The $x, y$ coordinates of the center of mass of a uniform $L -$ shaped lamina of mass $3 kg$ is

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$(\frac{5}{6} m, \frac{5}{6} m)$

$(1 m, 1 m)$

$(\frac{6}{5} m, \frac{6}{5} m)$

$(2 m, 2 m)$

answer is A.

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Detailed Solution

Choosing the $x$ and y axes as shown in the figure. The coordinates of the vertices of the $L -$ shaped lamina is as shown in the figure. Divide the L-shaped lamina into three squares each of side $1 m$ and mass $1 kg$ (because the lamina is uniform). By symmetry, the center of mass $C_{1}, C_{2} a n d C_{3}$ of the squares are their geometric centers and have coordinates $c_{1} (\frac{1}{2}, \frac{1}{2}), C_{2} (\frac{3}{2}, \frac{1}{2})$ and $C_{3} (\frac{1}{2}, \frac{3}{2})$ respectively.

The coordinates of the center of mass of the $L -$ shaped lamina is $X_{CM}$ = $\frac{m_{1} C_{1} + m_{2} x C_{2} + m_{3} C_{3}}{m_{1} + m_{2} + m_{3}}$ = $\frac{1 \times 1 / 2 + 1 \times 3 / 2 + 1 \times 1 / 2}{1 + 1 + 1}$ = $\frac{5}{6} m$

$Y_{CM} = \frac{m_{1} y_{1} + m_{2} y_{2} + m_{3} y_{3}}{m_{1} + m_{2} + m_{3}} = \frac{1 \times \frac{1}{2} + 1 \times \frac{1}{2} + 1 \times \frac{3}{2}}{1 + 1 + 1} = \frac{5}{6} m$