Can You Pivot 5?

Setting up a coordinate system so that $X$ has coordinates $(0,0)$ , $Y$ has coordinates $(6,3)$ and $Z$ has coordinates $(18,0)$ , the balance beam (assumed to be a uniform lamina) has centre of mass at the centroid of $XYZ$ , which has coordinates $(8,1)$ , and so the centre of mass of the balance beam and the two masses is $G$ with coordinates $(\xi,\eta)$ , where $\xi \; = \; \tfrac{0\times10 + 8\times5+18\times1000}{1015} \; = \; \tfrac{3608}{203} \qquad \eta \; = \;\tfrac{0 \times 10 + 1 \times 5 + 0 \times 1000}{1015} \; = \; \tfrac{1}{203}$ In equilibrium, the system will hang with $G$ vertically below $Y$ , and so $\theta \; = \; 180^\circ - \angle XYG \; = \; 180^\circ - \tan^{-1}2 - \tan^{-1}\tfrac{\xi-6}{3-\eta} \; = \; \boxed{40.84^\circ}$