Halliday, Resnick & Walker — Equilibrium and Elasticity: Problem Problem_74

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 mass \( m_1 \) placed in the right pan; when the mass \( m \) is placed in the right pan, it is balanced by a mass \( m_2 \) in the left pan. Show that \( m = \sqrt{m_1 m_2} \).