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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?
Solution
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, zcm = 0; i.e., its centre of mass lies in the X–Y plane.
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