Two plates of some mass are attached rigidly to the two ends of a spring as shown in figure. One of the two ends of a spring s shown in figure. One of the plates resets on a horizontal surface to the other rusults a compression `y` of the spring when it is in equilibrium state. The further minimum compression required, so that when the force causing compression is removed the lower plate is lifted off the surface, will be :

Options

A. `0.5 y`
B. `3y`
C. `2y`
D. `y`

Answer

C

Explanation

Let upper block is pushed down by `x`, at equilibrium `mg = ky`, i.e, weight of upper of upper block is balanced by spring When it is deformed bu `y`, upper block will perform `SHM` with amplitude `x` equilibrium position, lower block will leave surface when spring is extended by `y`, means upper block is at distance `2y` from its mean position. That should be upper extreme position of upper block. So amplitude `x = 2y`
Alternate
at equilibrium of upper block `mg = ky`
lower plate will leave the surface if the extension in spring is `y`
Let upper plate is displaaced by `x` downward and left
so by energy conservation between compressed to extended positions
`0 + (1)/(2) k (x + y)^(2) = mg(x + 2y) = (1)/(2) ky^(2)`
`rArr (1)/(2) kx^(2) + (1)/(2) ky^(2) + kxy = mgx + mg2y + (1)/(2) ky^(2) rArr x = 2y`