A block of mass m hangs by means of a string which goes over a pulley of mass m and moment of inertia l, as shown in the diagram. The string does not realtive to the pulley. Find the frequency of small oscillations.

Explanation

Suppose the block is depresed by x. The pulley (owing to the constant) is depresed by `x//2`. Suppose the tension in the string are T and T' on both sides . We can write

for block `mg - T = mx` (i)
For pulley, `T + T' + mg - k (x + x_(0)) = m (x)/(2)` (ii)
The angular acceleration of the pulley ,
`alpha = (x//2)/(2 R)` (iii)
`(T - T' ) R = l (x)/(2 R)` (iv)
From the Eqs. (i), (ii), (iii) and (iv), we get
`3 mg - k (x + x_(0)) = ((5 m)/(2) + (l)/(2 R^(2))) x`
The frequency of small oscllation
`f = (1)/(2 pi) sqrt((k)/((5m)/(2) + (l)/(2 R^(2)))`