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]

Question

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]

Smart Achievers · Accepted Answer

(16)

From conservation of angular momentum

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

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

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

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

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

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]

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Solution

Q. If the Earth suddenly shrinks to 164th of its original volume with its mass remaining the same, the period of rotation of the Earth becomes 24xh. The value of x is ________ . [2023]

Ans. (16) From conservation of angular momentum 25MR2ω1=25M(R4)2ω2 ⇒MR2ω1=MR216ω2 ⇒ω1ω2=116 ⇒T2T1=116⇒T2T1=161=t1x ∴ t1=24⇒161=24t2⇒x=16

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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]