I tried to solve the following exercise in theoretical mechanics: Two Planets A and B with equal mass $m$and equal angular velocity $w_0$(constant) move around the sun $S$ with mass $M$ in circular concentric orbits with radius $r_1$ for Planet A and radius $r_2$ for Planet B, but such that they are always on the straight line SAB. Find the angular velocity and the mass of the planets.
My idea:
If you look at Planet A: There is a gravitational force towards the sun but also a gravitational force towards B. The addition of both forces must be the centripetal (radial) force because the planet is moving on a circle, so we can write:
$mr_1w_0^2= \frac{GmM}{r_1^2}-\frac{Gm^2}{(r_2-r_1)^2} $
The minus can be explained by the direction of the forces: the gravitational force towards the Sun ans planet B are in opposite directions.
This can be also done for Planet B but here the gravitational force towards the sun and the planet A are in same direction, thus:
$mr_2w_0^2= \frac{GmM}{r_2^2}+\frac{Gm^2}{(r_2-r_1)^2} $
I can use both of the equations to determine $w_0$ and $m$ but are they even correct? Am I even on the right track? I’d be very thankful if someone could take a look :)