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

Yes

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Never

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Some times

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Not theoretically but practically

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Solution

The correct option is A
Yes

We know for the system of point particles,

$x_{C O M} = \frac{m_{1} x_{1} + m_{2} x_{2} + . . .}{m_{1} + m_{2} + . . .}$
$y_{C O M} = \frac{m_{1} y_{1} + m_{2} y_{2} + . . .}{m_{1} + m_{2} + . . .}$
$z_{C O M} = \frac{m_{1} z_{1} + m_{2} z_{2} + . . .}{m_{1} + m_{2} + . . .}$
If all particles are in x - y plane
$\Rightarrow z_{1} = z_{2} = z_{3} = 0$
$\Rightarrow$ COM will also lie in x - y plane.

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

The correct option is A Yes We know for the system of point particles, xCOM = m1x1 + m2x2 + ...m1 + m2 +... yCOM = m1y1 + m2y2 + ...m1 + m2 +... zCOM = m1z1 + m2z2 + ...m1 + m2 +... If all particles are in x - y plane ⇒ z1 = z2 = z3 = 0 ⇒ COM will also lie in x - y plane.