Find the minimum value of the quantity

[gourish]

Homework Statement

There are two cubes of mass m. They are initially tied by a string tightly. They are kept from joining into each other because of a spring (which in this tied up state has a compression of ε) of coefficient of stiffness k. Find the minimum value of the quantity εk/mg so that when the string is cut and the spring is let loose to act, the lower cube jumps off the ground

Homework Equations

consider the natural length as "l" of the spring and the center of mass of lower cube as the reference line
total mechanical energy (initial)=1/2 k ε^2+mg (l-ε)
total mechanical energy (after)=1/2kx^2+mg(l-x)

The Attempt at a Solution

i found that k(e+x)/mg=2 but i did not get to the quantity to become εk/mg and did not understand how to relate the equations to the condition "the lower cube jumps off the ground" can anybody please help me

 

[tiny-tim]

hi gourish! welcome to pf!

… did not understand how to relate the equations to the condition "the lower cube jumps off the ground" can anybody please help me

hint: it won't jump unless it's pulled! …

with what (minimum) force must it be pulled? ​

 

[rude man]

1. Use energy conservation. What is the initial energy in the spring?
2. What is the criterion on x for the bottom mass to just start moving?
Let x = 0 is when the spring is relaxed. x is the distance of the top m above the level where the spring is relaxed.

initial spring energy = final spring energy + gain in potential energy of top mass.