The great downhill race

4 objects \(A, B, C, D\) are made up of the same material. They are released, simultaneously from rest, from the starting line on the top of a rough inclined plane, so that they roll without slipping. \(A\) and \(B\) are cylinders, and \(C\) and \(D\) are spheres.

What is the order relation of the times taken by the objects to cross the finish line?
(A, B, C, D in the answer options denote the times taken by corresponding objects.)

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Details and Assumptions:

  • The objects are released with all their centers above the same starting line, which is the top edge of the ramp. They are considered to have crossed the finish line--the red line near the bottom edge--only if their centers have crossed the line when viewed from the side.

  • Moment of inertia of a cylinder is \(\frac{1}{2} MR^2\) and that of a sphere is \(\frac{2}{5} MR^2\), where \(M\) is the mass and \(R\) is the radius of the respective objects.

  • \(B\) has twice the radius of \(A,\) and \(D\) has twice the radius of \(C\).

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