If a planet of mass .m. is revolving around the sun in a circular orbit of radius .r. with time period T, then the mass of the sun is

Options

A. `(4pi^2r^3)/(GT)`
B. `(4pi^2r^3)/(GT^2)`
C. `(4pi^2r)/(GT)`
D. `(4pi^2r^3)/(G^2T^2)`

Answer

To find the mass of the sun (M) when a planet of mass (m) is revolving around it in a circular orbit of radius (r) with a time period (T), we can use the concepts of gravitational force and centripetal force.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Planet:**
The planet experiences two main forces:
- The gravitational force (F_g) acting towards the sun.
- The centripetal force (F_c) required to keep the planet in circular motion.

2. **Write the Expression for Gravitational Force:**
The gravitational force between the sun and the planet is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot M \cdot m}{r^2} \]
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.

3. **Write the Expression for Centripetal Force:**
The centripetal force required to keep the planet moving in a circular orbit is given by:
\[ F_c = \frac{m v^2}{r} \]
where \( v \) is the tangential velocity of the planet.

4. **Relate Tangential Velocity to Time Period:**
The tangential velocity \( v \) can be expressed in terms of the time period \( T \):
\[ v = \frac{2\pi r}{T} \]
Substituting this into the centripetal force equation gives:
\[ F_c = \frac{m \left(\frac{2\pi r}{T}\right)^2}{r} = \frac{m \cdot 4\pi^2 r}{T^2} \]

5. **Set Gravitational Force Equal to Centripetal Force:**
Since the gravitational force provides the necessary centripetal force, we can set these two forces equal to each other:
\[ \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot 4\pi^2 r}{T^2} \]

6. **Cancel Out the Mass of the Planet:**
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G \cdot M}{r^2} = \frac{4\pi^2 r}{T^2} \]

7. **Rearrange to Solve for the Mass of the Sun (M):**
Now, we can solve for \( M \):
\[ M = \frac{4\pi^2 r^3}{G T^2} \]

### Final Answer:
The mass of the sun is given by:
\[ M = \frac{4\pi^2 r^3}{G T^2} \]