Question

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

• A. $\left(\frac{5a}{9}, \frac{2a}{3\sqrt{3}}\right)$
• B. $\left(\frac{2a}{3\sqrt{3}}, \frac{5a}{9}\right)$
• C. $\left(\frac{5a}{9}, \frac{2a}{\sqrt{3}}\right)$
• D. $\left(\frac{2a}{\sqrt{3}}, \frac{5a}{9}\right)$

Answer 1

Answer: (A)

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

Answer 2

A uniform sphere of mass $M$ and radius $R$ is placed on a rough horizontal surface (Figure). The sphere is struck horizontally at a height $h$ from the floor. Match Column $I$ with Column $II$. 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. The coordinates of centre of mass are $X_{CM} = \frac{m_{1}x_{1} +m_{2}x_{2} +m_{3}x_{3}}{m_{1}+m_{2}+m_{3}}$ $= \frac{\left[1\times0+\frac{3}{2} \times a+2 \times\frac{a}{2}\right]}{\left[1+ \frac{3}{2}+2\right]} = \frac{5a}{9}$ $Y_{CM} = \frac{m_{1}y_{1} +m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}$ $= \frac{\left[1\times 0+\frac{3}{2} \times 0+2 \times \frac{\sqrt{3}a}{2}\right]}{\left[1+ \frac{3}{2}+2\right]}$ $= \frac{2\sqrt{3}a}{9}$ $= \frac{2a}{3\sqrt{3}}$