Problem — Fermi estimate, measurements of g

Problem

Estimate, using Fermi-style reasoning, how many independent measurements of the gravitational constant $g$ could be made in one hour of lab work in a class of $25$ students, using the bouncing-ball method. What would be a plausible uncertainty on the final mean value?

Solution

Fermi set-up. No precise data is needed: reasonable orders of magnitude, combined with common sense, are enough.

Time per single measurement. Each measurement requires: preparing and releasing the ball, timing the bounce (or measuring the heights), noting it down and doing the calculation. We estimate on the order of a few minutes per measurement, say $\sim 10$–$15$ min at first but dropping to a few minutes with practice. Taking an average of $\sim 12$ min $\to$ in $60$ min each student manages about $4$–$5$ measurements.

Total number of measurements. With $25$ students working in parallel: $N \approx 25 \times 4\text{–}5 \approx 100\text{–}125 \approx 10^2 \text{ measurements}$

Uncertainty on the mean value. A single home-made bounce measurement of $g$ is crude: relative uncertainty on the order of $10$–$20\%$ (reaction times, poorly read heights, air resistance). Averaging over $N \approx 100$ independent measurements, the random error on the mean shrinks by a factor $\sqrt{N}$: $\frac{\sigma_{\bar{x}}}{\bar{x}} \approx \frac{10\text{–}20\%}{\sqrt{100}} = \frac{10\text{–}20\%}{10} \approx 1\text{–}2\%$

Caveat. This reduction only holds for the random part of the error. If all measurements share the same systematic error (for example, a mis-calibrated metre stick), averaging does not help: the $\sim 1$–$2\%$ limit is optimistic and assumes systematics are kept under control.

$\ev{N \sim 10^2 \text{ measurements} \qquad \dfrac{\sigma_{\bar{x}}}{\bar{x}} \sim 1\text{–}2\%}$

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Topics: Metodo sperimentale Skills: Stime di Fermi · Propagazione degli errori