Two bodies, a ring and a solid cylinder of the same material, are rolling down without slipping an inclined plane. The radii of the bodies are the same. The ratio of the velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is (√x)/2. Then the value…

Question

Two bodies, a ring and a solid cylinder of the same material, are rolling down without slipping an inclined plane. The radii of the bodies are the same. The ratio of the velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is (√x)/2. Then the value of x is .

Accepted Answer

3. Rolling without slipping from rest through a height h: v²=(2gh)/(1+k) with k=I/(MR²). Ring: k=1, so vᵣ²=gh. Solid cylinder: k=1/2, so v_c²=(4gh)/3. (vᵣ²)/(v_c²)=3/4, so (vᵣ)/(v_c)=(√3)/2. Comparing with (√x)/2 gives x=3. "Of the same material" tells you the densities match, so the masses differ — but mass cancels out of v, so it never matters.

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