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.*

*Therefore,*

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