A block `(B)` is attached to two unstriched sprig `S_(1)` and S_(2) with spring constant `K` and `4K` , respectively (see fig 1) The other ends are atteched in identical support `M_(1)` and `M_(2)` not attached in the walls . The springs and supports have negligible mass . There is no friction anywhere . The block `B` is displaced toword wall `1` by a small distance `z` (figure (ii)) and released . THe block return and moves a maximum displacements `x` and `y` are musured with reoact to the equalibrum of the block `B` and the ratio `y//x` is

Options

A. 4
B. 2
C. `(1)/(2)`
D. `(1)/(4)`

Answer

C

Explanation

When the block B is moved towards the wall 1, through a dis"tan"ce x, the spring `S_(1)` is compressed and the P.E. stored in the spring is increased. It is
`U_(1)=(1)/(2)K_(1)x^(2)=(1)/(2)Kx^(2)`
When the block B is released, it starts moving towards left, and compresses the spring `S_(2)`, through a maximum dis"tan"ce y. The P.E. stored in the spring `S_(2)` is
`U_(2)=(1)/(2)K_(2)y^(2)=(1)/(2)(4K)y^(2)=2Ky^(2)`
From the principle of conservation of energy
`U_(1)=U_(2)`
`therefore (1)/(2)Kx^(2)=2Ky^(2) therefore x^(2)=4y^(2) or x=2y`
`therefore (y)/(x)=(1)/(2)`