Answer
We can rank the forces according to the magnitudes of the torques they produce on the particles about the origin:
$1 = 4 \gt 2 = 3 = 5 = 6$
Work Step by Step
We can write a general expression for the magnitude of the torque:
$\tau = r ~F~sin~\phi$
We can find an expression for the magnitude of the torque produced by each force:
Force 1:
$\tau_1 = (\sqrt{2})(F)~sin~90^{\circ} = \sqrt{2}~F$
Force 2:
$\tau_2 = (\sqrt{2})(F)~sin~135^{\circ} = F$
Force 3:
$\tau_3 = (\sqrt{2})(F)~sin~45^{\circ} = F$
Force 4:
$\tau_4 = (\sqrt{2})(F)~sin~90^{\circ} = \sqrt{2}~F$
Force 5:
$\tau_5 = (\sqrt{2})(F)~sin~135^{\circ} = F$
Force 6:
$\tau_6 = (\sqrt{2})(F)~sin~45^{\circ} = F$
We can rank the forces according to the magnitudes of the torques they produce on the particles about the origin:
$1 = 4 \gt 2 = 3 = 5 = 6$
