Three particles of masses 50 g, 100 g and 150 g are placed at the vertices of an equilateral triangle of side 1 m (as shown in the figure). The (x, y) coordinates of the centre of mass will be:

$$(x_1, y_1) = (0, 0)$$

$$(x_2, y_2) = (1, 0)$$

$$(x_3, y_3) = (1\cos60^\circ, 1\sin60^\circ) = \left(0.5, \frac{\sqrt{3}}{2}\right)$$

$$X_{\text{cm}} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{M_{\text{total}}}$$

$$X_{\text{cm}} = \frac{50(0) + 100(1) + 150(0.5)}{300} = \frac{100 + 75}{300} = \frac{175}{300} = \frac{7}{12}\text{ m}$$

$$Y_{\text{cm}} = \frac{m_1y_1 + m_2y_2 + m_3y_3}{M_{\text{total}}}$$

$$Y_{\text{cm}} = \frac{50(0) + 100(0) + 150\left(\frac{\sqrt{3}}{2}\right)}{300} = \frac{75\sqrt{3}}{300} = \frac{\sqrt{3}}{4}\text{ m}$$