In the arrangement shown in the figure, a pulley of mass M is suspended from a light spring of stiffness constant k and a mass m is suspended form a string tightly wound on the pulley. If the mass m is given a small downward displacement and released, the time period of its small oscillations is (Assume that the string does not slip on pulley)

Step-by-step solution

Let the initial elongation of the spring in the equilibrium state be e. Then K e=M g+2 T₀ aligned & T₀=m g & K e=M g+2 m g aligned When the mass m is pulled and released, let the instantaneous displacement of m be x and that of pulley be x / 2 . Since the total energy during SHM remains conserved. V= speed of m at that instant 1 2 m v^2+ 1 2 M ( v 2 )^2+ 1 2 I ^2+ 1 2 K ( x 2 +e )^2-M g x 2 -m g x= constant I= M R^2 2 and for no slipping V 2 +R =v W= V 2 R 1 2 [m v^2+ M 4 v^2+ M 8 v^2 ]+ 1 2 K ( x 2 +e )^2-M g x 2