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Step-by-Step Solution
Step 1: Identify the Force Providing Centripetal Motion
Each mass, of mass $m$, moves in a circular orbit under the influence of the gravitational attraction of the other mass. This mutual gravitational force acts as the centripetal force for each mass.
Step 2: Write the Expression for the Gravitational Force
The Newtonian gravitational force between two masses $m$ and $m$ separated by distance $r$ is:
$F_{\text{grav}} = \frac{G\,m\,m}{r^{2}} = \frac{G\,m^{2}}{r^{2}},$
where $G$ is the universal gravitational constant.
Step 3: Determine the Separation Between Masses
Since each mass revolves in a circle of radius $a$ around their common center, the separation between the two masses is $r = 2a$.
Step 4: Express the Required Centripetal Force
For uniform circular motion with angular speed $\omega$, the centripetal force needed by each mass is:
$F_{\text{centripetal}} = m\,\omega^{2}\,a.$
Step 5: Equate the Gravitational Force to the Centripetal Force
Because the gravitational force provides the centripetal force, we set:
$\frac{G\,m^{2}}{(2a)^{2}} = m\,\omega^{2}\,a.$
Which simplifies to:
$\frac{G\,m^{2}}{4\,a^{2}} = m\,\omega^{2}\,a.$
Step 6: Solve for $\omega$
Divide both sides by $m\,a$:
$\frac{G\,m}{4\,a^{3}} = \omega^{2}.$
Taking the square root:
$\omega = \sqrt{\frac{G\,m}{4\,a^{3}}}.$
Final Answer
The angular speed of each particle is
$\displaystyle \sqrt{\frac{G\,m}{4\,a^{3}}}.$
