spring of force constant k clamped rigidly at onoe end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure 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) The maximum extension of the spring in both the cases will be `=F//k`, where k is the spring constant of the spring used.
(b) In figure , if x is the extension in the spring, when mass mis returning to its mean position after being released free, then restoring force on the mass is `F=-kx, i.e., Fpropx`
As, this F is directed towards mean position of the mass, hence the mass attached to the spring will execute SHM.
Here, spring factor `=` spring constant `=`k.
inertia factor `=` mass of the given mass `=` m
As, time period, `T=2pisqrt((i n ert i a fact o r)/(spri ng fact o r)) ` `:. T=2pisqrt((m)/(k))`
In figure, we have a two body system of spring constant k andn reduced mass, `mu=(mxxm)/(m+m)=(m)/(2)`
Here, inertia factor`=m//2` and spring factor `=k`
`:.` time period, `T=2pisqrt(((m//2))/(k))=2pisqrt((m)/(2k))`