# 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.)


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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