Question

Three particles of masses $1kg,\frac{3}{2}kg$, and $2kg$ 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)

Let the masses $1kg,\frac{3}{2}kg$ and $2kg$ 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
$X_{CM}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$
$=\frac{\left[1\times 0+\frac{3}{2}\times a+2\times \frac{a}{2}\right]}{\left[1+\frac{3}{2}+2\right]}=\frac{5a}{9}$
$Y_{CM}=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3}{m_1+m_2+m_3}$
$=\frac{\left[1\times 0+\frac{3}{2}\times 0+2\times \frac{\sqrt{3}a}{2}\right]}{\left[1+\frac{3}{2}+2\right]}$
$=\frac{2\sqrt{3}a}{9}$
$=\frac{2a}{3\sqrt{3}}$