Let the rotational inertia of an object that rotates about its center of mass be

Question

Let the rotational inertia of an object that rotates about its center of mass be given with some fractional coefficient represented as K (so for a sphere it's K= 2/5, disk K = 1/2, hollow cyinder K=1 and so forth) multiplied by its mass and radius squared in the usual fashion. Now, for rolling without slipping starting from rest down an inclined plane of a given angle theta, find the time it takes for the object to roll down a distance D along the plane (noting at the end that this is independent of the object's radius or mass!). Then (and here's the real question) find which of these three objects (a sphere, a disk, or a hollow cylinder) will roll down in the least time.

Explanation / Answer

Loss in potential energy= gain in kinetic energy

mg D sin (theta) = (kmR^2)2

--> is prportionate to 1/sqrt k

---> the larger the , the faster the rotation, the least time taken to roll down

---> the body with the smallest k will take the least time

---> ball will roll down fastest with smallest k=2/5

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