Case Study 2 — Measuring Gravity With a Pendulum, and Where the Error Bar Comes From

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Case Study 2 — Measuring Gravity With a Pendulum, and Where the Error Bar Comes From

Field: Experimental physics and measurement (metrology) Calculus used: Differentials (§11.4), error propagation and the power-law rule (§11.5), the small-angle linearization $\sin\theta \approx \theta$ (§11.3, §11.10)

Every measured number in science arrives with a companion: its uncertainty, the $\pm$ that says how much to trust it. Where does that $\pm$ come from? Almost always, it is manufactured by the calculus of this chapter — differentials propagating the wobble in raw measurements through to the final result. This case study walks through one complete experiment, the classic determination of the gravitational acceleration $g$ with a pendulum, and shows exactly how the error bar is born.

The experiment and its hidden linearization

A simple pendulum of length $\ell$ swings under gravity. Newton's second law for the angle $\theta$ gives the exact equation $$\ddot\theta = -\frac{g}{\ell}\sin\theta,$$ which has no elementary solution. The whole experiment rests on a single act of linear approximation. For small swings, replace $\sin\theta$ by its linearization at $0$ — the rule $\sin\theta \approx \theta$ from §11.3 — and the equation collapses into simple harmonic motion, $$\ddot\theta = -\frac{g}{\ell}\theta,$$ whose period is the familiar $$T = 2\pi\sqrt{\frac{\ell}{g}}.$$ This is the formula every physics student uses, and it is already an approximation: the chapter's tangent-line idea is baked into the apparatus before any number is read. Solving for $g$ gives the working equation $$g = \frac{4\pi^2 \ell}{T^2}.$$ We will measure $\ell$ and $T$, plug in, and — crucially — propagate the measurement errors to attach an honest $\pm$ to $g$.

The raw measurements

Suppose we measure:

Length: $\ell = 1.000$ m, read with a tape to $\pm 0.002$ m. That is a relative error $\Delta\ell/\ell = 0.002/1.000 = 0.2\%$.
Period: Here is the first piece of experimental wisdom. We do not time a single swing. A single period is about $2$ s, and a human stopwatch reaction is good to maybe $\pm 0.1$ s — a ruinous $5\%$ error. Instead we time 50 complete swings and divide. The total time comes to $100.3$ s, again with a single $\pm 0.1$ s reaction error on the whole batch, so $T = 100.3/50 = 2.006$ s and $\Delta T = 0.1/50 = 0.002$ s. The relative error is now $\Delta T/T = 0.002/2.006 = 0.10\%$.

Timing many swings is the experiment's central trick, and §11.5 tells us why it pays off so handsomely — but to see that, we first need the propagation rule.

Computing $g$ and propagating the error

The central value is immediate: $$g = \frac{4\pi^2 (1.000)}{(2.006)^2} = \frac{39.478}{4.0240} = 9.811\ \mathrm{m/s^2}.$$

Now the uncertainty. We have a function of two measured inputs, $g = g(\ell, T)$, so we use the multivariable form of error propagation previewed in §11.11. Taking the differential, $$dg = \frac{\partial g}{\partial \ell}\,d\ell + \frac{\partial g}{\partial T}\,dT = \frac{4\pi^2}{T^2}\,d\ell + \left(-\frac{8\pi^2\ell}{T^3}\right)dT.$$ Dividing through by $g = 4\pi^2\ell/T^2$ turns this into the far cleaner relative-error statement, where everything telescopes: $$\frac{dg}{g} = \frac{d\ell}{\ell} - 2\,\frac{dT}{T}.$$ For a worst-case error bar we take absolute values and add (the errors might conspire in the same direction): $$\boxed{\,\frac{\Delta g}{g} \approx \frac{\Delta \ell}{\ell} + 2\,\frac{\Delta T}{T}.\,}$$ This is the power-law rule of §11.5 in action: $\ell$ appears to the first power, so it contributes its relative error once; $T$ appears squared (in the denominator), so it contributes twice. The exponent is the amplification factor, exactly as the chapter promised.

Plugging in our numbers: $$\frac{\Delta g}{g} \approx 0.2\% + 2(0.10\%) = 0.2\% + 0.20\% = 0.40\%.$$ Converting back to absolute terms, $\Delta g \approx 0.0040 \times 9.811 = 0.039\ \mathrm{m/s^2}$. So the experiment reports $$g = 9.81 \pm 0.04\ \mathrm{m/s^2}.$$ That $\pm 0.04$ is not a guess and not a convention. It is the differential $dg$, evaluated at the measured point, fed by the measured uncertainties in $\ell$ and $T$. The accepted value $g = 9.81\ \mathrm{m/s^2}$ sits comfortably inside the interval — the error bar did its job.

