$n$ particles of the same mass are placed on the $x$ - axis. If the particles are placed at $x = 1, 2, 3, . . . . . . . . . . n$ respectively, then find the location of COM.

\frac{(n + 1)}{2}

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\frac{n (n + 1)}{2}

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\frac{n (n - 1)}{2}

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None of the above

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Solution

The correct option is A $\frac{(n + 1)}{2}$
We know,
$X_{c o m} = \frac{m_{1} x_{1} + m_{2} x_{2} + m_{3} x_{3} + - - - - - + m_{n} x_{n}}{m_{1} + m_{2} + m_{3} + - - - - - m_{n}}$

$\Rightarrow X_{c o m} = \frac{m .1 + m .2 + m .3 + - - - - - - - m . n}{m + m + - - - - - - m}$

$= \frac{m [1 + 2 + 3 - - - - - n]}{n m} = \frac{n (n + 1)}{2 n} = \frac{(n + 1)}{2}$

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