Two black holes (the remains of exploded stars), separated by a distance of 10.0
ID: 2078829 • Letter: T
Question
Two black holes (the remains of exploded stars), separated by a distance of 10.0 AU (1 AU = 1.50 times 10^11 m), attract one another with a gravitational force of 5.80 times 10^25 N. The combined mass of the two black holes is 4.90 times 10^30 kg, what is the mass of each black hole? Step 1 We first create an equation for the mass of one of the black holes in terms of the other and then write the equation for the gravitational force between the black holes. We solve a quadratic equation for the mass in the first equation to find the two possible solutions for the masses of the black holes. Step 2 Let M be the combined mass of the two black holes. Solving for m_2 in terms of m_1, we have m_2 = M - m_1. Substituting this expression for m_2 into the formula for the magnitude F_G of the gravitational force F_G = G m_1 m_2/r^2 = G m_1(M - m_1)/r^2 = G m_1 M - m_1^2/r^2, where G = 6.67 times 10^-11 N middot m^2/kg^2, and r is the separation between the two black holes. Rearranging the equation above into the quadratic form gives Gm_1^2 - GMm_1 + F_G^r^2 = 0. Solving for m_1 using the quadratic formula, we have m_1 = GM plusminus Squareroot (GM)^2 - (4 GF_G^r^2)/2G = M plusminus Squareroot M^2 - (4 F_G^r^2/G)/2 = (2450000000000 (kg) plusminus (2449489743000 kg).Explanation / Answer
Gravitational Force = G m1 m2 / r^2
5.80 x 10^25 = (6.67 x 10^-11)(m1 m2) / (10 x 1.50 x 10^11)^2
m1 m2 = 1.9565 x 10^60 ...... (i)
and m1 + m2 = 4.90 x 10^30
m1 (4.90 x 10^30 - m1) = 1.9565 x 10^60
m1^2 - 4.90 x 10^30 m1 + 1.9565 x 10^60 = 0
m1 = 4.46 x 10^30 Or 4.39 x 10^29 kg ....These are two masses of black holes. ( ---------Ans)
[ If you want that square root term.
that will be = 2.01 x 10^30 kg ]