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

Answer

To find the mass of the sun (Ms) given that a planet of mass (m) is revolving around it in a circular orbit of radius (r) with a time period (T), we can follow these steps:

### Step 1: Identify the Forces Acting on the Planet
The planet experiences two main forces:
1. **Centripetal Force (Fc)**: This is required to keep the planet in circular motion.
2. **Gravitational Force (Fg)**: This is the attractive force between the planet and the sun.

### Step 2: Write the Expression for Centripetal Force
The centripetal force required for circular motion is given by:
\[ F_c = \frac{m v^2}{r} \]
where \(v\) is the orbital speed of the planet.

### Step 3: Relate Orbital Speed to Time Period
The orbital speed \(v\) can be expressed in terms of the radius \(r\) and the time period \(T\):
\[ v = \frac{2\pi r}{T} \]
Substituting this into the centripetal force equation gives:
\[ F_c = \frac{m \left(\frac{2\pi r}{T}\right)^2}{r} = \frac{m (4\pi^2 r)}{T^2} \]

### Step 4: Write the Expression for Gravitational Force
The gravitational force acting on the planet due to the sun is given by Newton's law of gravitation:
\[ F_g = \frac{G M_s m}{r^2} \]
where \(G\) is the gravitational constant and \(M_s\) is the mass of the sun.

### Step 5: Set the Forces Equal
Since the centripetal force is provided by the gravitational force, we can set these two expressions equal to each other:
\[ \frac{m (4\pi^2 r)}{T^2} = \frac{G M_s m}{r^2} \]

### Step 6: Simplify the Equation
We can cancel \(m\) from both sides (assuming \(m \neq 0\)):
\[ \frac{4\pi^2 r}{T^2} = \frac{G M_s}{r^2} \]

### Step 7: Solve for the Mass of the Sun
Rearranging the equation to solve for \(M_s\):
\[ M_s = \frac{4\pi^2 r^3}{G T^2} \]

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

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