Question

Three identical spheres A, B, C each of radius $R$ are placed touching one another on a horizontal table. Where is the centre of mass of the system located relative to A (say)?

Answer 1

Let the mass of each sphere is 'm' and center of mass of the system is at a distance 'x' m away from the center of A.

Now,

$x=\frac{\sum m_i{x}_i}{\sum m_i}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$

$x=\frac{m\left(x\right)+m(2R-x)+m(4R-x)}{m+m+m}$

$x=\frac{m(-x+6R)}{3m}$

$x=-\frac{x}{3}+2R$

$x+\frac{x}{3}=2R$

$x=1.5R$

So the center of mass will be at a point 1.5 times of radius distant from the center of sphere A.