Measuring g by free fall

What you are trying to do

Determine the acceleration due to gravity by timing an object in free fall over a range of measured heights.

Apparatus

• Electromagnet holding a steel ball, on a tall clamp stand
• Trapdoor switch at the bottom, wired so the falling ball stops the clock
• Electronic timer started by the release circuit
• Metre rule; set squares help measure to the right points

Variables

• Independent: the drop height h
• Dependent: the fall time t
• Control: the same ball throughout, released from rest each time

Method

1. Switching off the electromagnet releases the ball and starts the timer in the same instant; the ball breaking the trapdoor contact stops it.
2. Measure the height from the bottom of the ball to the trapdoor. Measuring to the wrong point on the ball shifts every height by the same amount, which is a systematic error.
3. Time three drops at each height and take the mean, then repeat over a range of heights from around 0.4 m to over a metre.

Analysis

1. From rest, h = ½gt², so a graph of h against t² is a straight line of gradient g/2, passing through the origin if there is no timing or distance offset.
2. Plot h against t² and determine the gradient from a best-fit line. The gradient uses all the data points; a statistically significant intercept or curvature may indicate a timing or distance offset. Doubling the gradient gives g.
3. What the gradient will support. Twice the gradient landing on 9.8 m s⁻², with no significant intercept, supports h = ½gt² and measures g to about a per cent. Twice the gradient landing on 9.2 m s⁻² alongside a positive intercept on the t² axis supports a different conclusion: the recorded times are all too long by a fixed amount, which is what a timer starting before the ball does. The intercept, not the gradient, is what distinguishes the two.
4. What this timing cannot reach. The clock is electronic, so reaction time never enters, but you still cannot see either end of the fall: the timer starts when the current stops rather than when the ball moves, and stops when the trapdoor opens rather than when the ball arrives. Air resistance is not the limit here, because a dense steel ball falling a metre barely notices it. The release is. A light gate at the top, broken by the ball itself, times from the ball instead of from the circuit and removes the delay rather than correcting for it.

A worked set of readings

Mean of three electronically timed drops at each height:

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 h against t² is 4.905 m s⁻², and doubling it gives g = 9.81 m s⁻². These idealised points give no intercept, so no release delay is present; in a real run a small positive intercept on the t² axis indicates one.

Evaluating the result

Idealised readings with one drop allowed to go wrong. At 0.800 m the three timings were 0.404 s, 0.404 s and 0.464 s, and all three went into the mean without anyone looking at them separately. The third column divides h by t², which the model says is half of g at every height.

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 unaffected rows give 4.905 m s⁻² and the remaining one gives 4.453, so that height on its own would give g = 8.91 m s⁻². Look at the individual timings before the graph: two drops agree to a millisecond and the third is 60 ms longer, a gap larger than the scatter at the other heights. Competing explanations include the trapdoor sticking, the ball clipping the frame, and a genuinely slower release. Repeat the measurement at 0.800 m and inspect the apparatus. Exclude the reading only if repeat measurements and an identified fault justify doing so; record the reason, and keep the original timings in the table.

Where the uncertainty comes from

• Release delay: The electromagnet's field takes a moment to collapse, so the ball leaves slightly after the timer starts. That inflates every t equally, curving the small-h end and pulling g down: a systematic error, invisible to repeats.
• Timing resolution: Electronic timing is good to a millisecond; the shortest drops are under 0.3 s, so prefer larger heights where the percentage error is smaller.
• Height measurement: Bottom of ball to trapdoor, with the rule vertical; a set square against the stand helps.
• Random scatter: Small: release is electrical, not human. The mean of three still guards against the occasional glancing trapdoor hit.

What earns the marks

Safety