Two black holes (the remains of exploded stars), separated by a distance of
10.0 AU
(1 AU = 1.50 ✕ 10^11 m),
attract one another with a gravitational force of
6.50 ✕ 10^25 N.
The combined mass of the two black holes is
5.40 ✕ 10^30 kg.
What is the mass of each black hole?
largest value-----kg
smallest value-----kg
To find the mass of each black hole, we can use Newton's law of gravitation. The formula for gravitational force is:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant (6.67 × 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses of the two objects (in this case, the black holes), and r is the distance between the centers of the objects.
We're given the gravitational force (F = 6.50 × 10^25 N) and the distance (r = 10.0 AU = 1.50 × 10^12 m). We'll also assume that the mass of the two black holes is the same (m1 = m2 = m).
Substituting these values into the formula, we can solve for the mass (m):
6.50 × 10^25 N = (6.67 × 10^-11 m^3 kg^-1 s^-2) * (m * m) / (1.50 × 10^12 m)^2
Simplifying,
6.50 × 10^25 N = (6.67 × 10^-11 m^3 kg^-1 s^-2) * m^2 / (1.50 × 10^12 m)^2
Rearranging the equation,
m^2 = (6.50 × 10^25 N * (1.50 × 10^12 m)^2) / (6.67 × 10^-11 m^3 kg^-1 s^-2)
Calculating,
m^2 ≈ 3.9 × 10^48 kg^2
To find the mass of each black hole, we take the square root of this value:
m ≈ √(3.9 × 10^48 kg^2)
Using a calculator,
m ≈ 6.24 × 10^24 kg
Therefore, the mass of each black hole is approximately 6.24 × 10^24 kg.
It's important to note that the calculation assumes equal masses for both black holes.
To find the mass of each black hole, we can use the gravitational force equation:
F = (G * m1 * m2) / r^2
where:
F is the gravitational force,
G is the gravitational constant (6.67 x 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the black holes,
and r is the distance between the black holes.
Given:
F = 6.50 x 10^25 N,
r = 10.0 AU = 10.0 x 1.50 x 10^11 m = 1.50 x 10^12 m,
G = 6.67 x 10^-11 N m^2/kg^2.
Using the formula, we can rearrange it to solve for the mass of one black hole:
m1 * m2 = (F * r^2) / G
Substituting the given values:
m1 * m2 = (6.50 x 10^25 N) * (1.50 x 10^12 m)^2 / (6.67 x 10^-11 N m^2/kg^2)
Calculating this, we get:
m1 * m2 = 3.675 x 10^50 kg^2
Since the combined mass of the two black holes is 5.40 x 10^30 kg, we can express it as:
m1 + m2 = 5.40 x 10^30 kg
Now we have a system of equations to solve. We can use the quadratic formula to find the masses.
Let's assume that the masses of the black holes are m1 and m2, where m1 is the larger mass.
From the equation m1 + m2 = 5.40 x 10^30 kg, we can express m2 in terms of m1:
m2 = 5.40 x 10^30 kg - m1
Substituting m2 in the equation m1 * m2 = 3.675 x 10^50 kg^2:
m1 * (5.40 x 10^30 kg - m1) = 3.675 x 10^50 kg^2
Expanding and rearranging the equation:
5.40 x 10^30 m1 - m1^2 = 3.675 x 10^50
Rearranging it as a quadratic equation:
m1^2 - 5.4 x 10^30 m1 + 3.675 x 10^50 = 0
Now we can apply the quadratic formula:
m1 = [-(-5.4 x 10^30) ± sqrt((-5.4 x 10^30)^2 - 4(1)(3.675 x 10^50))] / (2(1))
Calculating this equation, we get two solutions for m1:
m1 = 5.423 x 10^30 kg
m1 = 6.771 x 10^20 kg
Since we assumed that m1 is the larger mass, the largest value for the mass of each black hole is 5.423 x 10^30 kg, and the smallest value is 6.771 x 10^20 kg.