A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed v. The sphere and the cylinder reach up to maximum heights h? and h?, respectively, above the initial level. The ratio is . The value of

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Solution

Q.
A solid sphere and a hollow cylinder roll up without slipping on the same inclined plane with the same initial speed v. The sphere and the cylinder reach up to maximum heights $h_{1}$ and $h_{2}$ , respectively, above the initial level. The ratio $h_{1} : h_{2}$ is $\frac{n}{10}$ . The value of n is _____. [2024]

Ans.
(7)

Energy conservation, Gain in P.E. = Loss in K.E.

$\frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = m g h$

For pure rolling; $v = ω R$

$\frac{1}{2} m v^{2} + \frac{1}{2} I \frac{v^{2}}{R^{2}} = m g h$

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

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

$\frac{h_{1}}{h_{2}} = \frac{1 + \frac{2}{5}}{1 + 1} = \frac{7}{5 \times 2} = \frac{7}{10}$

Hence, $n = 7$

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