I want to know a brief about centre of mass.

We know that centre of mass defines the whole trajectory of system, but I came to a question while solving one derivation regarding it in its equation, that, as centre of mass of the thin semicircular wire of radius r [edit for clarity: a thin wire bent into a semi-circular shape] is $(0,2r/\pi)$. I am assuming that the center of the ring lies at the origin of coordinates and the ring itself lies completely above the $x$-axis.

When I assume $r$ some value, suppose 4, then its centre of mass became $(0,2.54).$ But considering this semi-circular shape has no mass of wire at that location, because its radius is 4, then how that point be its centre of mass? How it will define in physical manner? Also, is centre of mass a balance mass point?



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