# Cavendish Experiment Simulator - measure G, torsion balance, optical lever

> Virtual Cavendish torsion balance: damped oscillation, optical-lever readings, turning-point analysis, G with corrections, weighing the Earth, Coulomb balance.

- URL: https://qilingtools.com/en/physics/cavendish
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

The Cavendish experiment in your browser: two small lead spheres on a rod hang from a thin fibre, and swinging the large spheres to position A or B makes the rod twist in a damped oscillation under a tiny gravitational torque, which a mirror turns into a light spot moving several centimetres on a scale. You record the turning points of the spot as in the lab; the tool finds the equilibrium and period for each position, applies G = 2π²Lr²θ/(MT²) with corrections for the far spheres, the rod’s inertia and damping, compares with CODATA and weighs the Earth. A second tab is Coulomb’s torsion balance, repeating his 1785 test of the inverse-square law for charges.

## How to use

1. Pick the teaching balance or Cavendish’s original, press Start and speed up time to watch the spot
2. With the spheres at A record turning points for 4–5 periods, then swing to B and keep recording
3. Read the equilibria and periods, the resulting G with corrections, the CODATA comparison and the Earth’s mass
4. Turn on noise, drift or the blind run to see where real experimental errors come from
5. Switch to the Coulomb balance, turn the torsion head, halve charges by contact and fit the force exponent

## Features

- Torque from the exact positions of all four sphere pairs, RK4 damped torsional motion, optional noise and fibre drift
- Optical lever: a mirror turn φ moves the spot D·tan 2φ, with an exaggerated rotation view
- Automatic or manual turning points, equilibrium (s₁ + 2s₂ + s₃)/4, period and logarithmic decrement
- Ideal formula for G plus far-sphere β, rod inertia and damped-period corrections
- Blind mode with a hidden true G, and weighing the Earth from your G
- Coulomb balance with torsion head, charge sharing and exponent fit (small-angle vs exact geometry)

## FAQ

### How does a torsion balance measure G?

The large spheres pull on the small ones with a torque 2·GMm/r²·(L/2), balanced by the fibre’s κθ. κ need not be measured separately because the oscillation period is T = 2π√(I/κ) with I = 2m(L/2)². Eliminating κ gives G = 2π²Lr²θ/(MT²), so you need only the deflection θ, the period T and L, r, M; the small mass m cancels.

### Why swing the large spheres from A to B?

The fibre’s zero position is hard to know and drifts slowly. At the two mirror positions the torque is equal and opposite, so the difference between the two equilibria is exactly 2θ whatever the zero is. The optical lever doubles it again, giving a spot shift S = 4Dθ, and many systematic errors cancel in the difference.

### What is the turning-point method?

The amplitude decays all the time, so averaging two neighbouring turning points is biased towards the earlier one. Three consecutive turning points combined as (s₁ + 2s₂ + s₃)/4 cancel a linearly decaying amplitude, so the equilibrium hardly depends on damping and you need not wait for the motion to stop (a real balance takes an hour or two). Neighbouring turning points are half a period apart.

### Which corrections are needed and how big are they?

The ideal formula counts only the near pair. The far large sphere also attracts each small sphere, giving an opposing torque fraction β = r³/(r² + L²)^{3/2}, about 7% for a teaching balance. The rod and mirror add moment of inertia, about 3–4%, and damping makes the measured period slightly longer than the undamped one. With all three the simulated data come back to CODATA within about 0.1%. Real experiments also handle the rod’s own attraction and sphere offsets.

### Why is it called weighing the Earth?

Surface gravity is g = GM⊕/R², and g and R were already known; only G was missing. With G, M⊕ = gR²/G ≈ 6×10²⁴ kg and the mean density ρ = 3g/(4πGR) ≈ 5.5 g/cm³, far denser than surface rock, which points to a dense core. Cavendish found 5.48 g/cm³ in 1798, within about 1% of today’s 5.51.

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