Particles of masses m, 2m, 3m, ldots, nm gram | vedclass

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(A) The distance of the centre of mass $x_{cm}$ from the fixed point is given by the formula:$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$Substituting the

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

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(A) The distance of the centre of mass $x_{cm}$ from the fixed point is given by the formula:$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$Substituting the given values:$x_{cm} = \frac{m(l) + 2m(2l) + 3m(3l) + \ldots + nm(nl)}{m + 2m + 3m + \ldots + nm}$$x_{cm} = \frac{ml(1^2 + 2^2 + 3^2 + \ldots + n^2)}{m(1 + 2 + 3 + \ldots + n)}$Using the standard summation formulas $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum_{i=1}^n i = \frac{n(n+1)}{2}$:$x_{cm} = \frac{l \cdot \frac{n(n+1)(2n+1)}{6}}{\frac{n(n+1)}{2}}$$x_{cm} = l \cdot \frac{n(n+1)(2n+1)}{6} \cdot \frac{2}{n(n+1)}$$x_{cm} = \frac{l(2n+1)}{3}$

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

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