Simple Pendulum
The mathematics of swinging: length, gravity and period
Summary
A simple pendulum is a small mass hanging from the end of a light string that does not stretch. For small angles, its period is given by T = 2π√(L/g): the time of a swing does not depend on the mass, only on the length of the string and the acceleration due to gravity. Drag the bob and let go; change the length, the planet and the angle to see for yourself where the formula holds and where it starts to drift.
In depth
As the story goes, at the age of 19 Galileo Galilei timed the swings of a chandelier in Pisa Cathedral against his own pulse and noticed that each swing took the same time, however wide it was (isochronism). This is only true for small angles. With the small-angle approximation (sin θ ≈ θ), the equation of motion becomes θ¨ + (g/L)·θ = 0, which is a simple harmonic oscillator. The angular frequency is ω = √(g/L) and the period is T = 2π√(L/g). At large angles the real period gets longer: at 90° it is 18% longer than the formula predicts. Christiaan Huygens invented the pendulum clock in 1656, improving clock accuracy from minutes to seconds; in 1851 Léon Foucault used a giant pendulum to show that the Earth rotates.
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