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The centre of mass is the mass-weighted average of positions, so it always lies nearer the heavier body. With 5 kg at 0.5 m and 3 kg at 2.5 m the balance point is at 1.25 m - 0.75 m from the heavy mass and 1.25 m from the light one, in inverse proportion to their masses.
Centre of mass
x(cm) = (m1 x1 + m2 x2) / (m1 + m2)
Lever relation
m1 x d1 = m2 x d2 about the centre of mass
No. A ring or a boomerang has its centre of mass in empty space, which is perfectly allowed.
Because moments must cancel. The heavier mass needs a shorter arm to produce the same moment.