Question

A 19 kg sphere is at the origin and a 12 kg sphere is at (x,y)=( 18 cm , 0 cm) . At what point or points could you place a small mass such that the net gravitational force on it due to the spheres is zero? Express your answers using two significant figures separated by a comma.

Answer 1

Let x be the position of the mass from the origin.

Then, the net gravitational field strength on it will be:

$F_{net} = G\left( -\frac{19}{x^2} - \frac{-12}{(0.18-x)^2} \right ) = 0$

=> $\frac{19}{x^2} = \frac{12}{(0.18-x)^2}$

12x² = 0.6156 - 6.84x + 19x²

=> 7x² - 6.84x + 0.6156 = 0

solving this gives x = 0.8768 m and x = 0.1003 m

since x = 0.8768 m will be to the right of both the spheres, it will be unstable and so it must be discarded.

Therefore, x = 0.1003 m = 10.03 cm is where the small mass must be for the net gravitational force to be zero on it.