Three identical stars of mass M form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length L. What is the speed of the stars?
To find the speed of the stars, we can start by considering the forces acting on one of the stars. Since the stars are moving in a circle about the center of the equilateral triangle, there must be a net inward force acting on each star, providing the necessary centripetal force to keep them in circular motion.
The magnitude of the centripetal force can be found using the equation:
F = m * v^2 / r
where F is the centripetal force, m is the mass of one of the stars, v is the speed of the star, and r is the distance from the star to the center of the circle.
Since the three stars are identical and form an equilateral triangle, the distance from each star to the center of the circle is L/√3. Therefore, we can rewrite the equation as:
F = m * v^2 / (L/√3)
Next, we consider the gravitational force between two stars. The magnitude of the gravitational force is given by:
F = (G * m^2) / r^2
where G is the gravitational constant and r is the distance between the stars.
Since the masses of the stars are equal, we can simplify the equation as:
F = (G * m^2) / r^2 = (G * m^2) / (L^2)
To find the net inward force, we multiply the gravitational force by the cosine of the angle formed by the line connecting two stars and the normal to the circular path. In an equilateral triangle, this angle is 30 degrees. Therefore, the net inward force is:
F_net = (G * m^2) / (L^2) * cos(30°)
Since there are three stars in the system, the net inward force must be three times greater than the centripetal force:
3 * F = (G * m^2) / (L^2) * cos(30°)
Now, we can substitute the centripetal force equation into this equation:
3 * (m * v^2 / (L/√3)) = (G * m^2) / (L^2) * cos(30°)
Simplifying further, we can cancel the mass term and rearrange the equation to solve for v:
v^2 = (G * √3) / (L * cos(30°))
Finally, taking the square root of both sides, we can find the speed of the stars:
v = √[(G * √3) / (L * cos(30°))]
You can substitute the numerical values for G, √3, and cos(30°) to calculate the speed of the stars.