Problem — Designing the measurement of g

Problem

You want to measure $g$ in a school laboratory with uncertainty $\le 2\%$. Choose a method, state the instruments, the main sources of error with their respective remedies, and estimate the minimum number of repetitions.

Solution

Method: free fall. A sphere released from rest travels the height $h$ in time $t$ according to $h = \tfrac{1}{2}g t^2$, from which $g = \frac{2h}{t^2}$

Instruments. Two photogates (start and stop) to measure $t$ electronically, avoiding human reaction time; a tape measure for the height $h$; a dense sphere (e.g. steel) to make air resistance negligible.

Sources of error and remedies.

• Reaction time in hand-operated timing: eliminated by using photogates.
• Air resistance: reduced with a dense, compact sphere and with $h$ not too large, so the speed remains modest.
• Measurement of $h$: care in positioning the photogate markers and in reading the tape measure.

Uncertainty budget. From the propagation of errors on a product/quotient: $\frac{\Delta g}{g} \approx \frac{\Delta h}{h} + 2\,\frac{\Delta t}{t}$ The time carries double the weight: to stay under $2\%$ it is essential to control $t$ above all (photogates) and to measure $h$ carefully.

Repetitions. The random error on the mean scales as $1/\sqrt{N}$: about $N \approx 10$ repetitions are enough to average out the residual fluctuations and estimate the spread.

$\ev{g = \dfrac{2h}{t^2}\ ;\quad \text{photogates} + \text{dense sphere},\ \ N \approx 10}$

Links

Topics: Metodo sperimentale Concepts: Cifre significative e incertezza Skills: Propagazione degli errori