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

Answer

A

Explanation

`r_(CM)= (m_(1) r_(1) + m_(2)r_(2) + m_(3)r_(3)+ ....)/(m_(1) + m_(2) + m_(3) + ....)`
`= (mxx l + 2m xx 2l + 3m xx 3l+ .....)/(m+ 2m+ 3m+ ....)`
`= (m[1+ 2^(2) + 3^(2) + ...])/(m[1 + 2+ 3+ ....])= (l [(n(n+1) (2n+1))/(6)])/((n(n+1))/(2))= (l(2n+1))/(3)`