# Simple Pendulum

A simple pendulum is a small spherical bob suspended by a long cotton thread held between the two halves of a clamped split cork in a stand. The bob is considered a point mass and the string is taken to be in-extensible. The Pendulum can oscillate freely about the point of suspension.

When the pendulum is displaced through a small distance from its equilibrium position and then let free, it executes angular oscillations in a vertical plane about its equilibrium position. The distance between the point of suspension and the centre of gravity of the bob defines the length of the pendulum.

The forces acting on the bob of the pendulum in the displaced position are:

• the weight of the bob mg vertically downwards
• tension in the string T acting upwards along the string

The weight mg is resolved in two components: (a) mg cosθ along the string but opposite to T and (b) mg sinθ perpendicular to the string.

The component mg cosθ balances the tension T and the component mg sinθ produces acceleration in the bob in the direction of the mean position. The restoring force is mg sinθ.

For small displacement x of the bob, the restoring force is

F = mgθ = mg x/l

The force per unit displacement k = mg/l

ω = √(k/m) = √(g/l)

2π/T = √(g/l)

T = 2π√(g/l)