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

In figure (a), on applying force F to the end of a spring, an extension l is produced
Therefore, F = kl
`l = (F)/(k)`
in figure (b), extension` = (l)/(2)`
and `F = 2k x = (2kl)/(2)`
` l = (F)/(k)`
(b) For one block system, if mass is released free
Then, `F = ma = m (d^(2)x)/(dt^(2)) = -kx`
`(d^(2)x)/(dt^(2)) = -(k)/(m)x = -omega^(2)x`
`omega^(2) = (k)/(m)`
`omega = sqrt((k)/(m))`
`therefore` Time period `T = (2pi)/(omega) = 2pi sqrt((m)/(k))`
for two block system.
`F = m(d^(2)x)/(dt^(2)) = -2kx`
`(d^(2)x)/(dt^(2)) = (-2k)/(m) x = -omega^(2)x`
`omega = (2k)/(m)omega, omega = sqrt((2k)/(m))`
`T = (2pi)/(omega) = 2pisqrt((m)/(2k))`