Question

If the earth were to suddenly shrink to \( \dfrac{1}{64} \) of its present volume, keeping mass constant, then what would be the new duration of the day?

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

Hint:

Use conservation of angular momentum when no external torque acts.
Volume \(\propto R^3\), so radius scales as cube root.
When \( R \downarrow \), \( \omega \uparrow \), leading to shorter day duration.

Answer 1

The Correct Option is B

Step 1: Use conservation of angular momentum: \[ I_1 \omega_1 = I_2 \omega_2 \] Since mass remains constant, \( I \propto R^2 \) Step 2: Volume reduces by \( \frac{1}{64} \Rightarrow R \to \frac{R}{4} \) \[ I_2 = I_1 \cdot \left( \frac{1}{4} \right)^2 = \frac{I_1}{16} \Rightarrow \omega_2 = 16 \omega_1 \] Step 3: Time period \( T = \frac{2\pi}{\omega} \Rightarrow T_2 = \frac{T_1}{16} = \frac{24}{16} = 1.5\,\text{hours} \)