Figure a) shows a spring of force constant k clamped rigidly at once end and a mass m attached to its free end. A force F applied at the free end stretches the spring. 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 figure is stretched by the same force F.
(a) What is the maximum extension of the spring in the two cases ?
(b) If the mass in figure and the two masses in figure are released free, what is the period of oscillation in each case?

Explanation

(a) Let y be the maximum extension produced in the spring in Fig. (a)
Then F = ky (in magnitude) `therefore y = (F)/(k)`
(b) ln fig. (a), F = - ky
`implies ma=-ky implies a=-(k)/(m)"y "therefore omega^(2)=(k)/(m)i.e., omega=sqrt((k)/(m))`
Therefore, period `T = (2pi)/(omega)=2pisqrt((m)/(k))`
In fig. (b), we may consider that the centre of the system is O and there are two springs each of length `(l)/(2)` attached to the two masses, each m, so that K is the spring factor of each of the springs.
Then, K = 2k
`therefore T = 2pi sqrt((m)/(K))`
`=2pi sqrt((m)/(2k))`