Question

two bodyes A and B of mass m and 2m respectively are kept at a distance d apart. where should be place a third body in between them so that the net gravitational force due to bodies on it is 0?

Answer 1

Here,

m₁ = m

m₂ = 2m

To make the net gravitational force between them 0 let a mass x be placed at a distance r from the mass m

Force between mass x and m will be

F_x,m = Gxm/r²

Force between mass x and 2m will be

F_x,2m = G2mx/(d-r)²

Now force between masses 2m and x must be equal to the force between x and m

So,

F_x,m = F_x,2m

Gxm/r² = G2mx/(d-r)²

=> 2r² = (d –r)²

=> d² – 2rd + r² = 2r²

=> r² + 2rd – d² = 0

=> r = {-2d +√(4d² + 4d²)}/2

=> r = (√2- 1)d

The 2nd mass x must be placed at a distance (√2- 1)d from the small mass m