Question

If the earth suddenly shrinks to \(\dfrac{1}{64}\) of its original volume, while keeping the same mass, then the duration of the day will be:
[Assume earth is a perfect sphere]

• A. \(24\) hours
• B. \(1.5\) hours
• C. \(16\) hours
• D. \(48\) hours

Hint:

If no external torque acts on a rotating body, angular momentum remains conserved: \[ I\omega=\text{constant} \] A decrease in radius reduces the moment of inertia and increases the angular speed.

Answer 1

The Correct Option is B

Step 1: Find the new radius.
Volume shrinks to \( V/64 \), so \( R'^3 = R^3/64 \Rightarrow R' = R/4 \). Mass is unchanged.

Step 2: Apply conservation of angular momentum.
\( I\omega = I'\omega' \Rightarrow \frac{2}{5}MR^2 \cdot \omega = \frac{2}{5}M\left(\frac{R}{4}\right)^2\omega' \Rightarrow \omega' = 16\omega \). New period: \[ T' = \frac{T}{16} = \frac{24}{16} = 1.5\,\text{hours} \] \[ \boxed{1.5\text{ hours}} \]