Cavendish Torsion Balance 3D

What the experiment measures

Newton's law says two masses attract with F = G M m / r². For a lead sphere of 1.5 kg and a 15 g ball 4.6 cm apart the force is about 7×10-10 N, so the arm of a torsion balance is used as an extremely soft spring. Two small balls sit at the ends of a light rod of half-length l hung by a fibre; two big spheres sit beside them. The gravity torque twists the fibre until the fibre's restoring torque balances it.

Fibre, period and equilibrium

The torsion constant of a round fibre is k = π Gs d&sup4; / (32 L) (shear modulus Gs, diameter d, length L). The arm obeys I θ'' = τG(θ, φ) - k(θ - θ0) - bθ' + τd with I = 2 m l² plus the rod and mirror. Without damping the period is T = 2π √(I/k), so a measured period gives k = 4π² I / T² without knowing Gs or d. For damping ratio ζ the true values are ω0² = ωd² + (c/2)², which the page uses with the measured decay.

At rest k(θeq - θ0) = τG(θeq). The textbook estimate of the torque is τ = 2 G M m l / r0², so G = k Δθ r0² / (2 M m l), with Δθ half the change when the big spheres are swapped from position A to B (this cancels the unknown zero θ0 and doubles the signal).

Exact geometry

The page sums all four sphere pairs as point masses (Newton's shell theorem for the spheres): τG = G M m Σi,j (si × rij)z / |rij|³. The far spheres pull the other way and the line between the centres is not exactly tangential, so for the default geometry the textbook formula is about 7 % too high; the "G (exact geometry)" readout removes that error.

Turning-point analysis

The arm takes many minutes to settle, so the equilibrium is estimated from three successive extremes x1, x2, x3 of a damped swing: xeq = (x1x3 - x2²) / (x1 + x3 - 2x2). The period is the time between x1 and x3.

Optical lever

A mirror turned by θ turns the reflected beam by 2θ, so a scale at distance D sees the spot move about 2θD. The page traces the actual reflected ray (the laser is a little off-axis), not the small-angle formula.

The three numerical checks

(1) A free swing of the arm without gravity against 2π √(I/k). (2) The settled deflection of the time integration against a bisection root of k(θ - θ0) = τG(θ). (3) A complete simulated measurement of G (both positions, turning-point equilibria, k from the period) against the value of G that drives the model.

Honest simplifications

Spheres are point masses; the arm and fibre have a single torsional mode (no pendulum swing, no anelastic creep or fibre drift); damping is linear and viscous; the "disturbance" is a correlated random torque standing for convection currents, not a computed flow; the thermal (Brownian) torque is far smaller than the signal and is not included; gravitational pull of the walls, case and observer, and the electrostatic and magnetic effects that limit real experiments are not modelled. G = 6.6743×10^-11 m³ kg^-1 s^-2 is used as the model's true value (CODATA 2018). Dimensions and the 1798 preset are approximate.