Two stars of masses $m$ and $2 m$ at a distance $d$ rotate about their common center of mass in free space. The period of revolution is:

$2 π \sqrt{\frac{d^{3}}{3 G m}}$
$\frac{1}{2 π} \sqrt{\frac{3 G m}{d^{3}}}$
$\frac{1}{2 π} \sqrt{\frac{d^{3}}{3 G m}}$
$2 π \sqrt{\frac{3 G m}{d^{3}}}$

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Solution

The correct option is A
$2 π \sqrt{\frac{d^{3}}{3 G m}}$

Step 1: Given
Mass of the first star: $m_{1} = m$
Mass of the second star: $m_{2} = 2 m$
Distance between stars is $d$
Let, $r_{1}$ and $r_{2}$ and be the orbital radius of the first and second star
We know,
$\begin{matrix} r_{1} + r_{2} = d \\ \Rightarrow & r_{1} = d - r_{2} \end{matrix}$
Step 2: Formula Used
Gravitational force between two objects is
$F_{G} = \frac{G m_{1} m_{2}}{r^{2}}$
Where $G$ is the gravitational constant, $m_{1}$ and $m_{2}$ are the masses of the two objects and $r$ is the distance between them.
Centripetal force on an object is
$F_{C} = m r ω^{2}$
where $m$ is the mass of the object, $r$ is the radius of orbit and $ω$ is the angular velocity.
Time period is
$T = \frac{2 π}{ω}$
Where $ω$ is the angular velocity.
From the center of mass formula, we have,
If we are making measurements from the center of mass point for a two-mass system then the center of mass condition can be expressed as
$m_{1} r_{1} = m_{2} r_{2}$
Step 3: Find the orbital radius of the stars
From the center of mass formula,
$\begin{matrix} m r_{1} = 2 m r_{2} \\ \Rightarrow & r_{1} = 2 (d - r_{1}) \\ \Rightarrow & r_{1} = 2 d - 2 r_{1} \\ \Rightarrow & r_{1} = \frac{2}{3} d \end{matrix}$
Step 4: Find an expression for angular velocity
Equate the gravitational force and centripetal force acting on the first mass, since the gravitational force between masses provides the necessary centripetal force.
$\begin{matrix} F_{C} = F_{G} \\ \Rightarrow & m_{1} r_{1} ω^{2} = \frac{G m_{1} m_{2}}{d^{2}} \\ \Rightarrow & m \frac{2 d}{3} ω^{2} = \frac{G m (2 m)}{d^{2}} & [∵ r_{1} = \frac{2 d}{3}] \\ \Rightarrow & ω^{2} = \frac{3 G m}{d^{3}} \\ \Rightarrow & ω = \sqrt{\frac{3 G m}{d^{3}}} \end{matrix}$
Step 4: Calculate the time period of revolution using the formula
$\begin{array}{rcl} T & = & \frac{2 π}{ω} \\ = & \frac{2 π}{\sqrt{\frac{3 G m}{d^{3}}}} \\ = & 2 π \sqrt{\frac{d^{3}}{3 G m}} \end{array}$
Hence, the time period of revolution is $2 π \sqrt{\frac{d^{3}}{3 G m}}$ .

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Two stars of masses m and 2m at a distance d rotate about their common center of mass in free space. The period of revolution is:2πd33Gm12π3Gmd312πd33Gm2π3Gmd3

Two stars of masses $m$ and $2 m$ at a distance $d$ rotate about their common center of mass in free space. The period of revolution is:

$2 π \sqrt{\frac{d^{3}}{3 G m}}$
$\frac{1}{2 π} \sqrt{\frac{3 G m}{d^{3}}}$
$\frac{1}{2 π} \sqrt{\frac{d^{3}}{3 G m}}$
$2 π \sqrt{\frac{3 G m}{d^{3}}}$