A block of mass `m` rigidly attached with a spring `k` is compressed through a distance `A`. If the block is released, the period of oscillation of the block for a complete cycle is equal to

Options

A. `(4pi)/(3)sqrt((m)/(k))`
B. `(pi)/(sqrt(2))sqrt((m)/(k))`
C. `(2pi)/(3)sqrt((m)/(k))`
D. None of these

Explanation

The period of motion from `A` to `O` is equal to quarter of the time period `T` of oscillation of mass spring system.
`rArr t_(AO)=(T)/(4)=(1)/(4)[2pisqrt((m)/(k))]=(pi)/(2)sqrt((m)/(k))`
Since the motion is simple harmonic
`OB=OAsin"(2pi)/(T)t_(OB)`, where `t_(OB)` is the time of motion from `O` to `B`.
`rArr t_(OB)=(T)/(2pi)sin^(-1)"(A//2)/(A)=(T)/(2pi)((pi)/(6))`
`=(T)/(12)=(2pi)/(12)sqrt((m)/(k))=(pi)/(6)sqrt((m)/(k))`
`:. ` The total time of motion for a complete cycle `=t=2(t_(AO)+t_(OB))`
`rArr t=2[(pi)/(2)sqrt((m)/(k))+(pi)/(6)sqrt((m)/(k))]=(4pi)/(3)sqrt((m)/(k))`.