A swing that never quite stops
Once the great weights swing into place, the arm doesn't just move to a new position and stop. It overshoots and oscillates around the new rest point with a period of about 15 minutes, losing a little amplitude each swing. A run could take hours.
Cavendish didn't wait for it to settle. He wrote down the scale reading at each extreme of the swing and took the rest point as the midpoint. Because each swing is a little smaller than the last, he averaged in a way that corrects for the decay:
x₁, x₂, x₃ are three successive turning points. The green ticks on the chart apply this to the simulated swing. They land on the true rest line almost immediately, long before the motion dies out.
Minutes of the run, one simulated minute per second of playback. Click the chart to jump the model to that moment.
Lead, mahogany and a very thin wire
The design came from the geologist John Michell, who built a torsion balance for the purpose but died in 1793 before he could use it. The apparatus passed to Cavendish, who rebuilt much of it and ran the experiment in 1797–98 in an outbuilding at his house on Clapham Common. Select a part to see it in the model.
| Part | Specification | Why it matters |
|---|
From a twist to the constant of gravity
The experiment balances two torques: the gravitational pull of the big balls and the springback of the twisted wire. You can't calibrate a wire that weak against a known force, so the arm's own swing period stands in for the calibration.
- Gravity between a pair of balls. Big mass M, small mass m, centres a distance r apart.F = G M mr²
- Torque on the arm. Two pairs, each pulling at a lever arm of L/2, so the torques add.τ = 2 · F · L2 = F L
- The wire pushes back in proportion to how far it's twisted. κ is the wire's torsion constant.τ = κ θ
- The period measures κ. A torsion pendulum swings with period T. The arm's moment of inertia is dominated by the two small balls.T = 2π √(I / κ), I = 2m(L/2)² = ½ m L² ⇒ κ = 2π² m L²T²
- Set the torques equal and solve for G. The small mass cancels. Writing the twist as the linear shift d of each small ball (θ = 2d/L), the arm length cancels too.G = 2π² L r² θM T² = 4π² r² dM T²
- Weigh the Earth. Surface gravity g = G M⊕ / R². Divide the Earth's mass by its volume:ρ⊕ = 3 g4π G R
Cavendish didn't write G at all. He compared the big ball's pull directly with the Earth's pull on the same small ball and reported density relative to water. Physicists named and extracted G from his numbers much later.
Run the numbers
Defaults use his first wire's 15-minute period and a deflection that fits his published result. Change any value.
This ignores small corrections Cavendish did make: the far big ball pulling each small ball the wrong way, the pull of the rods, the arm and the case. Together they shift the answer by a few percent.
Fighting forces bigger than gravity
The signal is about 1.5 × 10⁻⁷ newtons, roughly one fifty-millionth of the small balls' weight. Almost anything else in the room could swamp it, and most of Cavendish's paper is about ruling these effects out.
Heat and air currents
A temperature difference of a fraction of a degree across the case sets up convection that pushes the balls harder than gravity does. That is why the room was sealed, the lamps and observer stayed outside, and the weights were moved by rope. He tested this directly by warming the weights and watching the arm drift.
Drift of the wire
A freshly loaded wire slowly relaxes, so the zero point creeps. Reversing the weights and measuring the shift between the two sides doubles the signal and cancels a steady drift.
Magnetism
Iron in the fittings or magnetic impurities could pull like gravity does. He checked the size of any magnetic effect and found it negligible next to the gravitational one.
Everything else attracts too
The rods holding the weights, the mahogany case and the arm itself all have mass. He estimated each pull and corrected for it, along with the far ball on each side.
An arithmetic slip
His paper reports a mean density of 5.48 times that of water. Averaging his own results gives 5.448. The error was spotted decades later.
The stiffer wire
Partway through, he switched to a stiffer wire. The period dropped from about 15 to 7.5 minutes and the deflection shrank to around 0.03 inches. The results agreed, a check that the answer didn't depend on the wire.
Within one percent, in 1798
G is still the least precisely known of the fundamental constants, and the best modern measurements disagree with each other by more than their stated uncertainties. Many of them still use a torsion balance descended from Michell's design, now with fibres of tungsten or fused silica, laser readout and vacuum chambers.