Question

a 20kg sphere is at the origin and a 10kg sphere is at x=20 cm. at what position on the x-axis could you place a small mass such that the net gravitational force on it due to the spheres is zero. I can get all the way to the point where r needs to be solved for. please show how to solve for r.

Answer 1

To zero out, the two masses M = 20 kg and m = 10 kg must exert equal but opposite forces on the third object. This will happen when the respective gravity fields g are equal.

Thus g = GM/R^2 = Gm/r^2 = g must be true (these are the respective gravity fields g). Thus, M/m = (R/r)^2 and R^2 = (M/m)r^2 so that R = r sqrt(M/m) = r sqrt(20/10).

As the distance between the two masses is D = r + R = 20 cm = r (1 + sqrt(2)), we have r = D/(1 + sqrt(2))

= 20/(1 + sqrt(2)) = 8.28 cm then R = 20 - 8.28 = 11.72 cm

Then the point from M at the origin is p(8.28,0) ANS.