Two identical stars of mass $M$orbit around their centre of mass. Each orbit is circular and has radius $R$, so that the two stars are always on opposite sides of the circle, the velocity of each body is..

Solution:

Given:

• Each star has a mass $M$.
• Each orbit has a radius $R$.
• The two stars are always on opposite sides of the circle, implying that the distance between the two stars is $2R$.

In a two-body system like this, where both bodies have identical masses and orbit around their common center of mass, the gravitational force between the two stars provides the necessary centripetal force to keep each star in its circular orbit. The gravitational force $F$ between the two stars is given by Newton's law of universal gravitation: $$F = \frac{G M^2}{(2R)^2}$$ where $G$ is the gravitational constant. The centripetal force needed to keep each star in its circular orbit is: $$F_c = \frac{M v^2}{R}$$ Setting the gravitational force equal to the centripetal force (since they are the same in this case) and solving for the velocity $v$: $$\frac{G M^2}{(2R)^2} = \frac{M v^2}{R}$$ Simplify and solve for $v$: $$v^2 = \frac{G M}{4R}$$ $$v = \sqrt{\frac{G M}{4R}}$$ This expression gives us the velocity of each body in their orbit. Let's calculate this velocity considering the provided variables $M$, $R$, and $G$.