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 is the classic 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 through the origin with gradient g/2.
2. Doubling the gradient gives g. The graph is the whole point: it averages every reading at once, and an intercept exposes a systematic error that a single calculation would silently absorb.

A worked set of readings

Mean of three electronically timed drops at each height:

The gradient of h against t² is 4.905 m s⁻², and doubling it gives g = 9.81 m s⁻². The line passes through the origin, so no release delay crept in here; in a real run a small positive intercept on the t² axis is the delay's signature.

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