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40. 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 the same, the ratio of the maximum heights they climb is

1. 4: 3: 2
2. 10: 15: 7
3. 14: 15: 20
4. 2: 3: 4

Correct answer: 14: 15: 20

Solution

Energy conservation gives h = (v^2/2g)(1 + k^2/R^2). For ring, solid cylinder and solid sphere, (1 + k^2/R^2) = 2 : 3/2 : 7/5 = 20 : 15 : 14, i.e. the heights are in the ratio 14 : 15 : 20.

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