two binary stars moving in a perfect circle orbit
The stars have the same mass, the distance between them is one billion km (1x10^9 km), and the time each takes for one orbit is 10.4 Earth years.
Determine the mass of each star (m1=m2).
(this problem DOESN'T uses the gravity formula with the G constant.
this problem must be solve using circular dynamics ecuations so please help!!!!)
Actually, the solution will involve the G constant. You will have to use it to compute the mass, unless you use Kepler's third law in a different form the involves the sum of the masses of the two objects, in terms of solar mass. "G" is already "built in" to that solution
Equal-mass stars revolve in orbits (circular in this case) about a point midway between the stars. The radius of each star's orbit is d/2, where d is the interstellar separation, 10^12 m.
Centripetal force = Gravity force
Let either mass be m.
G*m^2/d^2 = m*V^2/(d/2)= 2m*V^2/d
or G*m/d = 2*V^2
V*Period = 2*pi*d/2 = pi*d
Eliminate V from the first equation, using V from the second equation, and solve for the mass, m
V = pi*d/Period = 9573 m/s
m = 2*d*V^2/G
To determine the mass of each star in this scenario, we can use circular dynamics equations that relate the mass of the stars, the distance between them, and the time taken for one orbit.
In a circular orbit, the centripetal force required to keep an object moving in a circle is provided by the gravitational force between the two objects. The centripetal force (F_c) can be expressed as:
F_c = G * m1 * m2 / r^2
Where G is the gravitational constant, m1 and m2 are the masses of the stars, and r is the distance between them.
The gravitational force between the two objects can also be expressed as:
F_g = G * m1 * m2 / r^2
Since both stars have the same mass (m1 = m2), we can rewrite the equations as:
F_c = G * m^2 / r^2
F_g = G * m^2 / r^2
In this case, the centripetal force (F_c) is provided by the gravitational force (F_g), so we can equate them:
F_c = F_g
G * m^2 / r^2 = G * m^2 / r^2
Now we can substitute the given values to solve for the mass (m):
G * m^2 / (1x10^9 km)^2 = G * m^2 / (1x10^9 km)^2
The distance between the stars is one billion kilometers, so we can convert it to meters:
G * m^2 / (1x10^9 km)^2 = G * m^2 / (1x10^9 km)^2
Since the equation is balanced on both sides, we can cancel out the G, (1x10^9 km)^2, and m^2 terms:
m^2 = m^2
This shows that the mass of each star is equal (m1 = m2). Therefore, the mass of each star in this binary system is the same.