First slide

Location of centre on mass for multiple point masses

Question

Three particles of masses 1 kg, $\frac{3}{2}$ kg, and 2kg are located the vertices of an equilateral triangle of side a. The x, y coordinates of the centre of mass are

Moderate

A

$(\frac{5 a}{9}, \frac{2 a}{3 \sqrt{3}})$
B

$(\frac{2 a}{33}, \frac{5 a}{9})$
C

$(\frac{5 a}{9}, \frac{2 a}{\sqrt{3}})$
D

$(\frac{2 a}{\sqrt{3}}, \frac{5 a}{9})$

Solution

Let the masses 1 kg, $\frac{3}{2} kg$ and 2 kg are located at the vertices A, B and C as shown in figure above. The coordinates of points A, B and C are (0, 0), (a, 0), ( $\frac{a}{2}, \frac{\sqrt{3} a}{2}$ ) respectively.
The coordinates of centre of mass are
$X_{CM} = \frac{m_{1} x_{1} + m_{2} y_{2} + m_{3} x_{3}}{m_{1} + m_{2} + m_{3}}$
= $\frac{[1 \times 0 + \frac{3}{2} \times a + 2 \times \frac{a}{2}]}{[1 + \frac{3}{2} + 2]} = \frac{5 a}{9}$
$Y_{CM} = \frac{m_{1} + y_{1} + m_{2} y_{2} + m_{3} y_{3}}{m_{1} + m_{2} + m_{3}}$
= $\frac{[1 \times 0 + \frac{3}{2} \times 0 + 2 \times \frac{\sqrt{3} a}{2}]}{[1 + \frac{3}{2} + 2]} = \frac{2 \sqrt{3} a}{9} = \frac{2 a}{3 \sqrt{3}}$

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