# Centre of mass of a system of particles

If the centre of mass of a system of particles is at the origin, does this mean that if there is a particle on the positive X-axis, there must be atleast one particle on the negative X-axis?

I feel that this is true, since to get $X_{COM}$ as zero there should be a particle on the negative X -axis.

Am I on the right track?

• Think of a triangle with one vertex on the positive X-axis, symmetrical with respecto the the X-axis. Edit: I'm talking about the axis, not the negative half-plane. – anderstood Apr 27 '15 at 15:27

No, a particle on the positive $x$-axis does not imply a particle on the negative $x$-axis, only that there are particles whose $x$-coordinate is negative.
For example, take three particles with equal mass, and take one positioned on the negative $x$-axis, and the two others symmetrically about the $x$-axis in the $x-y$-plane, so that the sum of their two position vectors is on the $x$-axis with the length of the third one. No particle on the negative $x$-axis, but the center of mass is in the origin.
$$x_{\mathrm{CM}} = \sum_{i=1}^{n}m_ix_i$$
So if your system has two particles and the center of mass is the origin indeed if $x_1>0$ then $x_2<0$.