Question

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

Answer 1

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_i is located at the point (x_i,y_i,z_i).
The co-ordinates of the centre of mass are given as:

\[X = \frac{1}{M} \sum\nolimits_{i = 1}^{i = n} \left( ∆ m_i \right) x_i \]
\[Y = \frac{1}{M} \sum\nolimits_{i = 1}^{i = n} \left( ∆ m_i \right) y_i \]
\[Z = \frac{1}{M} \sum\nolimits_{i = 1}^{i = n} \left( ∆ m_i \right) z_i\]

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