Mechanics problem -- 2 masses & spring on a surface

[Tanya Sharma]

Thanks you very much for clearing my doubts.

I am glad you were able to solve the problem

 

[ehild]

The minimum velocity of block 1 can be 0.
x(F -$\mu$1M1g-$\mu$2M2g/2)=0
or F =g($\mu$1M1+$\mu$2M2/2).
YES I got the answer.But I still have confusion that why my first approach to the solution(using Newton's Law) was wrong.

Your approach assumed that both masses are moving. Only in that case are the forces of friction μmg. If one of the mass is in rest, friction is static. Can be less than μmg.

The masses in the problem can move in a peculiar way. Initially both are in rest, so the force F has to overcome μ₁m₁g and the elastic force of the spring. But the other mass is in rest, till the elastic force overcomes the frictional force acting to it. At the same time, the first mass can get into rest, and the elastic force alone drives the second mass forward. So the average applied force can be less than μ₁m₁g+μ₂m₂g.

ehild