Capstone: Measure g with a String
A pendulum's swing time depends on just two things — its length and the planet it hangs on. Time it, and the planet is the unknown you solve for.
Builds on: 0.2 Acceleration: When Velocity Itself Changes2.1 Energy: The Quantity That Only Changes Costume
Why a pendulum knows g
A swinging weight is gravity’s metronome. Pull it aside and gravity hauls it back, overshooting through the middle, forever trading height for speed — Unit 2’s energy swap on a string. How quickly each swing completes depends on how hard gravity pulls, and — remarkably — on almost nothing else:
Not in the formula: the mass (heavier bobs are pulled harder but are harder to move — the same cancellation that makes hammer and feather fall together) and, for small swings, the amplitude (wider swings travel farther but faster; below ~15° the trade balances to within a fraction of a percent). Galileo reportedly noticed this in a swinging cathedral lamp, timing it against his pulse.
Rearranged, it is a g-meter
The T² is your accuracy lever and your accuracy trap at once: timing errors get squared. The fix is old and beautiful — time twenty swings, not one, and divide by twenty. Your ±0.2 s reaction error stays ±0.2 s, but is now spread over ~28 s of measurement: a one-percent error instead of twenty.
What you need
- A string, 1 m or longer — thin and non-stretchy (shoelace, kitchen twine)
- A small heavy weight — a nut, a padlock, a ring of keys
- A tape measure and your phone’s stopwatch
- Somewhere to hang it: a door frame, a shelf edge, a broom across two chairs
The technique that makes it accurate
- Measure L from the pivot to the centre of the weight — the largest single error source.
- Keep the swing narrow: a hand-width of displacement on a metre of string.
- Start counting at “zero” (not one!) as it passes an extreme, and time 20 full periods.
- Repeat three times; if a count disagrees wildly, you miscounted — drop it and redo.
Done with care, a shoelace and a phone deliver g to within one or two percent of 9.81 m/s² — a measurement Galileo would have traded years for, performed before breakfast.
From Huygens (1656) to the 1930s, the best clocks on Earth were pendulums — the formula you are using kept trains scheduled and longitude found for three centuries. And because T depends on g, a pendulum clock taken up a mountain runs measurably slow: your experiment, run in reverse, is a gravity survey.
⚗️ Lab — Digital Twin: The Pendulum
Rehearse the experiment before doing it for real — and see why the rules matter.
- Time 20 swings with the on-screen stopwatch, then check your g against the slider’s true value.
- Push the amplitude past 40° and watch the small-angle formula start to lie.
- Set lunar gravity: same string, swings six times… well, √6 times slower. T² sees it.
Work it by hand — 0 / 3
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±1% count.