One end of a spring of force constant K is fixed to a vertical wall and the other to a body of mass m resting on a smooth horizontal surface. There is another wall at a distance `x_0` from the body. The spring is then compressed by `3x_0` and released. The time taken to strike the wall from the instant of release is (given `sin^-1((1)/(3))=((pi)/(9))`)

Options

A. `(pi)/(6)sqrt((m)/(K))`
B. `(2pi)/(3)sqrt((m)/(K))`
C. `(pi)/(4)sqrt((m)/(K))`
D. `(11pi)/(18)sqrt((m)/(K))`

Answer

D

Explanation

When spring is compressed by `3x_0`. Amplitude `A=3x_0` The time taken from extreme compressed position to mean position `t_1=(T)/(4)`
If time taken `(t_2)` from mean position to `x=x_0` is given
`x=Asin(2pit_2)/(T)impliesx_0=3x_0sin(2pit_2)/(T)`
`sin((2pit_2)/(T))`
`T=(1)/(3)implies(2pit_2)/(T)=(pi)/(9)impliest_2=(T)/(18)`
`t_1+t_2=(T)/(4)+(T)/(18)=(11)(18)T=(11)/(18)2pisqrt((m)/(K))=(11)/(9)pisqrt((m)/(K))`