When ‘n’ number of particles each of mass ‘m’ are at distances x1 = a, x2 = ar, x3 = ar^2 , ……. xn = ar^n , units from origin…

Q: When ‘n’ number of particles each of mass ‘m’ are at distances x₁ = a, x₂ = ar, x₃ = ar² , ……. x_n = arⁿ , units from origin one the X – axis, then find the distance of their centre of mass from origin.

Sol: $\large x_{cm} = \frac{m \times a + m \times ar + …. + m \times ar^n}{m + m + m + ….+ n terms} $

$\large = \frac{m(a+ ar + ar^2 + …+ ar^n)}{n m}$

If r >1 ;

$\large x_{cm} = \frac{1}{n} [\frac{a(r^n -1)}{r-1}]$

If x < 1 $\large x_{cm} = \frac{1}{n} [\frac{a(1- r^n)}{1-r}]$