Mass Center of a sector of a uniform circular plate
Find location of mass center of a sector of a thin uniform plate.

Explanation

Consider a sector of a thin uniform plate of surface mass density mass density `sigma` (mass per unit area) and radius r subtending angle `2theta` on its center
Let a thin arc of radius r and width dr be an infintely small part of the sector. Mass dmof the arc AB equals to product of mass per unit area and area of the arc. `dm=sigma(2rthetadr)=2sigmarthetadr` Due to symmetry mass center of this arc must be on the angle bisector i.e. on x-axis at distance `x=(rsintheta)/theta`.
Using above two information in eq. , we obtain the mass center of the sector.
`x_(c ) = (intxdm)/(M)rarr " " x_(c ) = (int_(0)^(R )((r sin theta)/(theta)) (sigma r theta dr))/(sigma("Area of the sector")) = (int_(0)^(R )((r sin theta)/(theta)) (2 sigma r theta dr))/(sigma r theta^(2)) = (2 r sin theta)/(3 theta)`