Question

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: (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 bottom of the incline is same, the ratio of the maximum heights they climb is :
(A) 10 : 15 : 7 (B) 14: 15: 20
(C) 4 : 3 : 2 (D) 2 : 3 : 4

Answer 1

Given: The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: (i) a ring of radius R, (ii) a solid cylinder of radius R/2  and (iii) a solid sphere of radius R/4

To find: The ratio of the maximum heights they climb?

Solution:

• Now we have given (i) a ring of radius R, (ii) a solid cylinder of radius R/2  and (iii) a solid sphere of radius R/4.

• We have given that the speed of the center of mass at bottom of the incline is same.

• Let h1 be the hieght climbed by a ring, h2 be the hieght climbed by a solid cylinder and h3 be the hieght climbed by a solid sphere.

• So by energy conservation, we have:

                1/2 mv² x (1 + k²/r²) = mgh

• Here h ∝ 1 + k²/r²

• So:

                h1 : h2 : h3 = (1 + 1) : (1 + 1/2) : (1 + 2/5)

                h1 : h2 : h3 = 2 : 3/2 : 7:5

                h1 : h2 : h3 = 20 : 15 : 14

Answer:

            So the ratio is 20 : 15 : 14