Question

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

Answer 1

The plate has uniform density and same thickness everywhere. So its $CM$ coincide with the centroid.
Divide the given plate into two parts of area $A_1$ and $A_2$ as shown in the figure. We have
$A_1=2\times 1m^2$ with its centroid $C_1(1,1/2)$ and
$A_2=2\times 1m^2$ with its centroid $C_1(1/2,3/2)$
The centroid of the whole plate can be defined as
$\overline{x}=\frac{A_1{x}_1+A_2{x}_2}{A_1+A_2}=\frac{2\times 1+1\times \frac{1}{2}}{2+1}=\frac{5}{6}m$
$\overline{y}=\frac{A_1{y}_1+A_2{y}_2}{A_1+A_2}=\frac{2\times \frac{1}{2}+1\times \frac{3}{2}}{2+1}=\frac{5}{6}m$