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Q.

When 100 particles of masses 1,2,3………….100 kg are at distances 1m,2m,3m………….100m respectively from the origin on the x-axis. The distance of center of mass of system of particles from the origin is

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a

54 m

b

62 m

c

67 m

d

70 m

answer is C.

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Detailed Solution

$x_{cw} = \frac{m_{1} x_{1} + \dots \dots \dots \dots \dots . \dots m_{n} x_{n}}{m_{1} + m_{2} + \dots \dots \dots \dots \dots + m_{n}}$
$\begin{array}{l} = \frac{1 X 1 + 2 X 2 + \dots \dots \dots \dots \dots nXn}{1 + 2 + \dots \dots \dots \dots . . + n} \\ = \frac{1^{2} + 2^{2} + \dots \dots \dots \dots + n^{2}}{1 + 2 + \dots \dots \dots \dots . + n} \\ = \frac{\frac{n (n + 1) (2 n + 1)}{6}}{\frac{n (n + 1)}{2}} = \frac{2 n + 1}{3} \end{array}$
Hence =100
$∴ x_{cw} = \frac{201}{3} = 67 m$

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