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

Options

A. ` 4:3:2`
B. `10:15:7`
C. `14:15:20`
D. `2:3:4`

Answer

c

Explanation

`mgh= (1)/(2)mv_(cm)^(2)+ (1)/(2)I_(cm)omega^(2)`
`= (1)/(2)mv_(cm)^(2)+ (1)/(2)I_(cm)((v_(cm))/(R))^(2)`
`= (1)/(2) ( m+ I_(cm)/(R^(2)))v_(cm)^(2)= (1m)/(2)[1+(k^(2))/(R^(2))]`
`:. h prop 1+ (K^(2))/(R^(2))`
For ring `h prop 2 ( because K= R)`
for solid cylinder, `h prop (3)/(2)( because K= (R)/(sqrt(2)))`
for solid sphere, `h prop (7)/(5)( because K= sqrt((2)/(5)))`
Ratio of heights `2: (3)/(2): (7)/(5) implies 20 : 15 : 14`