Keeping the mass of the earth as constant, if its radius is reduced to 1/4th of its initial value, then the period of revolution of the earth about its own axis and passing through the centre, (in hours) is (assume the earth to be a solid sphere and its initial period of rotation as 24 h)

Options

A. 12
B. 3
C. 6
D. 1.5

Explanation

To solve the problem, we need to analyze how the period of revolution of the Earth changes when its radius is reduced to 1/4th of its initial value while keeping its mass constant.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The initial period of rotation of the Earth (T) is given as 24 hours.
- We denote the initial radius of the Earth as \( R_e \).

2. **Using the Formula for the Period of Rotation**:
- The period of rotation (T) of a solid sphere is given by the formula:
\[ T = 2\pi \sqrt{\frac{R}{g}} \]
where \( R \) is the radius and \( g \) is the acceleration due to gravity.

3. **Finding the New Radius**:
- If the radius is reduced to 1/4th of its initial value, the new radius \( R' \) is:
\[ R' = \frac{R_e}{4} \]

4. **Calculating the New Period of Rotation**:
- We can denote the new period of rotation as \( T' \). Using the formula for the period of rotation with the new radius:
\[ T' = 2\pi \sqrt{\frac{R'}{g'}} = 2\pi \sqrt{\frac{R_e/4}{g'}} \]
- Since the mass of the Earth remains constant, the acceleration due to gravity \( g' \) at the new radius can be expressed as:
\[ g' = \frac{GM}{(R_e/4)^2} = \frac{GM \cdot 16}{R_e^2} = 16g \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

5. **Substituting \( g' \) into the New Period Formula**:
- Now substituting \( g' \) into the equation for \( T' \):
\[ T' = 2\pi \sqrt{\frac{R_e/4}{16g}} = 2\pi \sqrt{\frac{R_e}{64g}} = 2\pi \cdot \frac{1}{8} \sqrt{\frac{R_e}{g}} = \frac{T}{4} \]
- Since the initial period \( T = 24 \) hours, we find:
\[ T' = \frac{24}{4} = 6 \text{ hours} \]

### Final Answer:
The new period of revolution of the Earth about its own axis, when its radius is reduced to 1/4th of its initial value, is **6 hours**.