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  4. Particles of masses m, 2m, 3m, ldots ., nm are placed on the same line at distance L, 2L, 3L,..nL from O. The distance of the centre of mass from O is

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Q. Particles of masses $m, \, 2m, \, 3m,\ldots ., \, nm$ are placed on the same line at distance $L, \, 2L, \, 3L,..nL$ from $O.$ The distance of the centre of mass from $O$ is

NTA AbhyasNTA Abhyas 2020

Solution:

Solution L
$\because \, \, \, x_{C M}=\frac{m L + 2 m \times 2 L + \ldots + n m \times n L}{m + 2 m + \ldots + n m}$
$=\frac{L \left(1^{2} + 2^{2} + \ldots + n^{2}\right)}{\left(1 + 2 + \ldots + n\right)}$
$=L\frac{\displaystyle \sum n^{2}}{\displaystyle \sum n}=L\frac{\frac{n \left(n + 1\right) \, \left(2 n + 1\right)}{6}}{\frac{n \left(n + 1\right)}{2}}$
$=\frac{\left(2 n + 1\right) L}{3}$