# Fundamental Physics/Motion/Pendulum Oscillations

## Pendulum Oscillations

Animation of a pendulum showing the velocity and acceleration vectors.

A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolated system using the following assumptions:

• The rod or cord on which the bob swings is massless, inextensible and always remains taut;
• The bob is a point mass;
• Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
• The motion does not lose energy to friction or air resistance.
• The gravitational field is uniform.
• The support does not move.

The differential equation which represents the motion of a simple pendulum is

${\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\sin \theta =0}$ Eq. 1

where g is acceleration due to gravity, l is the length of the pendulum, and θ is the angular displacement.