A simple pendulum of length L and mass m has a spring of force constant k connected to it at a distance h below its point of suspension. Find the frequency of vibrations of the system for small values of amplitude.

Explanation

Let pendulum is given a small angular displacement `theta`,

Torque about O, `tau=mgLsintheta+kxh`
`=mgLsintheta+kh^2tantheta`
For small `theta`, `sin theta`_-^~tantheta`_-^~theta`
`tau=-(mgL+kh^2)theta`
`Ialpha=-(mgL+kh^2)theta`
`alpha=-((mgL+kh^2)/(mL^2))theta=-omega^2theta`
`f=(1)/(2pi)sqrt((mgL+kh^2)/(mL^2))`