The Torsion Pendulum: The Twist That Weighed the Earth

The rotational twin of a spring

Hang a disk from a thin wire or fibre and twist it. The wire resists, producing a restoring torque proportional to the twist angle — exactly the rotational counterpart of Hooke's law, F = -kx. Every term in the ordinary spring-mass-damper equation has a direct rotational analogue: mass becomes moment of inertia I, spring constant becomes the torsion constant κ, and position becomes angle θ.

linear:      m·x'' + c·x' + k·x = 0
rotational:  I·θ'' + b·θ' + κ·θ = 0

I  = moment of inertia of the suspended disk about the wire's axis
b  = rotational damping (air resistance, internal friction of the wire)
κ  = torsion constant of the wire — its stiffness against twisting

The period, and why gravity is nowhere in it

For the undamped case, the angular frequency and period fall out immediately by analogy with the linear oscillator, ω = √(k/m):

ω₀ = √(κ / I)
T  = 2π √(I / κ)

Notice what's missing: g. A swinging pendulum's period depends on gravity because gravity is its restoring force; a torsion pendulum's restoring force comes entirely from the wire's elasticity, so its period is the same on Earth, on the Moon, or floating in the middle of the ISS. That property — a period set by mechanical stiffness rather than local gravity — is exactly why mechanical wristwatches historically used a torsion-like balance wheel instead of a swinging pendulum: a pocket watch that tips over shouldn't change its timekeeping.

Damping and the decay envelope

With damping included, the same underdamped/critical/overdamped classification from the linear oscillator applies, governed by the rotational damping ratio ζ = b / (2√(Iκ)). A lightly damped torsion pendulum (small b — a thin wire in still air) oscillates for a long time with an exponentially decaying envelope e^(-ζω₀t), swing after swing losing the same fraction of its remaining amplitude — precisely the behaviour visible in the live demo above, and the same signature used to measure a wire's material properties from a recorded ringdown.

Cavendish, 1798: measuring the unmeasurable

The torsion pendulum's real claim to fame is Henry Cavendish's 1798 experiment. He suspended a light horizontal rod with two small lead spheres at its ends from a very fine torsion wire, then brought two much larger lead spheres close to them. The gravitational attraction between the sphere pairs — a force too weak to notice by almost any other method available at the time — twisted the wire by a small but very measurable angle, because κ could be made astonishingly small for a fine enough fibre.

From the deflection angle, the known masses and separations, and the pendulum's measured oscillation period (which gives κ via the formula above), Cavendish extracted the gravitational constant G to within about 1% of the modern value — and from G, calculated the mass and mean density of the Earth itself, becoming the first person in history to actually weigh the planet.

Frequently asked questions

Try it live

Everything above runs in your browser — open Torsion Pendulum and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.