Why timing many swings matters so much

Return to the timing trick with the propagation rule now in hand. Because $T$ enters squared, its relative error counts double — it is the more dangerous of the two inputs. Had we timed a single swing, $\Delta T/T$ would have been $0.1/2.0 = 5\%$, and the rule would give $\Delta g/g \approx 0.2\% + 2(5\%) = 10.2\%$: the result would be $g = 9.8 \pm 1.0$, almost worthless. Timing $N = 50$ swings divides the per-swing timing error by $50$, dragging that $5\%$ down to $0.1\%$ and the whole $g$ error down to $0.4\%$.

This is the design payoff of error analysis: the differential told us in advance which knob to turn. There was no reason to chase the length to micrometer precision — at $0.2\%$ it was already the smaller contributor. The leverage was all in the timing, because of the exponent $2$. A physicist who understands §11.5 spends their effort where the amplification factor is largest.

# Propagating measurement error into g = 4*pi^2 * L / T^2.
import math
L, dL = 1.000, 0.002          # length and its uncertainty (m)
T, dT = 2.006, 0.002          # period and its uncertainty (s), from 50 swings
g  = 4 * math.pi**2 * L / T**2
rel = dL/L + 2 * dT/T         # power-law propagation rule
dg = rel * g
print(f"g = {g:.3f} +/- {dg:.3f} m/s^2   ({100*rel:.2f}% error)")
# Hand-computed output:
#   g = 9.811 +/- 0.039 m/s^2   (0.40% error)

The deeper point: two approximations, one chapter

Notice that two distinct ideas from Chapter 11 are doing the work, and they are not the same idea. First, the small-angle linearization $\sin\theta \approx \theta$ (§11.3) is what made the pendulum solvable at all — it is a modeling approximation, replacing a hard differential equation with an easy one, and it introduces its own systematic error (real pendulums at larger amplitude swing slightly slower than $2\pi\sqrt{\ell/g}$ predicts). Second, the differential $dg$ (§11.4–11.5) is a measurement approximation, translating input uncertainties into an output uncertainty. The first determines whether your formula is right; the second determines how precisely your numbers are known. A careful experimenter must account for both — and both are tangent lines.

What this case study shows

The humble $\pm$ on a lab result is calculus made visible. The differential converts the wobble in a ruler and a stopwatch into a single number, the power-law rule reveals which measurement to obsess over, and the small-angle linearization is the reason the experiment has a clean formula in the first place. From a tape measure and a stopwatch, the tangent line delivers $g = 9.81 \pm 0.04\ \mathrm{m/s^2}$ — value and honesty in one stroke.

Discussion Questions

The rule $\Delta g/g \approx \Delta\ell/\ell + 2\,\Delta T/T$ weights the period error by $2$. Derive this weighting from $g = 4\pi^2\ell\,T^{-2}$ using the power-law rule of §11.5, identifying the exponent on each variable.
Suppose you could halve either the length error or the period error, but not both. Using the numbers in this study, which choice shrinks $\Delta g$ more? Why?
The small-angle approximation makes real large-amplitude pendulums swing slightly slower than the formula predicts. Is this a random error (reduced by averaging) or a systematic error (not reduced by averaging)? How would it bias the measured $g$?
A student times only 1 swing but repeats it 50 times and averages the 50 single-swing times. Is that statistically equivalent to timing one batch of 50 swings? Explain why the batch method also suppresses the per-trial reaction error, not just random scatter.

Short Annotated Reading

Stewart, Calculus: Early Transcendentals, §3.10 (Linear Approximation and Differentials). Contains the volume-of-a-sphere error example that this study parallels for a two-variable formula.
OpenStax Calculus Volume 1, §4.2 (Linear Approximations and Differentials). Free treatment of differentials and the relative/percentage-error vocabulary used here.
Taylor, An Introduction to Error Analysis (2nd ed., University Science Books). The standard undergraduate reference; Chapters 3–4 derive the propagation rules this study applies, including the multivariable form previewed in §11.11.
Stewart §3.10 worked "pendulum" / period exercises. Good companion problems for practicing the squared-period weighting before attempting Exercise C5.