Chapter 12, Problem 74P

A pan balance is made up of a rigid, massless rod with a hanging pan attached at each end. The rod is supported at and free to rotate about a point not at its center. It is balanced by unequal masses placed in the two pans. When an unknown mass m is placed in the left pan, it is balanced by a miss m₁, placed in the right pan; when the mass m is placed in the right pan, it is balanced by a mass m₂ in the left pan. Show that $m=\sqrt{m_1{m}_2}$.