Blocks spring system

Two blocks of masses $m_1$ and $m_2$ are connected by a mass less spring of constant $k$. The blocks rest on a rough floor with the spring in its equilibrium length. Coefficient of friction between the blocks and the floor is $\mu$. Here we want to find the minimum horizontal force that must be applied on $m_1$ to just move $m_2$. Let $F$ be the force applied to $m_1$ and $X$ the the displacement of the spring when the block $m_2$ starts to move. Hence we have equality $$kX=\mu m_2g.$$ The work done by $F$ is partly dissipated by friction and the rest is stored in the spring as a potential energy. That is

$$FX=\mu m_1gX+\frac{1}{2}kX^2,$$ or equivalently $$F=\mu m_1g+\frac{1}{2}kX.$$ Using the first and the last equation we have $$F=\mu g(m_1+\frac{1}{2}m_2).$$

This solution can be found here. Is this the correct solution? For me this is rather counter intuitive. Let $m_2=10m_1$ then $$F=\frac{3}{5}\mu gm_2.$$ The maximum friction force of $m_2$ alone is $\mu gm_2$, how can a force less than this move the entire system? Furthermore, since $F$ is independent of $k$, we can choose a spring with very large $k$ so that we can view the spring as a rod or very small $k$ so that $m_1$ touches $m_2$. In this case, shouldn't we have $F=\mu g(m_1+m_2)?$ Of course, my intuition could be wrong.

• For block 1, you are using $\mu$ for the coefficient of kinetic friction and for block 2, you are using $\mu$ for the coefficient of static friction. This aspect of the problem statement doesn't make sense to me. Do they really want you to do that? – Chet Miller Feb 29 '16 at 2:44
• For simplification, lets assume the static and kinetic friction coefficient are identical. – Sukan Feb 29 '16 at 2:49

• "... it applies a greater force to both blocks than the force you used to compress it." Doesn't this violate Newton's 3rd Law? If the solution is correct, then for $m_1\ll m_2$ we will have $F\approx\frac{1}{2}\mu gm_2$. That is, using a light object and a spring we can move a heavy object for only half force. I must say this is hard to accept. Anyway, thank you for the explanation. – Sukan Feb 29 '16 at 3:26