First we look at the forces acting on the block:
Now we look at the forces acting on the pulley.
By examining the mechanical goings on of this rope and pulley system, it can be seen that the block will be accelerating twice as fast as the pulley. This can be explained by considering the following: assume that the block moves 1 unit of length to the right. To keep the same tension in the rope, the pulley will have to move, but for every unit it moves to the right, it uses up 2 units of rope (one for the bottom, and one for the top). Therefore, for one unit moved by the block, the pulley only moves half of a unit. We find an equation relating F, T, and acceleration and replace the acceleration of the pulley with half the acceleration of the block and plug in what we got for T above and end up with an equation which we simplify and solve for a.
