Question

Three particles, each of mass 'm', are placed at the corners of a right angled triangle as shown in Fig. If OA = a and OB = b, the position vector of the centre of mass is (here i and j are unit vectors along x and y axes respectively).

• A.
  
• B.
• C.
• D.

Answer 1

The correct option is A


The (x, y) co-ordinates of the masses at 0, A and B respectively are (refer to fig. 5.4)
$(x_1=0,y_1=0),(x_2=a,y_2=0)and(x_3=0,y_3=b)$
The (x,y) co - ordinates of the centre of the centre of mass are $x_{CM}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}=\frac{m\times 0+m\times a+m\times 0}{m+m+m}=\frac{a}{3}$
$y_{CM}=\frac{m_1{y}_1+m_2{y}_2+m_3{y}_3}{m_1+m_2+m_3}=\frac{m\times 0+m\times 0+m\times b}{m+m+m}=\frac{b}{3}$
The position vector of the centre of mass is $x_{CM}i+y_{CM}j$
$=\frac{a}{3}i+\frac{b}{3}j=\frac{1}{3}(ai+bj)$, which is choice (a).