If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes.

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Solution

Q.
If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be _______ hours 30 minutes. [2024]

Ans.
(13) $By angular momentum conservation, I_{1} ω_{1} = I_{2} ω_{2}$

   $\frac{2}{5} m r^{2} \cdot ω = \frac{2}{5} m {(\frac{3}{4} r)}^{2} \cdot ω^{'} \Rightarrow ω^{'} = \frac{16}{9} ω$

   $\frac{2 π}{T^{'}} = \frac{16}{9} \times \frac{2 π}{T} \Rightarrow T^{'} = \frac{9 T}{16} = \frac{9}{16} \times 24 hours = \frac{9 \times 3}{2} hours$

   $\Rightarrow T^{'} = \frac{27}{2} hr = 13 h r 30 mins$

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