The barycenter of a gravitating system of masses is the center of mass of the system, where to a distant observer, it appears that the mass of the system is concentrated. As seen from the vantage point of the Sun, for instance, the Earth-Moon system looks like a single object orbiting the Sun at the distance between the Sun's center and the Earth-Moon barycenter. The motion of the barycenter defines the mean orbit of the Earth-Moon system. If you replaced the Earth and Moon by point masses of the appropriate sizes, at their centers, at the barycenter, the gravitational force on a test mass by the Earth would be the same force of gravity as by the Moon. The barycenter location is DEFINED by the masses in the system ( m(earth) + m(moon) ): where D is the Earth-Moon distance, the barycenter point is located at:
m(moon)
R = D x --------------------
m(earth) + m(moon)
from the center of the Earth. At this point, the gravitational forces
of the Earth and Moon are equal in magnitude to:
2
G m(e) m(m) G m(earth) (m(earth) + m(moon))
f = ------------- = ----------------------------------
2 2
R D m(moon)
There is a NET force of zero because the forces are equal
and opposite to each other.