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

How to use Cavendish Experiment Simulator

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

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.