Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it.
A simply method for solving this problem is energy method. Initial extension of the spring in the mean position is
During oscillation, at any position x below the equilibrium position, let the velocity of m be v and angular velocity of the pulley be . If r is the radius of the pull, then .
At any instant, total energy for the system executing SHM is constant. Therefore
Substitute the value of
Taking derivative of both sides with respect to t,
The above equation is a second order differential equation of a system executing SHM.