Pendulum and Spring

For small swings, T = 2π√(L/g), where L is the length from the pivot to the centre of the bob and g is the acceleration due to gravity. A 1 m pendulum on Earth takes about 2 seconds for one full swing.

It does not depend on the mass of the bob. For small amplitudes it hardly depends on the amplitude either, which is why pendulums keep good time. For large swings the period gets longer: at 60° it is about 7% longer than 2π√(L/g).

Simple harmonic motion is a back-and-forth motion where the restoring force is proportional to the displacement and points toward the middle, F = −kx. The position follows a sine wave, x = A sin(ωt + φ), with a period that doesn't depend on the amplitude.

T = 2π√(m/k), where m is the mass and k is the spring constant (stiffness). A heavier mass or a softer spring makes it bounce more slowly. Gravity does not affect the period.

A seconds pendulum has a period of exactly 2 seconds, so each swing from one side to the other takes 1 second. On Earth its length is about 99.4 cm. Old grandfather clocks use one.

Hooke's law says that the force needed to stretch or squeeze a spring is proportional to how far it is stretched: F = kx, where k is the spring constant. The spring pulls back with an equal and opposite force, −kx.