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Center of Mass as an Average Point

Three identic...

Question

Three identical. spheres of mass in each are placed at the corners of an equilateral triangle of side 2 in. Taking one of the corner as the origin, the position vector of the centre of mass is

A

\sqrt{3} (^i -^j)

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B

\frac{^i}{\sqrt{3}} +^j

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C

\frac{^i +^j}{3}

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D

^i + \frac{^j}{\sqrt{3}}

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Solution

The correct option is C $^i + \frac{^j}{\sqrt{3}}$
The x coordinate of centre of mass is
$¯ x = \frac{\sum m_{i} x_{i}}{\sum m_{i}} = \frac{m \times 0 + m \times 1 + m \times 2}{m + m + m} = 1$
$¯ y = \frac{\sum m_{i} y_{i}}{\sum m_{i}} = \frac{m \times 0 + m (2 s i n 60^{o}) + m \times 0}{m + m + m} = 1$
$¯ y = \frac{m \sqrt{3}}{3 m} = \frac{1}{\sqrt{3}}$
Position vector of centre of mass is $(^i + \frac{^j}{\sqrt{3}})$

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Q. Three identical spheres of mass

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each are placed at the corners of an equilateral triangle of side

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