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.
Best Answer
You are right to be suspicious of an answer that contradicts your intuition - but in this case the solution is right
The key (not clearly explained) is that you apply a constant force that is greater than the force needed to move one block: consequently that block will accelerate. As the spring compresses, the first block will accelerate less, and eventually it will decelerate. When it stops, the compression of the spring is maximum - and it applies a greater force to both blocks than the force you used to compress it.
So the solution is correct. If you still have difficulty believing this, think about a hammer: when you swing it, you apply a little bit of force over a long distance; when it hits the nail, it feels a lot of deceleration over a short distance. The force applied to the nail is greater than the force you applied to the hammer. Similar (not same) concept... You do work over a large distance with a little bit of force, to allow you to do the same work over a short distance (with a large force)
Does that help?