Find the centre of mass of a uniform $L$-shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is $3 \; kg$.

(N/A) Choosing the $X$ and $Y$ axes as shown in the figure,we have the coordinates of the vertices of the $L$-shaped lamina. We can think of the $L$-shape as consisting of $3$ squares,each of side length $1 \; m$. Since the lamina is uniform,the mass of each square is $1 \; kg$. The centres of mass $C_{1}$,$C_{2}$,and $C_{3}$ of the squares are,by symmetry,their geometric centres. Their coordinates are $(0.5, 0.5) \; m$,$(1.5, 0.5) \; m$,and $(0.5, 1.5) \; m$ respectively. We take the masses of the squares to be concentrated at these points. The centre of mass $(X, Y)$ of the whole $L$-shape is the centre of mass of these mass points.
Hence,
$X = \frac{[1(0.5) + 1(1.5) + 1(0.5)] \; kg \cdot m}{(1 + 1 + 1) \; kg} = \frac{2.5}{3} \; m = \frac{5}{6} \; m$
$Y = \frac{[1(0.5) + 1(0.5) + 1(1.5)] \; kg \cdot m}{(1 + 1 + 1) \; kg} = \frac{2.5}{3} \; m = \frac{5}{6} \; m$