JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Rotational Mechanics

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 earth becomes

\frac{24}{x} h

. The value of

x

is ____ .

Answer: 16

Solution

Volume scales as

R^{3}

, so shrinking to

\frac{1}{64}

of the volume gives

R_{2} = \frac{R}{4}

. With the mass fixed and no external torque, angular momentum is conserved,

I_{1} ω_{1} = I_{2} ω_{2}

with

I = \frac{2}{5} M R^{2}

. Thus

ω_{2} = ω_{1} (\frac{R}{R _{2}})^{2} = 16 ω_{1}

. Since

T \propto \frac{1}{ω}

, the new period is

T_{2} = \frac{T _{1}}{16} = \frac{24}{16}

h, so

x = 16

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Integer answer

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