1. **Problem statement:** Three identical stars, each of mass $M$, orbit in a circle of radius $R$ around their center of mass (CM), positioned at the vertices of an equilateral triangle. We need to find: a) The magnitude and direction of the force $F$ on each star. b) The period $T$ of their orbit. 2. **Setup and important rules:** - The stars form an equilateral triangle inscribed in a circle of radius $R$. - Each star experiences gravitational forces from the other two stars. - The net force on each star provides the centripetal force for circular motion. - Newton's law of gravitation: $$F_g = G_N \frac{M^2}{r^2}$$ where $r$ is the distance between two masses. - The distance between two stars (side of the equilateral triangle) is $$s = \sqrt{3} R$$ because the side length of an equilateral triangle inscribed in a circle of radius $R$ is $s = \sqrt{3} R$. 3. **Calculate the gravitational forces on one star:** - Each star feels force from the other two stars. - Magnitude of force from one star on another: $$F_g = G_N \frac{M^2}{s^2} = G_N \frac{M^2}{(\sqrt{3} R)^2} = G_N \frac{M^2}{3 R^2}$$ 4. **Find the net force magnitude and direction:** - The two forces from the other stars are equal in magnitude and separated by $60^\circ$ (angle of equilateral triangle). - Using vector addition, the resultant force magnitude is: $$F = 2 F_g \cos(30^\circ) = 2 \times G_N \frac{M^2}{3 R^2} \times \frac{\sqrt{3}}{2} = G_N \frac{M^2}{3 R^2} \sqrt{3} = \frac{G_N M^2 \sqrt{3}}{3 R^2}$$ - Direction: The net force points toward the center of the circle (the CM), providing the centripetal force. 5. **Find the period $T$ of the orbit:** - The centripetal force on each star is: $$F = M \frac{v^2}{R}$$ - Equate gravitational net force to centripetal force: $$\frac{G_N M^2 \sqrt{3}}{3 R^2} = M \frac{v^2}{R} \implies v^2 = \frac{G_N M \sqrt{3}}{3 R}$$ - The orbital period $T$ is related to velocity by: $$T = \frac{2 \pi R}{v} = 2 \pi R \sqrt{\frac{3 R}{G_N M \sqrt{3}}} = 2 \pi \sqrt{\frac{3 R^3}{G_N M \sqrt{3}}}$$ - Simplify: $$T = 2 \pi \sqrt{\frac{\sqrt{3} R^3}{G_N M}}$$ **Final answers:** - a) Force magnitude on each star: $$F = \frac{G_N M^2 \sqrt{3}}{3 R^2}$$ directed toward the center (CM). - b) Orbital period: $$T = 2 \pi \sqrt{\frac{\sqrt{3} R^3}{G_N M}}$$