A system consists of two stars of equal masses that revolve in a circular orbit about their centre of mass . Orbital speed of each star is v and period is T . Find the mass M of each star . (G is gravitational constant )

Options

A. `( 2G v^(3))/( pi T)`
B. `( v^(3)T)/( pi G)`
C. `( v^(3)T)/( 2pi G)`
D. `( 2T v^(3))/( pi G)`

Answer

To find the mass \( M \) of each star in a system where two stars of equal mass revolve around their center of mass, we can follow these steps:

### Step 1: Understanding the System
In this system, we have two stars of equal mass \( M \) revolving around their common center of mass. The distance between the two stars is \( 2r \) (where \( r \) is the distance from each star to the center of mass).

### Step 2: Gravitational Force
The gravitational force \( F_g \) acting between the two stars can be expressed using Newton's law of gravitation:
\[ F_g = \frac{G M^2}{(2r)^2} = \frac{G M^2}{4r^2} \]
where \( G \) is the gravitational constant.

### Step 3: Centripetal Force
The centripetal force required to keep each star in circular motion is given by:
\[ F_c = \frac{M v^2}{r} \]
where \( v \) is the orbital speed of each star.

### Step 4: Setting Forces Equal
Since the gravitational force provides the necessary centripetal force, we can set these two forces equal to each other:
\[ \frac{G M^2}{4r^2} = \frac{M v^2}{r} \]

### Step 5: Simplifying the Equation
We can cancel \( M \) from both sides (assuming \( M \neq 0 \)):
\[ \frac{G M}{4r^2} = \frac{v^2}{r} \]
Multiplying both sides by \( 4r^2 \) gives:
\[ G M = 4 v^2 r \]

### Step 6: Expressing \( r \) in Terms of \( T \)
The period \( T \) of revolution is related to the radius \( r \) and the speed \( v \) by the formula:
\[ T = \frac{2\pi r}{v} \]
From this, we can express \( r \) as:
\[ r = \frac{v T}{2\pi} \]

### Step 7: Substituting \( r \) into the Mass Equation
Now, substituting \( r \) back into the equation \( G M = 4 v^2 r \):
\[ G M = 4 v^2 \left( \frac{v T}{2\pi} \right) \]
This simplifies to:
\[ G M = \frac{4 v^3 T}{2\pi} = \frac{2 v^3 T}{\pi} \]

### Step 8: Solving for \( M \)
Now, we can solve for \( M \):
\[ M = \frac{2 v^3 T}{G \pi} \]

### Final Result
Thus, the mass \( M \) of each star is:
\[ M = \frac{2 v^3 T}{G \pi} \]