Question

Ten solid spheres of same radius having densities $\rho ,2\rho ,3\rho$... $10\rho$ respectively are put on the $x$ - axis at positions $x=1,x=2,x=3$ ... $x=10$. Find the position of centre of mass of these ten spheres.

• A. $x=9$
• B. $x=7$
• C. $x=0$
• D. $x=6$

Answer 1

The correct option is B $x=7$


Let radius of each sphere be $r$. The radius of each sphere is same, therefore volume of each sphere will also be the same.
Hence, mass of the spheres is proportional to the densities only. Let the mass of sphere of density $\rho$ be $m$, then the masses of the spheres will $m,2m,3m..10m$
We know
$x_{com}=\frac{x_1{m}_1+x_2{m}_2+--x_n{m}_n}{m_1+m_2+m_3--m_n}$
$=\frac{1\times m+2\times 2m+3\times 3m+--10\times 10m}{m+2m+3m-----10m}$
$=\frac{m[1^2+2^2+3^2-----{10}^2]}{m[1+2+3-----10]}$
$=\frac{10(10+1)(20+1)}{6}\times \frac{2}{10(10+1)}$
$\therefore x_{com}=\frac{21}{3}=7\text{units}$
i.e the COM of system will be at $x=7$.