Lamina at equilibrium

If a lamina is allowed to freely rotate about a point it will rotate until the line between the point and the center of mass is vertical.

Example

Given the following shape is suspended from $A$, find the angle $AF$ makes with the vertical.

Find the total area

$$9 \times 8 + 4 \times 3 = 84$$

Find the center of mass

$$\begin{align} \frac{9 \times 8}{84} \times \begin{pmatrix} 4 \\ 4.5 \end{pmatrix} + \frac{4 \times 3}{84} \times \begin{pmatrix} 10 \\ 1.5 \end{pmatrix} &= \begin{pmatrix} x \\ y \end{pmatrix}\\ &= \begin{pmatrix} \frac{24}{7} + \frac{10}{7} \\ \frac{27}{7} + \frac{3}{14} \end{pmatrix} \\ &= \begin{pmatrix} \frac{34}{7} \\ \frac{57}{14} \end{pmatrix} \\ \end{align}$$

Find the angle

$$\begin{align} \tan(\theta) &= \frac{\frac{34}{7}}{\frac{57}{14}}\\ \theta &= 50.0^\circ\\ \end{align}$$