If the mass of earth were `2` times the present mass, the mass of the moon were half the present mass and the moon were revolving round the earth at the same present distance, the time period of revolution of the moon would be (in day)

Options

A. `56`
B. `28`
C. `14sqrt(2)`
D. `7`

Explanation

To solve the problem, we will use Kepler's third law of planetary motion, which relates the time period of revolution of a satellite to the mass of the central body and the radius of the orbit.

### Step-by-Step Solution:

1. **Understanding Kepler's Third Law**:
Kepler's third law states that the square of the time period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit around a central body (in this case, the Earth). The formula can be expressed as:
\[ T^2 \propto \frac{r^3}{M} \]
where \( M \) is the mass of the central body (Earth).

2. **Initial Conditions**:
Let's denote:
- The present mass of the Earth as \( M_e \).
- The present mass of the Moon as \( M_m \).
- The present distance between the Earth and the Moon as \( r \).
- The present time period of the Moon as \( T_0 = 28 \) days.

3. **New Conditions**:
According to the problem:
- The new mass of the Earth \( M_e' = 2M_e \).
- The new mass of the Moon \( M_m' = \frac{1}{2}M_m \).
- The distance \( r \) remains the same.

4. **Applying Kepler's Third Law with New Mass**:
The new time period \( T' \) can be expressed as:
\[ T'^2 = \frac{4\pi^2 r^3}{G \cdot M_e'} \]
Substituting \( M_e' = 2M_e \):
\[ T'^2 = \frac{4\pi^2 r^3}{G \cdot (2M_e)} = \frac{2\pi^2 r^3}{G \cdot M_e} \]

5. **Relating New Time Period to Old Time Period**:
Since \( T_0^2 = \frac{4\pi^2 r^3}{G \cdot M_e} \), we can relate \( T' \) to \( T_0 \):
\[ T'^2 = \frac{1}{2} T_0^2 \]
Taking the square root of both sides gives:
\[ T' = T_0 \cdot \frac{1}{\sqrt{2}} = 28 \cdot \frac{1}{\sqrt{2}} = \frac{28}{\sqrt{2}} = 14\sqrt{2} \text{ days} \]

6. **Final Answer**:
Thus, the time period of revolution of the Moon under the new conditions is:
\[ T' = 14\sqrt{2} \text{ days} \]