Dynamics | Part IA, 2001

Find the moment of inertia of a uniform solid cylinder of radius aa, length ll and total mass MM about its axis.

The cylinder is released from rest at the top of an inclined plane of length LL and inclination θ\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 32\sqrt{\frac{3}{2}} times the time taken to slip down the smooth plane.

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