A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights $\mathrm{h}_{1}$ and $\mathrm{h}_{2}$, respectively, above the initial level. The ratio $\mathrm{h}_{1}: \mathrm{h}_{2}$ is $\frac{\mathrm{n}}{10}$. The value of n is ______.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Rotational Mechanics

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed v. The sphere and the cylinder reaches upto maximum heights

h_{1}

and

h_{2}

, respectively, above the initial level. The ratio

h_{1} : h_{2}

\frac{n}{10}

. The value of n is ______.

Answer: 7

Solution

Rolling up the incline, all kinetic energy converts to potential energy:

m g h = \frac{1}{2} m v^{2} (1 + \frac{K ^{2}}{R ^{2}}) \Rightarrow h \propto (1 + \frac{K ^{2}}{R ^{2}}) .

Solid sphere:

\frac{K ^{2}}{R ^{2}} = \frac{2}{5} \Rightarrow 1 + \frac{2}{5} = \frac{7}{5}

.
Hollow cylinder:

\frac{K ^{2}}{R ^{2}} = 1 \Rightarrow 1 + 1 = 2

\frac{h _{1}}{h _{2}} = \frac{7/5}{2} = \frac{7}{10} .

n = 7

. Answer: 7.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Integer answer

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