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Q.
Two particles of mass $2\, kg$ and $1\, kg$ are placed at points $(1,-2)$ and $(-4,3)$. Where should a third particle of mass $5\, kg$ be placed so that centre of mass of three particle system lies at origin ?
Solution:
$X _{ cm }=\frac{ m _{1}\, x _{1}+ m _{2} \,x _{2}+ m _{3}\, x _{3}}{ m _{1}+ m _{2}+ m _{3}}$
$\Rightarrow O =\frac{(2)(1)+(1)(-4)+5 x _{3}}{2+1+5}$
$\Rightarrow 5 x _{1}+2-4$
$x _{1}=\frac{2}{5}$
$Y _{ cm }=\frac{ m _{1} \,y _{1}+ m _{2} \,y _{2}+ m _{3} \,y _{3}}{ m _{1}+ m _{2}+ m _{3}}$
$\Rightarrow O =\frac{2(-2)+1(3)+ m _{3} y _{3}}{2+1+ m _{3}}$
$\Rightarrow -4+3+5 y _{3}=0$
$\Rightarrow 5 y _{3}=1$
$y _{3}=\frac{1}{5}$