The simple pendulum

Galileo is said to have timed a swinging church chandelier against his pulse and found its motion periodic. A stone on an inextensible thread about 100 cm long swings with a period of about two seconds.

The idealised simple pendulum is a small bob of mass m on a massless, inextensible string of length L, fixed at a rigid support and swinging in a vertical plane.

The bob feels just two forces: its weight mg and the string's tension T. Split the weight into mg cos θ, acting along the string, and mg sin θ, acting at right angles to it along the arc.

The bob moves on a circle of radius L, so it has a radial acceleration supplied by T − mg cos θ and a tangential acceleration supplied by mg sin θ. The radial force passes through the support and gives no torque about it.

The restoring torque about the support is τ = −L(mg sin θ), negative because it acts to reduce θ. With τ = Iα, this gives α = −(mgL/I) sin θ.

sin θ = θ − θ³/3! + θ⁵/5! − …, with θ in radians. For small θ, sin θ ≈ θ, and the two stay close even up to about 20°.

Then α = −(mgL/I)θ, which has the form of the SHM equation for angular displacement, with ω = √(mgL/I) and T = 2π√(I/mgL).

For a massless string I = mL², so T = 2π√(L/g). The period depends only on the length and on g; it does not depend on the mass of the bob, nor (for small swings) on the amplitude.

A seconds pendulum has T = 2 s (one second each way). From L = gT²/4π² with g = 9.8 m s⁻², L = 9.8 × 4/(4π²) ≈ 1 m.