The time period of mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be…….

Options

A. `2T`
B. `(T)/(4)`
C. `2`
D. `(T)/(2)`

Answer

D

Explanation

Periodic time of mass suspended to whole spring
`T= 2pi sqrt((m)/(k)), 2pi, k` is constant.
If the spring is cut into four parts, the force constant of each part is different.
Suppose, length of each part `l_(1)` and force constant `k_(1)` and length of whole spring is l and force constant k
`therefore k_(1)l_(1) = kl`
`therefore k_(1) (l/4)= kl`
`therefore k_(1)= 4k`
Periodic time of each part of spring `T.= 2pi sqrt((m)/(k_1))`
`therefore T. = 2pi sqrt((m)/(4k))" ""........"(1)`
and periodic time of whole spring
`T= 2pi sqrt((m)/(k))" ""..........."(2)`
`therefore (T.)/(T)= (1)/(sqrt(4))= (1)/(2)`
`therefore T.= (T)/(2)`.