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The following bodies are made to roll up (without slipping) the same incline from a horizontal plane: (i) a ring of radius R, (ii) a solid cylinder of radius /R2, and (iii) a solid sphere of radius /R4. If in each case the centre-of-mass speed at the bottom is the same, the ratio of the maximum heights they climb is

Asked in JEE Main 9th April 1st Shift 2019 · Rolling on an incline

• (1) 20:15:14
• (2) 2:3:4
• (3) 4:3:2
• (4) 10:15:7

Step-by-step solution

Why the other options are wrong

• (2) 2:3:4 rises through the list, but the heights fall through it: the ring climbs highest because it carries the most energy in spin, and the sphere lowest. Any option that increases from (i) to (iii) has the order backwards.
• (3) 4:3:2 at least falls in the right direction, but the steps are wrong. The true steps are 2, 3/2, 7/5, which is 20:15:14 — the last two are very close together, because a solid cylinder and a solid sphere are not far apart in I/(MR²).
• (4) 10:15:7 is not even monotonic. A quick check: with the same speed at the bottom, the ring must climb higher than the cylinder, and the cylinder higher than the sphere.

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