The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R , (ii) a solid cylinder of radius R 2 and (iii) a solid sphere of radius R 4 . If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is:

Question

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane: (i) a ring of radius R , (ii) a solid cylinder of radius R 2 and (iii) a solid sphere of radius R 4 . If, in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is:, 2 : 3 : 4, 20 : 15 : 14, 4 : 3 : 2, 14 : 15 : 20

MARKS App · Accepted Answer

From conservation of mechanical energy. 1 2 m R v 2 + 1 2 I R ω r 2 = m R g h R ⇒ V 2 2 + V 2 2 = g h R 1 2 m d v 2 + 1 2 I d ω d 2 = m d g h d ⇒ V 2 2 + V 2 4 = g h d 1 2 m s v 2 + 1 2 I s ω s 2 = m s g h s ⇒ V 2 2 + V 2 5 = g h s ∴ h R : h d : h s = 1 : 3 4 : 7 10 = 20 : 15 : 14

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