Particles of masses m, 2m, 3m, dots, nm grams | vedclass

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(A) The position of the centre of mass $X_{CM}$ is given by the formula:$X_{CM} = \frac{\sum m_i x_i}{\sum m_i}$Substituting the given values:$X_{CM}

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

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(A) The position of the centre of mass $X_{CM}$ is given by the formula:$X_{CM} = \frac{\sum m_i x_i}{\sum m_i}$Substituting the given values:$X_{CM} = \frac{m(l) + 2m(2l) + 3m(3l) + \dots + nm(nl)}{m + 2m + 3m + \dots + nm}$Factoring out $m$ and $l$:$X_{CM} = \frac{ml(1^2 + 2^2 + 3^2 + \dots + n^2)}{m(1 + 2 + 3 + \dots + n)}$Using the standard summation formulas $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum_{k=1}^n k = \frac{n(n+1)}{2}$:$X_{CM} = \frac{l \cdot \frac{n(n+1)(2n+1)}{6}}{\frac{n(n+1)}{2}}$Simplifying the expression:$X_{CM} = l \cdot \frac{n(n+1)(2n+1)}{6} \cdot \frac{2}{n(n+1)}$$X_{CM} = \frac{(2n+1)l}{3}$

Column-$I$	Column-$II$
$(A)$ Blue-green algae	$(i)$ Spirogyra
$(B)$ Unicellular algae	$(ii)$ Nostoc
$(C)$ Filamentous algae	$(iii)$ Volvox
$(D)$ Colonial algae	$(iv)$ Chlamydomonas

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

Similar Questions

Match the following columns:
Column-$I$ Column-$II$
$(A)$ Blue-green algae $(i)$ Spirogyra
$(B)$ Unicellular algae $(ii)$ Nostoc
$(C)$ Filamentous algae $(iii)$ Volvox
$(D)$ Colonial algae $(iv)$ Chlamydomonas

In a fission reaction $_{92}^{236}U \to ^{117}X + ^{117}Y + n + n$,the binding energy per nucleon of $X$ and $Y$ is $8.5\, MeV$,whereas that of $^{236}U$ is $7.6\, MeV$. The total energy liberated will be about:

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