Balls are attached to light rods and can move in horizontal circles as shown in the figure. Rank in order, from smallest to largest, the torques T1 to T4 about the centers of the circles. Explain.

No need to worry. You don't need to confuse because of the masses. It does not matter. Torque is just FXr i.e. the radius multiply by the component of force perpendicular to it, or the Force multiplied by the component of radius perpendicular to it. In this case, F and r are perpendicular all the time. So the Torques are:
a)Fr

b)2Fr

c)F2r=2Fr

d)2F2r=4Fr

Now, the order can be easliy assigned. T4>T3=T2>T1