Find the moment of inertia of a uniform solid cylinder of radius $a$, length $l$ and total mass $M$ about its axis.

The cylinder is released from rest at the top of an inclined plane of length $L$ and inclination $\theta$ to the horizontal. The first time the plane is perfectly smooth and the cylinder slips down the plane without rotating. The experiment is then repeated after the plane has been roughened, so that the cylinder now rolls without slipping at the point of contact. Show that the time taken to roll down the roughened plane is $\sqrt{\frac{3}{2}}$ times the time taken to slip down the smooth plane.