If all the particle of a system lie in a cube, is it necessary that the centre of mass be in the cube?

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Solution

Yes. As a cube is a 3-dimensional body, all the particles of a system lying in a cube lie in the x,y and z plane.

Let the i^th element of mass ∆m_iis located at the point (x_i,y_i,z_i).
The co-ordinates of the centre of mass are given as:
$X = \frac{1}{M} \sum_{i = 1}^{i = n} (∆ m_{i}) x_{i} Y = \frac{1}{M} \sum_{i = 1}^{i = n} (∆ m_{i}) y_{i} Z = \frac{1}{M} \sum_{i = 1}^{i = n} (∆ m_{i}) z_{i}$

X, Y and Z lie inside the cube because it is a weighted mean.

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