Question

Find the c.m. of a uniform $L-$ shaped lamina (a thin flat plate) with dimensions as shown in figure. The mass of lamina is $3\text{kg}$

And report the sum of $x$ and $y$ coordinates. Answer upto 2 decimal places.

Answer 1

The plate is of uniform density and same thickness everywhere. So its C.M. will coincide with the centroid.
Divide the given plate into two parts of area $A_1$ and $A_2$ as shown in figure. We have
$A_1=2\times 1=2{\text{m}}^2$ with its centroid $C_1(1,1/2)$ and
$A_2=1\times 1=1{\text{m}}^2$ with its centroid $C_2(1/2,3/2)$
The centroid of whole plate can be defined as,
$x_{C.M.}=\frac{A_1{x}_1+A_2{x}_2}{A_1+A_2}=\frac{2\times 1+1\times \frac{1}{2}}{2+1}=\frac{5}{6}\text{m}$
and
$y_{C.M.}=\frac{A_1{y}_1+A_2{y}_2}{A_1+A_2}=\frac{2\times \frac{1}{2}+1\times \frac{3}{2}}{2+1}=\frac{5}{6}\text{m}$

Why this question? Tip: About the origin, mass is symmetrically distributed, so x and y coordinates will be same- don't need to calculate individually.