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  4. Three particles of masses 1 kg, (3/2) kg, and 2 kg are located at the vertices of an equilateral triangle of side a. The x, y coordinates of the centre of mass are

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

System of Particles and Rotational Motion

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. The coordinates of points $A, B$ and $C$ are$ (0,0), (a, 0),$$\frac{a}{2},$$ \frac{\sqrt{3a}}{2}$ respectively.
image
The coordinates of centre of mass are
$X_{CM} = \frac{m_{1}x_{1} +m_{2}x_{2} +m_{3}x_{3}}{m_{1}+m_{2}+m_{3}}$
$= \frac{\left[1\times0+\frac{3}{2} \times a+2 \times\frac{a}{2}\right]}{\left[1+ \frac{3}{2}+2\right]} = \frac{5a}{9} $
$Y_{CM} = \frac{m_{1}y_{1} +m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}} $
$ = \frac{\left[1\times 0+\frac{3}{2} \times 0+2 \times \frac{\sqrt{3}a}{2}\right]}{\left[1+ \frac{3}{2}+2\right]} $
$= \frac{2\sqrt{3}a}{9} $
$ = \frac{2a}{3\sqrt{3}}$