JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Rotational Mechanics

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Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is :

(A)The ring has the greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
(B)The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.✓ Correct
(C)The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.
(D)All of them will have same velocity.

Answer: (B)

Solution

For rolling without slipping down an incline, the velocity at the bottom depends on the moment of inertia. Using energy conservation:

m g h = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}

. For rolling,

v = R ω

, so

v = \frac{2 g h}{1 + k ^{2}}

where

k^{2} = I / (m R^{2})

. For a ring

k^{2} = 1

, cylinder

k^{2} = 1/2

, sphere

k^{2} = 2/5

. Since velocity is inversely related to

k^{2}

, the sphere (smallest

k^{2}

) has the greatest velocity and the ring (largest

k^{2}

) has the least velocity.

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Single correct option

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JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Rotational Mechanics

Solution

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