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 frequencyof small oscillations.

A block of mass m hangs by means of a string which goes over a pulley

1.	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 frequencyof small oscillations.
Answer» Solution :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)))`