Question 1)

Question

Consider two point-like objects, A and B, the first of which has four times the mass of the other (i.e. MA = 4MB). They are placed a certain distance d apart.
(i) Sketch (in 2-dimensions) the gravitational field lines that are present in the region of
space around the two masses. Show the direction of the field on the field lines.
(ii) Explain quantitatively where you should place a test particle such that it experiences a net gravitational force of zero from the two point-like objects. Indicate this position on your diagram.

Answer 1

d is distance apart
call x distance from 4 M
then (d-x) is distance from M

4M/x^2 = M/(d-x)^2
4(d-x)^2 = x^2
4(d^2 -2dx + x^2 ) = x^2
4 d^2 -8dx +3 x^2 = 0
(2d-3x)(2d-x)=0
x = 2 d (beyond M away from 4M)
x = (2/3) d (between them but closer to M)

Answer 2

In the answer, can you explain how you started at 4M/x^2 = M/(d-x)^2?
Thanks