Question

The dwarf planet, Pluto has 5 known moons. The largest moon, Charon, has a diameter half the size of pluto and a mass about one-eighth of Pluto’s mass. Let the distance between the centres of Charon and Pluto be d. Where along the line joining their centres (with respect to d) would a spaceship feel no net gravitational force? (Assume no other masses impact the spaceship’s gravitational force)

Answer 1

Pluto Mass: M Diameter: D | Charon: Mass: (1/8)M Diamater: (1/2)D

In order to find where there would be no net gravitational force we need to find where the force of gravity would be equal to each other

f(g) = Gm1(M)/r^2

f(p) = f(c) where p = Pluto and c = Charon

G(m1)(M)/r^2 = G(m1)(M/8)/(d-r)^2 --> start canceling variables

1/r^2 = (1/8)/(d-r)^2

1/r^2 = 1/(8)(d^2 - 2dr +r^2)

r^2 = 8d^2 - 16dr + 8r^2

7r^2 - 16dr + 8d^2 = 0

Solve for r using quadratic formula

r = 16d +- sqrt((16d)^2 - 4(7)(8d^2))/(14)

r = 16d +- sqrt(256d^2 - 224d^2) / 14

r = 16d +- sqrt(32d^2) / 14

r has to be a fraction of d because if it was greater than d, it would be on the other side of Charon

so

r = 16d - 5.6568d / 14 = 10.3432d/14 = ~0.7388d