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

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Question

$X_{C M}=\frac{m_{1} x_{1}+m_{2} x_{2}+\ldots .}{m_{1}+m_{2}+\ldots .}$

$=\frac{m \ell+2 m \cdot 2 \ell+3 m \cdot 3 \ell+\ldots}{m+2 m+3 m+\ldots}$

Vedclass · Accepted Answer

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

Vedclass · Answer

$X_{C M}=\frac{m_{1} x_{1}+m_{2} x_{2}+\ldots .}{m_{1}+m_{2}+\ldots .}$

$=\frac{m \ell+2 m \cdot 2 \ell+3 m \cdot 3 \ell+\ldots}{m+2 m+3 m+\ldots}$

$=\frac{m \ell(1+4+9+\ldots)}{m(1+2+3+\ldots)}$

$=\frac{\ell \frac{n(n+1)(2 n+1)}{6}}{\frac{n(n+1)}{2}}=\frac{\ell(2 n+1)}{3}$

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 the centre of mass of the particles from the fixed point (in centimetres) is

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