A pendulum is a physical object thatundergoes small angular oscillations under the restoring force of gravity. In its simplest form, a pendulum consists of a point mass at the end of a massless string attached to a suspending point, or a point mass attached to a massless rod that is set to pivot about a fixed point. if allowed to swing freely, the mass exhibits simple harmonic motion about its equilibrium position, with natural frequency of ω
Here g is the acceleration due to gravity of 9.8 m/sec and L is the length of the string or rod between the pivot point and the mass.
A more nuanced model is to view the pendulum as an extended mass having moment of inertia I pivoting about a fixed point that is not the center of mass, and which hangs downward from the point of suspension under the influence of gravity. The distance between the pivot point and the center of mass is L , and the mass of the object is m. Small excursions from the equilibrium position feel a restoring force due to torque from the force of gravity.In this case the natural frequency ω is given by
The restoring force is not truly linear in displacement, since the sine of the angle of displacement enters in, but if the displacements are small the small angle approximation makes the restoring force very nearly linear in angle, and the equations of motion for the pendulum become identical to those of a mass on a spring.