If the mass of a revolving around the sun is doubled and its frequency of revolution remains constant then the radius of its orbit will be

Options

A. Doubled
B. Thrice
C. Halved
D. Remain same

Answer

To solve the problem, we need to analyze the relationship between the mass of a planet revolving around the sun, its frequency of revolution, and the radius of its orbit.

### Step-by-Step Solution:

1. **Understanding the Forces**:
The gravitational force acting on the planet provides the necessary centripetal force for its circular motion. The gravitational force \( F_g \) is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the sun, \( m \) is the mass of the planet, and \( r \) is the radius of the orbit.

2. **Centripetal Force**:
The centripetal force \( F_c \) required to keep the planet in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the orbital speed of the planet.

3. **Equating Forces**:
Since the gravitational force provides the centripetal force, we can set them equal to each other:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]
Here, we can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]

4. **Rearranging the Equation**:
Multiplying both sides by \( r \) gives:
\[ \frac{GM}{r} = v^2 \]

5. **Finding the Orbital Speed in Terms of Frequency**:
The orbital speed \( v \) can also be expressed in terms of frequency \( f \):
\[ v = 2\pi rf \]
Substituting this into the equation \( \frac{GM}{r} = v^2 \):
\[ \frac{GM}{r} = (2\pi rf)^2 \]

6. **Squaring the Frequency**:
Expanding the right side gives:
\[ \frac{GM}{r} = 4\pi^2 f^2 r^2 \]

7. **Rearranging for Radius**:
Rearranging this equation to solve for \( r \):
\[ r^3 = \frac{GM}{4\pi^2 f^2} \]
Thus,
\[ r = \left(\frac{GM}{4\pi^2 f^2}\right)^{1/3} \]

8. **Considering the Changes in Mass**:
The problem states that the mass of the planet is doubled (\( m \to 2m \)), but the frequency \( f \) remains constant. Notice that the expression for \( r \) does not depend on the mass of the planet \( m \). Therefore, even when the mass of the planet is doubled, the radius \( r \) remains unchanged.

### Conclusion:
The radius of the orbit will remain the same when the mass of the planet is doubled and the frequency of revolution remains constant.

### Final Answer:
The radius of its orbit will remain the same.