Two plates of same mass are attached rigidly to the two ends of a spring as shown in figure. One of

Question

Two plates of same mass are attached rigidly to the two ends of a spring as shown in figure. One of the plates rests on a horizontal surface and the other results 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 :

Accepted Answer

Let upper block is pushed down by x. at equilibrium mg = ky, i.e., weight of upper block is balanced by spring when it is deformed by y. upper block will perform SHM with amplitude x about 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 displaced by x downward and left
so by energy conservation between compressed to extended positions
0 + k (x + y)² = mg (x + 2y) = ky²
⇒ kx² + ky² + kxy = mgx + mg²y + ky²
⇒ x = 2y

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