Two equal-mass stars maintain a constant distance apart of 6.0 1010 m and rotate about a point midway between them at a rate of one revolution every 12.0 yr.
(a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?
I will be happy to critique your thinking or work on this. Please don't post under multiple names.
(a) The two stars do not crash into one another due to the gravitational force between them because their motion is determined by the centripetal force acting on each star, which is provided by the gravitational force.
To understand why they don't crash, we can start by considering the gravitational force between the two stars. The gravitational force between two objects can be described by the equation:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In this case, since the stars have equal masses, we can simplify the equation to:
F = G * (m^2) / r^2
Now let's consider the centripetal force acting on each star. The centripetal force is the force required to keep an object moving in a circular path. It is given by the equation:
F = m * (v^2) / r
Where m is the mass of the star, v is the linear velocity of the star, and r is the distance from the star to the center of rotation.
Since the stars are rotating about a point midway between them, the linear velocity for each star is the same. We can assume that the distance between the stars remains constant.
Now, the crucial point to consider is that in order for the stars to maintain a constant distance apart and not crash into each other, the centripetal force acting on each star must be equal to the gravitational force between them.
Therefore, equating the equations for centripetal force and gravitational force, we can write:
m * (v^2) / r = G * (m^2) / r^2
Simplifying the equation further, we find:
(v^2) = G * m / r
Notice that the mass of the second star cancels out, indicating that the mass of each star does not affect their ability to maintain their distance apart. Therefore, the stars can maintain a constant distance because their rotation speed and the gravitational constant determine the necessary force to counteract the gravitational attraction.
(b) To find the mass of each star, we can rearrange the equation we derived earlier:
(v^2) = G * m / r
Now, we know that the stars rotate about a point midway between them at a rate of one revolution every 12.0 years. One revolution implies a full circle, and since the radius of rotation is half the distance between the stars (6.0 * 10^10 m / 2), we can calculate the linear velocity using the formula:
v = (2 * π * r) / T
Where T is the period of rotation, in this case, 12.0 years.
Plugging in the values, we can calculate the linear velocity. With the known values of G, r, and v, we can finally solve for the mass of each star:
(m^2) = (v^2) * r / G
Taking the square root of both sides, we find the mass of each star.