Cavenda

Cavenda peers through a telescope at a twitching dumbbell, with the whole Earth balanced on the other end of the beam.

The Physics

A torsion balance hangs a light arm with a small lead ball at each end. Large lead spheres brought close pull on the balls and twist the arm by a tiny angle; from that twist and the swing period you can work out the Earth's mean density. The gravity, twist and period relations below are the standard torsion-balance equations.

F = G M m / r², κθ = F L, T = 2π √(I / κ)

• F: gravitational force (N)
• G: gravitational constant (m³ kg⁻¹ s⁻²)
• M, m: large and small masses (kg)
• r: centre-to-centre distance (m)
• κ: torsion constant (N·m/rad)
• θ: twist angle (rad)
• L: lever arm (m)
• T: swing period (s)
• I: moment of inertia (kg·m²)

Henry Cavendish's paper was read to the Royal Society on 21 June 1798, using a torsion balance devised by John Michell. He printed Earth's density as 5.48 times water; his own data give 5.448, and today's value is about 5.51.

Honest caveat: He reported Earth's density, not G. Today's G (CODATA 2022) is 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻².

On the shirt

Front: Cavenda, the pixel-art character, with the title. Back: the three torsion-balance equations, Earth's density then and now, today's G and the 1798 paper. Printed on a unisex tee.