Mass center of a system of a segment of a uniform circular rod(arc)
Find location of mass center of a thin uniform rod bent into shape of an arc.

Explanation

Consider a thin rod of uniform line mass density lambda(mass per unit length) and radius r subtending angle 2theta on its center 0.
The angle bisector OP is the line of symmetry, and mass center lies on it. Therefore if we assume the angle bisector as one of the coordinate axes say x-axis, y-coordinate of mass center becomes zero. Let two very small segments A and B located to the line of symmetry (x-axis). Mass center of these two segments is on P at a distance `x=rcosprop` from center O.Total mass of these two elements is dm=2lambdardprop. Now using eq., we have`x_(c ) = underset(M)overset(intxdm)rarr " " x_(c )(int_(0)^(0)(r cos alpha) (2 lambdaraalpha))/(lambda r theta) = (r sin theta)/(theta)`
Mass center of a thin uniform are shaped rod of radius r subtending angle 2theta at the center lies on its angle bisector at distance OC from the center. `OC = (r sin theta)/(theta)`