Three identical massless rods $AC$, $CD$, and $DB$, each of length $L$, are connected by freely rota

Question

Three identical massless rods $AC$, $CD$, and $DB$, each of length $L$, are connected by freely rotating hinges at $C$ and $D$. The ends $A$ and $B$ are hinged to a rigid horizontal ceiling such that the distance $AB$ is twice the length of any rod ($AB = 2L$). The system is arranged such that $CD$ is horizontal and parallel to $AB$. A point mass $m$ is suspended from the hinge $C$. What is the minimum external force $F$ that must be applied at hinge $D$ to maintain the system in stationary equilibrium?

Professor PC Thomas Classes · Answer

Extend rods $AC$ and $BD$ downwards to intersect at point $O$. Since $AB = 2CD$ and lengths are $L$, geometry dictates $	riangle OAB$ is equilateral with side $2L$, and $OC = OD = L$. Since $AC$ and $BD$ are massless with free hinges, they only exert forces along their lengths. Taking torque about $O$, torques from $AC$ and $BD$ vanish. The lever arm of $mg$ at $C$ about $O$ is $L \sin(30^\circ) = L/2$. To minimize force $F$ at $D$, it must act perpendicular to $OD$ (lever arm $L$). Balancing torques: $F_{	ext{min}} \cdot L = mg \cdot (L/2) \implies F_{	ext{min}} = mg/2$.