Question

A block of mass M rests on a table. It is fastened to the lower end of a light vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg, so that the spring compression is 4mg/k, where k is the spring constant. In this configuration the upper block is released from rest. The spring lifts the lower block off the table. In terms of m, what is the greatest possible value for M?

Answer 1

the maximum value of M would be 3m
This can be thought of like this -

the m-M spring system is at equilibrium first with whatever spring compression it has.
an extra compression occurs with force = 3mg, this compression is x = 3mg/k

so, when the system is released, the system would execute oscillations with maximum amplitude = 3mg/k.

Thus when the spring is stretched to its maximum(3mg/k), there would be maximum force upwards on the mass M.
the equation for mass M would be ,

R + kx = Mg

where R is the normal reaction on M from the table

so, for the mass M to be lifted upwards, the limiting condition is R = 0
i.e. x = Mg/k
but x = 3mg/k

hence M <= 3m

Answer 2

When ext. F is removed the compression(x)=4mg/k
The condition so that the block lifts is that N(due to table)=0

So, kx = mg + Mg

3mg = Mg

M = 3m