Particles of masses m, 2m, 3m, ... , nm grams are placed on the same line at distances l, 2l, 3l, ..., nl cm from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres is

Options

A. `((2n + 1) l)/(3)`
B. `(l)/(n + 1)`
C. `(n (n^2 + 1) l)/(2)`
D. `(2l)/(n(n^2 + 1))`

Answer

A

Explanation

`X_(CM) = (m_(1) x_(1) + m_(2) x_(2) +… + m_(n) x_(n))/(m_(1) + m_(2) + … + m_(n)) = (ml + 2m * 2l + 3m * 3l + … + nm.nl)/(m + 2m + 3m + … +nm)`
`= (m l (1 + 4 + 9 + … + n^2))/(m (1 + 2 + 3+ …+ n)) = ( l "" (n (n +1) ( 2n +1))/(6))/((n(n+1))/(2)) = (l (2n + 1))/(3)`