Two spheres are placed so that their centers are 2.6 m apart. The force between the two spheres is 2.75 x 10^-12 N. What is the mass of each sphere if one sphere is twice the mass of the other sphere?(please use all the correct formula, and show all working because I want to understand how you arrived at your answer).
To determine the mass of each sphere, we can use the concept of gravitational force between two objects. Given that the force and distance between the spheres are known, we can use the formula for gravitational force:
F = (G * m1 * m2) / r^2
where:
F is the force between the two spheres
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the spheres
r is the distance between the centers of the spheres
Let's plug in the values into the formula and solve for the masses:
2.75 × 10^-12 N = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * m1 * m2 / (2.6 m)^2
To simplify further, let's divide both sides of the equation by (6.67430 × 10^-11 m^3 kg^-1 s^-2) and multiply by (2.6 m)^2:
[(2.75 × 10^-12 N) * (2.6 m)^2] / (6.67430 × 10^-11 m^3 kg^-1 s^-2) = m1 * m2
Now, we know that one sphere has twice the mass of the other. Let's assume the smaller mass is m and the larger mass is 2m:
[(2.75 × 10^-12 N) * (2.6 m)^2] / (6.67430 × 10^-11 m^3 kg^-1 s^-2) = m * (2m)
Simplifying further, we get:
(2.75 × 10^-12 N) * (2.6 m)^2 / (6.67430 × 10^-11 m^3 kg^-1 s^-2) = 2m^2
Now, let's solve for the mass m:
m^2 = [(2.75 × 10^-12 N) * (2.6 m)^2 / (6.67430 × 10^-11 m^3 kg^-1 s^-2)] / 2
m = square root of {[(2.75 × 10^-12 N) * (2.6 m)^2 / (6.67430 × 10^-11 m^3 kg^-1 s^-2)] / 2}
By calculating this expression using a calculator, we find that m is approximately 1.06 × 10^-9 kg.
Since the other sphere is twice the mass of m, its mass would be 2m:
2m ≈ 2 * 1.06 × 10^-9 kg ≈ 2.12 × 10^-9 kg
Therefore, the mass of each sphere is approximately 1.06 × 10^-9 kg and 2.12 × 10^-9 kg, respectively.