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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, and (iii) a solid sphere of radius R. If in each case, the speed of the centre of mass at the bottom of the incline is the same, the ratio of the maximum heights they climb is:

4:3:2

10:15:7

2:3:4

14:15:20

Solution:

Correct option is D. 14:15:20

1/2(m + I/R^2)v^2 = mgh

if radius of gyration is k, then
h = (1 + k^2/R^2)v^2/2g,

for ring Rring = 1,
for solid cylinder Rsolid cylinder = 1/2
for solid sphere Rsolid sphere = √(2/5)

h1:h2:h3 :: (1+1)(1+1/2):(1+2/5) :: 20:15:14