Question

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

• A. $\left( \frac{5}{6}m,\frac{5}{6}m \right)$
• B. (1m, 1m)
• C. $\left( \frac{6}{5}m,\frac{6}{5}m \right)$
• D. (2m, 2m)

Answer 1

Answer: (A)

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

Answer 2

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 1m and mass 1 kg ($\because$the lamina is uniform) by symmetry, the centres of mass $C_{1},C_{2}$and $C_{3}$of the squares are their geometric centres and have coordinates $C_{1}\left( \frac{1}{2},\frac{1}{2} \right),C_{2}\left( \frac{3}{2},\frac{1}{2} \right)andC_{3}\left( \frac{1}{2},\frac{3}{2} \right)$respectively.

The coordinates of the centre of mass of the L-shaped lamina is

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