Principle of Moments

Consider a light rod (of negligible mass) pivoted at a single point along its length, with two parallel forces $F_1$ and $F_2$ acting at its two ends, at distances $d_1$ and $d_2$ respectively from the pivot, along with the pivot's own normal reaction force $N$. For the rod to remain stationary and horizontal, it must be in both translational and rotational equilibrium — both the net force and the net torque on it must be zero.

Net force zero:

(the pivot's upward reaction exactly balances the two downward forces).

Net torque zero, taken about the pivot itself (so that $N$, passing through the pivot, contributes no torque there):

This is the principle of moments: for the rod to balance about its pivot, the turning moment on one side must exactly equal the turning moment on the other side. It can also be rearranged as $\dfrac{F_1}{F_2}=\dfrac{d_2}{d_1}$.

This principle is exactly what makes a beam balance used for weighing goods work, in the special symmetric case $d_1=d_2$ and $F_1=F_2$. …