Answer
We can rank the three possible locations according to the work done by the net gravitational force on the moving particle due to the fixed particles:
$b \gt a \gt c$
Work Step by Step
Note that at an infinite distance away: $~~U = 0$
We can write an expression for the change in gravitational potential energy if the particle is moved to location $a$:
$\Delta U_a = -\frac{G(2m)(m)}{d}-\frac{G(m)(m)}{3d}$
$\Delta U_a = -\frac{6Gm^2}{3d}-\frac{Gm^2}{3d}$
$\Delta U_a = -\frac{7}{3}~(\frac{Gm^2}{d})$
We can write an expression for the net work done on the moving particle by the net gravitational force due to the fixed particles:
$W_g = -\Delta U_a$
$W_g = \frac{7}{3}~(\frac{Gm^2}{d})$
We can write an expression for the change in gravitational potential energy if the particle is moved to location $b$:
$\Delta U_b = -\frac{G(2m)(m)}{d}-\frac{G(m)(m)}{d}$
$\Delta U_b = -\frac{2Gm^2}{d}-\frac{Gm^2}{d}$
$\Delta U_b = -3~(\frac{Gm^2}{d})$
We can write an expression for the net work done on the moving particle by the net gravitational force due to the fixed particles:
$W_g = -\Delta U_b$
$W_g =3~(\frac{Gm^2}{d})$
We can write an expression for the change in gravitational potential energy if the particle is moved to location $c$:
$\Delta U_c = -\frac{G(2m)(m)}{3d}-\frac{G(m)(m)}{d}$
$\Delta U_c = -\frac{2Gm^2}{3d}-\frac{3Gm^2}{3d}$
$\Delta U_c = -\frac{5}{3}~(\frac{Gm^2}{d})$
We can write an expression for the net work done on the moving particle by the net gravitational force due to the fixed particles:
$W_g = -\Delta U_c$
$W_g = \frac{5}{3}~(\frac{Gm^2}{d})$
We can rank the three possible locations according to the work done by the net gravitational force on the moving particle due to the fixed particles:
$b \gt a \gt c$
