Figure (a) shows a spring of force constant k clamped rigidly at one end and a mass attached to its free end. The spring is stretched by a force F at its free end. Figure (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig.(b) is stretched by the same force F.
(i) What is the maximum extension of spring in the two cases?
If the mass in (a) and the two masses in (b) are released free, what is the period of oscillation in each case ?

Explanation

For Fig. (a) there is a force of reaction Fat the clamped end. Hence the two systems are identical and hence the extension is given by
`x = F/k`
(ii) Period of oscillation for Fig. (a) `T = 2pi = sqrt(m/k)`
The centre of spring Fig. (b) can be assumed to be fixed as it does not move. Each end experiences a restoring force and hence the total restoring force is doubled. So the force constant becomes 2k.
The period `T = 2 pi sqrt(m/(2k))`