Determining g by free fall

Practical method

Aim

Determine the acceleration of free fall g by timing a falling object over several measured heights.

Variables

Independent variable: drop height h
Dependent variable: fall time t
Control variables: the same dense, small object; release from rest; the same timing method.

Apparatus

• Steel ball bearing
• Electromagnet release and trapdoor (or light gates)
• Electronic timer
• Metre rule and set square
• Clamp stand

Method

1. Clamp the electromagnet so the ball hangs above a trapdoor; switching off the magnet starts the timer and the ball opening the trapdoor stops it.
2. Measure the height h from the bottom of the ball to the trapdoor with a metre rule, using a set square to avoid parallax.
3. Release the ball and record the time t. Repeat three times and take the mean.
4. Repeat for at least five heights from 0.20 m to 1.00 m.
5. Plot h against t². From s = ½gt², the gradient is \(\dfrac{g}{2}\).

Sample results

Steel ball released from rest; times are means of three repeats.

Analysis

The graph of h against t² is a straight line. Gradient = \(\dfrac{1.000 - 0.200}{0.2052 - 0.0416}\) = 4.89 m s⁻², so g = 2 × 4.89 = 9.78 m s⁻². A small positive t-intercept shows a systematic delay in the magnet releasing the ball; using the gradient removes its effect.

Uncertainty

The repeat times at 0.200 m spread over about ±0.003 s, a percentage uncertainty of 1.5% in t and so 3% in t². Using the gradient of a best-fit line through all points, with steepest and shallowest lines, gives g = 9.8 ± 0.2 m s⁻².

Errors and improvements

Risk assessment

Common mistakes

• Plotting h against t instead of t² (a curve).
• Forgetting that the gradient is \(\dfrac{g}{2}\).
• Measuring h from the top of the ball.

Worked analysis

At h = 0.800 m, t = 0.405 s. g = \(\dfrac{2h}{t^{2}}\) = \(\dfrac{2 \times 0.800}{0.405^{2}}\) = 9.75 m s⁻². The graph value (9.78 m s⁻²) is more reliable because it uses every point.

Common misconception

A heavier ball does not fall faster in a vacuum; with negligible air resistance g is independent of mass.