The string and the spring depicted in the figure are light. Determine the time period of the mass m , assuming the pulley has a radius r and a moment of inertia I about its axis, and the string does not slip over it.

Step-by-step solution

Let the mass m be displaced downwards by a small distance x from its equilibrium position. The velocity of the mass is v = x . Since the string is inextensible and does not slip, the pulley rotates by an angle = x/r , and its angular velocity is = v/r . The spring is stretched by an additional amount x . The total mechanical energy of the system is: E = 1 2 mv^2 + 1 2 I ^2 + 1 2 k(x₀ + x)^2 - mgx where x₀ is the initial extension of the spring in equilibrium, satisfying kx₀ = mg . Substituting = v/r and expanding t