Mass remaining constant, if the earth suddenly contracts to one third of its present radius, the length of the day would be shorted by

Options

A. 8/3h
B. 12h
C. 8h
D. 64/3 h

Explanation

To solve the problem of how the length of the day would change if the Earth suddenly contracts to one third of its present radius while keeping its mass constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The mass of the Earth (M) remains constant.
- The original radius of the Earth (R) is reduced to \( R' = \frac{R}{3} \).

2. **Use the Conservation of Angular Momentum**:
- The angular momentum (L) of a rotating body is given by:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
- For a solid sphere, the moment of inertia is:
\[ I = \frac{2}{5} M R^2 \]
- The angular velocity is related to the time period (T) by:
\[ \omega = \frac{2\pi}{T} \]

3. **Set Up the Angular Momentum Equation**:
- For the initial state (before contraction):
\[ L_1 = I_1 \omega_1 = \frac{2}{5} M R^2 \cdot \frac{2\pi}{T_1} \]
- For the final state (after contraction):
\[ L_2 = I_2 \omega_2 = \frac{2}{5} M (R')^2 \cdot \frac{2\pi}{T_2} = \frac{2}{5} M \left(\frac{R}{3}\right)^2 \cdot \frac{2\pi}{T_2} \]

4. **Equate the Angular Momenta**:
- Since angular momentum is conserved:
\[ L_1 = L_2 \]
- This gives:
\[ \frac{2}{5} M R^2 \cdot \frac{2\pi}{T_1} = \frac{2}{5} M \left(\frac{R}{3}\right)^2 \cdot \frac{2\pi}{T_2} \]
- Canceling common terms:
\[ R^2 \cdot \frac{1}{T_1} = \left(\frac{R^2}{9}\right) \cdot \frac{1}{T_2} \]

5. **Simplifying the Equation**:
- Rearranging gives:
\[ \frac{1}{T_1} = \frac{1}{9T_2} \]
- Thus:
\[ T_2 = 9 T_1 \]

6. **Substituting the Original Time Period**:
- The original time period \( T_1 \) is 24 hours:
\[ T_2 = 9 \times 24 \text{ hours} = 216 \text{ hours} \]

7. **Calculate the Change in Length of Day**:
- The change in the length of the day (ΔT) is:
\[ \Delta T = T_1 - T_2 = 24 \text{ hours} - 8 \text{ hours} = 16 \text{ hours} \]

### Final Answer:
The length of the day would be shortened by **16 hours**.