First slide

Location of centre on mass for multiple point masses

Question

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

Moderate

A

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

$(1 m, 1 m)$
C

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

$(2 m, 2 m)$

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 ( $∵$ 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 $C_{1} (\frac{1}{2}, \frac{1}{2}), C_{2} (\frac{3}{2}, \frac{1}{2}) and C_{3} (\frac{1}{2}, \frac{3}{2})$ respectively.
The coordinates of the centre of mass of the L-shaped lamina is
$X_{CM} = \frac{m_{1} x_{1} + m_{2} x_{2} + m_{3} x_{3}}{m_{1} + m_{2} + m_{3}}$
= $\frac{1 \times \frac{1}{2} + 1 \times \frac{3}{2} + 1 \times \frac{1}{2}}{1 + 1 + 1} = \frac{5}{6} m$
$Y_{CM} = \frac{m_{1} y_{1} + m_{2} y_{2} + m_{3} y_{3}}{m_{1} + m_{2} + m_{3}}$
= $\frac{1 \times \frac{1}{2} + 1 \times \frac{1}{2} + 1 \times \frac{3}{2}}{1 + 1 + 1} = \frac{5}{6} m$

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