A block of mass m sits on top of another block of mass M , and the...

Given that the two blocks move together, they share the same acceleration. The acceleration of the system is given as:

$a = \dfrac{F}{2(m+M)}$

For the smaller block of mass $m$, the only horizontal force acting on it is the static friction force $f$ exerted by the larger block $M$. This friction force is responsible for the acceleration of the smaller block.

Using Newton's second law for the smaller block:

$f = ma$

Substituting the value of acceleration $a$:

$f = m \left( \dfrac{F}{2(m+M)} \right) = \dfrac{mF}{2(m+M)}$

The displacement of the system, and thus the smaller block, is $d$ in the direction of the acceleration (and the friction force).

The work done by the force of friction on the smaller block is:

$W = f \cdot d$

$W = \left( \dfrac{mF}{2(m+M)} \right) d = \dfrac{mFd}{2(m+M)}$