Particle of masses `m, 2m,3m,…,nm` grams are placed on the same line at distance `l,2l,3l,…..,nl cm` from a fixed point. The distance of centre of mass of the particles from the fixed point in centimeters is :

Options

A. `((2n + 1) L)/(4)`
B. `(L)/((2 n +1))`
C. `(n(n^(2) + 1) L)/(2)`
D. `((2n + 1) L)/(3)`

Answer

D

Explanation

`x_(c.m.) = (mL + 2m xx 2L + …+ nm xx nL)/(m + 2m + …+ nm)`
`= (L(1^(2) + 2^(2) + …+ n^(2)))/((1 + 2 + … + n))`
`= L (Sigma n^(2))/(Sigma n) = L ((n( n + 1)( 2n +1))/(6))/((n(n +1))/(2)) = ((2 n + 1) L)/(3)`