Measuring g with a falling object

With a steel ball, a ruler and a timer, you can measure a property of the entire Earth. The physics is one line of suvat. The marks are in the details of how you actually do it, so this lesson is mostly about the details.

The ball is released from rest, so $u = 0$. If it falls a height $h$ in time $t$:

$h = \tfrac{1}{2}gt^2$

So $h$ is proportional to $t^2$. Plot $h$ on the y-axis against $t^2$ on the x-axis and you get a straight line through the origin with

$\text{gradient} = \frac{g}{2} \quad\Longrightarrow\quad g = 2 \times \text{gradient}$

Why bother with a graph instead of one drop? The best-fit line uses every reading, so random errors partly cancel, anomalies stand out, and a line that misses the origin flags a systematic error (such as measuring every $h$ to the wrong point on the ball).

1. A small steel ball is held by an electromagnet above a trapdoor switch (or above a light gate), both connected to an electronic timer.
2. Measure the height $h$ from the bottom of the ball to the trapdoor with a metre rule. Keep the rule vertical and your eye level with the mark, to avoid parallax.
3. Switch off the electromagnet. The same switch starts the timer; the ball hitting the trapdoor stops it. Record $t$.
4. Repeat at the same height (say three times) and find the mean $t$.
5. Repeat for at least five more heights, spread from about $0.2\ \text{m}$ to $1.2\ \text{m}$.
6. Plot $h$ against $t^2$, draw the best-fit line, find the gradient, and calculate $g = 2 \times \text{gradient}$.