Year 13 Physics readiness: pendulum and uncertainty

A pupil checks whether a simple pendulum can be used to measure the gravitational field strength $g$. The pendulum length is $l = 0.500\,\mathrm{m}$. The pupil measures a time of $14.22\,\mathrm{s}$ for 10 complete oscillations. Calculate the period $T$ of the pendulum.

For small oscillations, the period of a simple pendulum is given by $T = 2\pi\sqrt{l/g}$. Use $l = 0.500\,\mathrm{m}$ and $T = 1.422\,\mathrm{s}$ to calculate $g$.

The absolute uncertainty in $l$ is $0.002\,\mathrm{m}$. The estimated absolute uncertainty in the measured time of $14.22\,\mathrm{s}$ is $0.10\,\mathrm{s}$. Calculate the percentage uncertainty in the calculated value of $g$. Use $g = 4\pi^2l/T^2$ and assume the percentage uncertainty in $T$ is the same as the percentage uncertainty in the measured time for 10 oscillations.

Explain why timing 10 complete oscillations and dividing by 10 gives a smaller percentage timing uncertainty than timing only one oscillation with the same stopwatch and reaction-time method.