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 $\frac{\mathrm{R}}{2}$ and (iii) a solid sphere of radius $\frac{R}{4}$. 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:

JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Rotational Mechanics

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

\frac{R}{2}

and
(iii) a solid sphere of radius

\frac{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) $20 : 15 : 14$ ✓ Correct
(C) $4 : 3 : 2$
(D) $2 : 3 : 4$

Answer: (B)

Solution

Climbing the incline, all the kinetic energy at the bottom turns into height,

\frac{1}{2} m v^{2} (1 + \frac{k ^{2}}{R ^{2}}) = m g h

, so

h \propto 1 + \frac{k ^{2}}{R ^{2}}

and the radii themselves drop out. For the ring

\frac{k ^{2}}{R ^{2}} = 1

gives

2

; for the solid cylinder

\frac{1}{2}

gives

\frac{3}{2}

; for the solid sphere

\frac{2}{5}

gives

\frac{7}{5}

. The heights are therefore in the ratio

2 : \frac{3}{2} : \frac{7}{5} = 20 : 15 : 14

JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Single correct option

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JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Rotational Mechanics

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