Question

Find the center of mass of a uniform L shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is 3 Kg.

Answer 1

Choosing the X and Y axes. we have the coordinates of the vertices of the L. shaped lamina as given in the figure. We can think of the L-shape to constant of 3 square each of length 1m. The mass of each square is 1 Kg. Since the lamina is uniform.

The center of mass of each square in X,Y coordinate $C_1(1/2,1/2),C_2(3/2,1/2).and{C}_3(1/2,3/2)$ of the square are by,
Formula for center of mass.

$X=\frac{m_a{x}_a+m_b{x}_b+m_c{x}_c}{m_a+m_b+m_c}$

$Y=\frac{m_a{y}_a+m_b{y}_b+m_c{y}_c}{m_a+m_b+m_c}$

$X=\frac{[1\left(1/2\right)+1\left(3/2\right)+1\left(1/2\right)]}{(1+1+1)}=\frac{5}{6}m$
$X=\frac{[1\left(1/2\right)+1\left(1/2\right)+1\left(3/2\right)]}{(1+1+1)}=\frac{5}{6}m$

Hence, center of mass of $L$ shape is $(\frac{5}{6},\frac{5}{6})$