Question

If the earth were to suddenly contract to half of its present radius, what would be the duration of the day?

• A. 6 h
• B. 18 h
• C. 24 h
• D. 30 h

Hint:

This is a common conservation of angular momentum problem. The key is that \( I \propto R^2 \), so if \( R \) halves, \( I \) becomes \( 1/4 \) of its original value. To conserve \( I\omega \), \( \omega \) must become 4 times larger, meaning the period \( T \) becomes \( 1/4 \) of its original value.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem describes a scenario where the Earth's radius suddenly changes while its mass remains constant. We need to find the new duration of a day, which implies a change in its angular velocity. This is a classic problem involving conservation of angular momentum.
Step 2: Key Formula or Approach:
1. Conservation of Angular Momentum: If no external torque acts on a system, its angular momentum ($L$) remains constant.
\[ L = I \omega = \text{constant} \]
2. Moment of Inertia of a Sphere: For a solid sphere, \( I = \frac{2}{5}MR^2 \).
3. Angular Velocity: Angular velocity \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of rotation (duration of a day).
Step 3: Detailed Explanation:
Let the initial radius be \( R_1 \) and the initial duration of the day be \( T_1 = 24 \text{ h} \).
Let the final radius be \( R_2 = R_1/2 \) and the final duration of the day be \( T_2 \).
By conservation of angular momentum:
\[ I_1 \omega_1 = I_2 \omega_2 \] Substitute \( I = \frac{2}{5}MR^2 \) and \( \omega = \frac{2\pi}{T} \):
\[ \left( \frac{2}{5}MR_1^2 \right) \left( \frac{2\pi}{T_1} \right) = \left( \frac{2}{5}MR_2^2 \right) \left( \frac{2\pi}{T_2} \right) \] Cancel common terms (\( \frac{2}{5}M(2\pi) \)):
\[ \frac{R_1^2}{T_1} = \frac{R_2^2}{T_2} \] Substitute \( R_2 = R_1/2 \):
\[ \frac{R_1^2}{T_1} = \frac{(R_1/2)^2}{T_2} = \frac{R_1^2/4}{T_2} \] \[ \frac{1}{T_1} = \frac{1}{4T_2} \] \[ T_2 = \frac{T_1}{4} \] Substitute \( T_1 = 24 \text{ h} \):
\[ T_2 = \frac{24 \text{ h}}{4} = 6 \text{ h} \]
Step 4: Final Answer:
The duration of the day would be 6 h.