If the Earth suddenly shrinks to th of its original volume with its mass remaining the same, the period of rotation of the Earth becomes h. The value of is ________ . [2023]

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Solution

Q.
If the Earth suddenly shrinks to $\frac{1}{64}$ th of its original volume with its mass remaining the same, the period of rotation of the Earth becomes $\frac{24}{x}$ h. The value of $x$ is ________ . [2023]

Ans.
(16)

From conservation of angular momentum

$\frac{2}{5} M R^{2} ω_{1} = \frac{2}{5} M {(\frac{R}{4})}^{2} ω_{2}$

$\Rightarrow M R^{2} ω_{1} = \frac{M R^{2}}{16} ω_{2}$

$\Rightarrow \frac{ω_{1}}{ω_{2}} = \frac{1}{16}$

$\Rightarrow \frac{T_{2}}{T_{1}} = \frac{1}{16} \Rightarrow \frac{T_{2}}{T_{1}} = \frac{16}{1} = \frac{t_{1}}{x}$

$∴$ $t_{1} = 24 \Rightarrow \frac{16}{1} = \frac{24}{t_{2}} \Rightarrow x = 16$

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