First slide

Law of conservation of angular momentum

Question

If earth suddenly shrinks to $\frac{1}{8}$ th of its original volume while mass remaining the same, then duration of day (in hour) will be

Moderate

A

6
B

8
C

12
D

3

Solution

Original volume of the earth,
$V = \frac{4}{3} π R^{3}$
Volume of earth after shrinking
$V^{'} = \frac{V}{4}$
$\Rightarrow \frac{4}{3} π R^{3} = \frac{1}{8} \times \frac{4}{3} π R^{3}$
$R^{'} = \frac{R}{2}$
By conservation of angular momentum,
$I^{'} ω^{'} = I ω$
$\Rightarrow \frac{2}{5} M R^{2} \times \frac{2 π}{T^{'}} = \frac{2}{5} M R^{2} \times \frac{2 π}{T}$
$\Rightarrow \frac{T}{T^{'}} = {(\frac{R}{R^{'}})}^{2}$
$\Rightarrow \frac{T}{T} = {(\frac{R}{R / 2})}^{2}$
$\Rightarrow \frac{T}{T^{'}} = 4$
$\Rightarrow T^{'} = \frac{T}{4} = \frac{24}{4}$ $[∵ T = 24 h]$
$= 6 h$

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