A block of mass m compresses a spring iof stifffness k through a distacne `l//2` as shown in the figure .If the block is not fixed to the spring the period of motion of the block is

Options

A. `2pisqrt((m)/(k))`
B. `(pi+4)sqrt((m)/(k))`
C. `(1+pi)sqrt((m)/(k))`
D. None of these

Explanation

The period of oscillation `=2pisqrt((m)/(k))`
`rArr ` The period of motion till the block is in contact with the spring is
`t=(T)/(2)=pisqrt((m)/(k))` then it leaves the spring with a speed `v=omegaA`
`v=(sqrt((k)/(m)))((l)/(2))`
Then it moves with constant velocity `v` for a distance `D=l+l=2l`
`rArr` The corresponding time of motion `=t_(2)=2l//v`
`rArr t_(2)=(2l)/((l)/(2)sqrt((k)/(m)))=4sqrt((m)/(k))`
`:. ` The time period of motion `=t=t_(1)+t_(2)=pisqrt((m)/(k))+4sqrt((m)/(k))`
`=sqrt((m)/(k))[pi+4]`.