Question

Three identical. spheres of mass in each are placed at the corners of an equilateral triangle of side 2 in. Taking one of the corner as the origin, the position vector of the centre of mass is

• A. $\sqrt{3}(\hat{i}-\hat{j})$
• B. $\frac{\hat{i}}{\sqrt{3}}+\hat{j}$
• C. $\frac{\hat{i}+\hat{j}}{3}$
• D. $\hat{i}+\frac{\hat{j}}{\sqrt{3}}$

Answer 1

Answer: (D)

$\hat{i}+\frac{\hat{j}}{\sqrt{3}}$

The x coordinate of centre of mass is
$\overline{x}=\frac{\sum m_i{x}_i}{\sum m_i}=\frac{m\times 0+m\times 1+m\times 2}{m+m+m}=1$
$\overline{y}=\frac{\sum m_i{y}_i}{\sum m_i}=\frac{m\times 0+m\left(2sin{60}^o\right)+m\times 0}{m+m+m}=1$
$\overline{y}=\frac{m\sqrt{3}}{3m}=\frac{1}{\sqrt{3}}$
Position vector of centre of mass is $(\hat{i}+\frac{\hat{j}}{\sqrt{3}})$