Page History

{excerpt} A pendulum is a physical object thatundergoes small angular oscillations under the restoring force of gravity. {excerpt} 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{*}

\\
{latex}\begin{large} \[ \omega = \sqrt{\frac{g}{L}} \]\end{large}{latex}
\\
Here [g|gee] 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|gravity (near-earth)]. 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|torque (single-axis) ] from the force of gravity.In this case the natural frequency {*}ω{*} is given by

 \\
{latex}\begin{large} \[ \omega = \sqrt{\frac{mLg}{I}} \]\end{large}{latex}
\\





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].