Question

Three identical spheres $A,B$ and $C$, each of radius $R$, are placed touching one other on a horizontal table. Where is the centre of mass of the system located relative to $A$ (say)?
Ans. $(0,R/\sqrt{3})$.

Answer 1

Consider the center of mass of sphere $A$ is $(0,0)$

So, the center of mass of sphere $B$ is at $(2R,0)$

The Center of mass of sphere $C$ is at

Now,

The center of mass of the system is

$r=\frac{m_1{r}_1+m_2{r}_2+m_3{r}_3}{m_1+m_2+m_3}r=\frac{m(0,0)+m(2R,0)+m(R,\sqrt{3}R)}{m+m+m}r=\frac{(0,0)+(2R,0)+(R,\sqrt{3}R)}{3}r=(0,\frac{R}{\sqrt{3}})$