Find the time period of mass M when displaced from its equilibrium positon and then released for the system shown in Fig 14.10.

Let the spring be extended by when loaded with mass m.

Due to extension there will be a restoring force applied by the spring on the pulley

Now let T be the tension on the pulley,

∴ where k’ is the spring constant.

pulley experiences two tension force due the two strings,

∴we get,

-----(1)

Where k is the spring constant.

Now let us pull the mass by , as we pull it by the spring extends by a distance of 2 because the string on one side is inextensible.

∴ the new tension force

pulley experiences two tension force due the two strings,

∴the net force acting,

-----(2)

Now putting equation (1) in (2) we get,

∴ we see that net force F is proportional to the the system performs a simple harmonic motion.

Comparing the above obtained net force with

We find that,

And for S.H.M the time period T is given as,

Hence the time period for the given system is,