A rigid rod of mass m with a ball of mass M attached to the free end is restrained to oscillate in a vertical plane as shown in the figure. Find the natural frequency of oscillation.

Answer

`(1)/(2 pi) sqrt((3k)/(27 M + 7m))`

Explanation

At equlibrium position deformation of the spring is `x_(0)`
`kx_(0) = (1)/(4) = Mg ((3)/(4) l) + mg ((1)/(4))`
When the rod is further rotated through an angle `theta` from equlibrium position, the restoring tarque.
`tau = - [k(x+ x_(0)) (1)/(4) cos theta - Mg ((3)/(4)) L cos theta] - mg ((L)/(4)) cos theta`
`= - [k(x+ x_(0)) (1)/(4) - Mg ((3)/(4)) L - mg ((L)/(4))] cos theta`
For small `theta, cos theta ~~ 1`
`tau = - (kl)/(4) x` `implies l alpha = - (kl^(2))/(4) theta`
`l = m ((3)/(4) L) ^(2) + (mL^(2))/(12) + m ((L)/(4))^(2)`
`f = (1)/(2 pi) sqrt((3k)/(27 M + 7m))`