# 4.2.3 Rotational Equilibrium (of hinged objects)

A body can remain non-rotating only if all the moments acting on it sum up to zero. For a body to be in rotational equilibrium, it is necessary that

$\displaystyle \sum{\tau }=0$

If we restrict ourselves to 2D problems, $\displaystyle \sum{\tau }=0$ simplifies into

Sum of clockwise moments = Sum of anti-clockwise moments

Example

A hinged uniform rod of length L is held stationary at an angle of 22° by two forces: F1 and F2 applied at the end and midpoint of the rod respectively. Calculate the ratio of the magnitude of F1 to F2.

Solution

\displaystyle \displaystyle \begin{aligned}\text{sum of CW moments}&=\text{sum of ACW moments}\\{{F}_{2}}\times \frac{{L\cos 22{}^\circ }}{2}&={{F}_{1}}\times L\sin 22{}^\circ \\\frac{{{{F}_{1}}}}{{{{F}_{2}}}}&=\frac{1}{2}\cot 22{}^\circ \\&=1.24\end{aligned}

Example Problem

Hinged Beam