Particles of masses m , 2 m , 3 m , … . , n m are placed on the same line at distance L , 2 L , 3 L , . . n…

1. $\frac{\left(2n+1\right) L}{4}$
2. $\frac{L}{\left(2n+1\right)}$
3. $\frac{n\left(n^2+1\right) L}{2}$
4. $\frac{\left(2n+1\right)L}{3}$

Solution

L

$\because x_{CM}=\frac{mL+2m\times 2L+\dots +nm\times nL}{m+2m+\dots +nm}$

$=\frac{L\left(1^2+2^2+\dots +n^2\right)}{\left(1+2+\dots +n\right)}$

$=L\frac{\sum n^2}{\sum n}=L\frac{\frac{n\left(n+1\right) \left(2n+1\right)}{6}}{\frac{n\left(n+1\right)}{2}}$

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

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