Simple harmonic motion: mass-spring and pendulum

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

1. 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.
2. For the pendulum, vary l over at least five values, measuring to the centre of the bob. Keep the swing under about ten degrees.

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

Analysis

1. 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.
2. Spring: T = 2π√(m/k), so T² against m gives gradient 4π²/k, and k follows.
3. In both cases the squared plot does the trick: it linearises the square root and lets every reading vote on the answer.
4. What the pendulum supports, and what the spring adds. A value of g within a per cent or two of 9.81 m s⁻² supports T = 2π√(l/g) over the lengths you tried, which is a stronger statement than any single timing could make. The spring gives k, and that one can be tested twice over: hang a load on the same spring, measure the extension it settles at, and k from F = kx should match the k from the timings. Two routes to one stiffness, one static and one dynamic, agree only if the model is right, because they fail in different ways.
5. The model's own limit. Isochrony is a small-angle result, so this procedure cannot test the pendulum at large amplitude without breaking the theory it is checking. By twenty degrees the period has crept up by about three quarters of a per cent, and by forty degrees by around three. The spring has its own version, since part of the spring oscillates with the load and shows up as a positive intercept on the T² against m graph. Neither is a mistake to be tidied away: report the intercept rather than forcing the line through the origin.

A worked set of readings: the pendulum

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

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

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:

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

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

Evaluating the result

Idealised pendulum readings with one timing miscounted. At 0.800 m the count started at one instead of zero, so nineteen swings were timed and the total was still divided by twenty. The third column divides T² by l, which the model says is 4π²/g at every length.

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

Three rows give 4.02 s² m⁻¹ and the miscounted one gives 3.63, low by 9.8 per cent, because a five per cent error in T becomes ten per cent in T². On the T² against l graph the point sits below the line, and the size of the drop is the giveaway: one swing missed in twenty always costs about a tenth of the plotted value, whatever the length. Time it again, starting the count at zero as the bob passes the marker, rather than deleting a point whose cause you have already worked out.

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

Safety