Time Period of a Simple Pendulum (HL IB Physics)

• A simple pendulum consists of a string and a bob at the end
  • The bob is a weight, generally spherical and considered a point mass
  • The bob moves from side to side
  • The string is light and inextensible remaining in tension throughout the oscillations
  • The string is attached to a fixed point above the equilibrium position
• The time period of a simple pendulum for small angles of oscillation is given by:

• Where:
  • T = time period (s)
  • L = length of string (from the pivot to the centre of mass of the bob) (m)
  • g = gravitational field strength (N kg^-1)

A simple pendulum

• The time period of a pendulum does depend on the gravitational field strength, meaning its period would be different on the Earth and the Moon

Small Angle Approximation

• This formula for time period is limited to small angles (θ < 10°) and therefore small amplitudes of oscillation from the equilibrium point
• The restoring force of the pendulum is the weight component acting along the arc of the circle towards the equilibrium position
• It is resolved to act at an angle θ to the horizontal x 
• When considering SHM because of small angle approximation it is assumed the restoring force acts along the horizontal
• So sin θ ≅ θ 

Forces on a pendulum when it is displaced. Assuming θ < 10°, the small angle approximation can be used to describe the time period of a simple pendulum such as this.