Question

Initially three bodies of mass $2m,2m$ and $3m$ are placed on the $x$-axis at position $x=1,x=2$ and $x=4$. If a body of mass $4m$ is placed at origin, then find the displacement of centre of mass from initial position.

• A. $-\frac{72}{77}$
• B. $\frac{72}{77}$
• C. $-\frac{10}{11}$
• D. $\frac{10}{11}$

Answer 1

The correct option is A $-\frac{72}{77}$

COM of system intially, $X_{\left(com\right)}=\frac{x_1{m}_1+x_2{m}_2+x_3{m}_3}{m_1+m_2+m_3}$
$X_{\left(com\right)}=\frac{1\times 2m+2\times 2m+4\times 3m}{2m+2m+3m}$
$=\frac{18m}{7m}=\frac{18}{7}$
COM when $4^{th}$ body is placed at origin,
$X_{(com{)}^1}=\frac{x_{1}m+x_2{m}_2+x_3{m}_3+x_4{m}_4}{2m_1+m_1+m_3+m_4}$
$X_{(com{)}^1}=\frac{1\times 2m+2\times 2m+4\times 3m+0\times 4m}{2m+2m+3m+4m}$
$=18m/11m=\frac{18}{11}$
Therefore, displacement of COM $=X_{(com{)}^1}-X_{com}$
$=\frac{18}{11}-\frac{18}{7}$
$=-\frac{72}{77}$