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

# 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

1. A

$\frac{\left(2\mathrm{n}+1\right)\mathrm{l}}{3}$

2. B

$\frac{\mathrm{l}}{\mathrm{n}+1}$

3. C

$\frac{\mathrm{n}\left({\mathrm{n}}^{2}+1\right)\mathrm{l}}{2}$

4. D

$\frac{2\mathrm{l}}{\mathrm{n}\left({\mathrm{n}}^{2}+1\right)}$

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### Solution:

Given, the masses of the particle be m, 2m , 3m .... nm

and line at distance l, 2l, 3l,.....nl

$\mathrm{COM}=\frac{{\mathrm{m}}_{1}{\mathrm{x}}_{1}+{\mathrm{m}}_{2}{\mathrm{x}}_{2}+{\mathrm{m}}_{3}{\mathrm{x}}_{3}.......+{\mathrm{m}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}{{\mathrm{m}}_{1}+{\mathrm{m}}_{2}+{\mathrm{m}}_{3}+.......+{\mathrm{m}}_{\mathrm{n}}}$

$\mathrm{COM}=\frac{{1}^{2}\mathrm{ml}+{2}^{2}\mathrm{ml}+{3}^{2}\mathrm{ml}+.......+{\mathrm{n}}^{2}{\mathrm{ml}}^{2}}{1\mathrm{m}+2\mathrm{m}+3\mathrm{m}+.......+\mathrm{nm}}$

$\mathrm{COM}=\mathrm{l}\left[\frac{\frac{\mathrm{n}\left(\mathrm{n}+1\right)\left(2\mathrm{n}+1\right)}{6}}{\frac{\mathrm{n}\left(\mathrm{n}+1\right)}{2}}\right]=\frac{\mathrm{l}.\left(2\mathrm{n}+1\right)}{3}$

Hence the correct answer is $\frac{\mathrm{l}\left(2\mathrm{n}+1\right)}{3}.$

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