If the only force exerted on a star far from the center of the Galaxy (r = 6.40 ✕ 1020 m) is the gravitational force exerted by the ordinary matter (Mord = 2.10 ✕ 1041 kg), find the speed of the star. Assume a circular orbit and assume all the Galaxy's matter is concentrated at the center.
To find the speed of the star, we can use the centripetal force equation, which states that the gravitational force acting on the star is equal to the centripetal force keeping it in orbit.
The gravitational force is given by the equation:
F_grav = G * (M1 * M2) / r^2
Where:
F_grav is the gravitational force
G is the gravitational constant (6.67430 * 10^-11 N*(m/kg)^2)
M1 is the mass of the star
M2 is the mass of the Galaxy's matter (assumed to be concentrated at the center)
r is the distance between the star and the center of the Galaxy
The centripetal force is given by the equation:
F_c = M1 * v^2 / r
Where:
F_c is the centripetal force
M1 is the mass of the star
v is the speed of the star
r is the distance between the star and the center of the Galaxy
Since the centripetal force is equal to the gravitational force, we can set these equations equal to each other:
G * (M1 * M2) / r^2 = M1 * v^2 / r
We can simplify this equation to solve for the speed v:
v^2 = G * M2 / r
Taking the square root of both sides, we get:
v = sqrt(G * M2 / r)
Now we can plug in the given values and calculate the speed:
G = 6.67430 * 10^-11 N*(m/kg)^2
M2 = 2.10 * 10^41 kg
r = 6.40 * 10^20 m
v = sqrt((6.67430 * 10^-11 N*(m/kg)^2) * (2.10 * 10^41 kg) / (6.40 * 10^20 m))
Calculating this expression will give us the speed of the star.