Question

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.

Answer 1

Correct Answer: 13

By conservation of angular momentum:

\[ I_1 \omega_1 = I_2 \omega_2 \]

Moment of inertia of a sphere:

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

Angular velocity relation:

\[ \omega = \frac{2 \pi}{T} \]

\[ \frac{2}{5} M R^2 \cdot \frac{2 \pi}{T_1} = \frac{2}{5} M \left( \frac{3}{4} R \right)^2 \cdot \frac{2 \pi}{T_2} \]

Simplifying:

\[ \frac{1}{T_1} = \frac{9}{16} \cdot \frac{1}{T_2} \]

\[ T_2 = \frac{16}{9} \cdot T_1 \]

Substituting $T_1 = 24$ hours:

\[ T_2 = \frac{9}{16} \cdot 24 = \frac{27}{2} = 13 \, \text{hours} \, 30 \, \text{minutes}. \]