A small block is connected to a mass less rod, which is turn is attached to a spring of force constant `k` as shown in Fig. The blcok is displaced down slightly, and left free. Find time period of small oscillations.

Explanation

Let angular displacement of the block is `theta`, then extension of the spring will be 'a theta'. If `F` is the force in spring, then restoring torque, `tau_(rest) =- Fa =- K (a theta) xxa`
`[ :' F = Kx = Ka theta]` and `=(tau_(rest))/(I) = (Ka^(2))/(I) (-theta)`
Now comparing with `alpha =- omega^(2) theta`, we get
`omega = sqrt((Ka^(2))/(I))` and `T = 2pi sqrt((I)/(Ka^(2)))`
Here `I = mb^(2) :. T =2pi sqrt((mb^(2))/(Ka^(2))) = 2pi(b)/(a) sqrt((m)/(K))`