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 , (iii) a solid sphere of radius R/4. If, in each case, the speed of the incline is same, theratio of the maximum heights they climb is :

Options

A. `4:3:2`
B. `14:15:20`
C. `20:15:14`
D. `2:3:4`

Answer

C

Explanation

Conservation of energy gives `1/2 I_(ROR)omega^(2) = m gh`
For ring `1/2 xx 2mR^(2) xx (V/R)^(2) = mgh_(1) rArr h_(1) = v^(2)/g`
For cylinder, `1/2 xx 3/2m(R/2)^(2) (v/(R//2))^(2) = mgh_(2) rArr h_(2) = 3/4v^(2)/g`
For sphere `1/2.7/5 m(R/4)^(2) xx (V/(R//4))^(2) = mgh_(3) therefore h_(3) =7/10 .v^(2)/g`
`therefore h_(1):h_(2):h_(3)=1:3/4:7/10 =20:15:14`