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Figure shows a system consisting of a massless pulley, a spring of force constant k and a block of mass m. If the block is slightly displaced vertically down from its equilibrium position and released, find the period of its vertical oscillation in cases (a) , (b) and (c).

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(a) In equilibrium, kx0 = mg....(i)

When further depressed by an amount x, net restoring force (upwards) is,

(b) In this case if the mass m moves down a distance x from its equilibrium position, then pulley will move down by x/2. So, the extra force in spring will be kx/2. Now, as the pulley is massless, this force kx/2 is equal to extra 2T or T = kx/4. This is also the restoring force of the mass. Hence,

(c) In this situation if the mass m moves down a distance x from its equilibrium position, the pulley will also move by x and so the spring will stretch by 2x. Therefore the spring force will be 2kx. The restoring force on the block will be 4kx. Hence,

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