Question

If all the particles of a system lie in X-Y plane, is it necessary that the centre of mass be in X-Y plane?

Answer 1

Yes, if all the particles of a system lie in the X–Y plane, then it's necessary that its centre of mass lies in the X–Y plane.

\[z_{cm} = \frac{m_1 z_1 + m_2 z_2 + . . . m_n z_n}{\sum_{} m}\]

As all the particles lie in the X–Y plane, their z-coordinates are zero.
Therefore, for the whole system, z_cm = 0; i.e., its centre of mass lies in the X–Y plane.