0.100	0.397	0.158
0.200	0.562	0.316
0.300	0.688	0.474
0.400	0.795	0.632
0.500	0.889	0.790

The gradient of T² against m is 1.579 s² kg⁻¹, so k = 4π²/gradient = 25.0 N m⁻¹.

Where the uncertainty comes from

Timing: Human reaction is about 0.2 s; spreading it over twenty cycles cuts it twentyfold, and the fiducial marker at the centre sharpens the start and stop.
Length: To the centre of the bob, not the top: half a bob diameter is a systematic error otherwise.
Amplitude: The pendulum's isochrony is a small-angle result; beyond about ten degrees the period creeps up.
The spring's own mass: A real spring carries some of its own mass along; it shows up as a small positive intercept on the T² against m graph, another reason not to force the line through the origin.

What earns the marks

Twenty oscillations, timed from the equilibrium position past a fiducial marker, repeated and averaged: state all three parts.
Plot T², not T. The gradient identifications (4π²/g and 4π²/k) are the standard follow-up.
Measure the pendulum to the bob's centre and keep the amplitude small, and say why for both.
Interpret intercepts physically (spring mass, length offset) rather than forcing lines through the origin.

Safety

A loaded spring can fly if it slips its support, and slotted masses land hard: clamp the stand, keep the stack modest, and keep eyes out of the line of a stretched spring.

Method and analysis here follow the standard approach; your school may vary the apparatus. Always follow your teacher’s risk assessment in the lab.