A uniform thin sheet of metal is cut in the shape of a semicircle of radius R and lies in the xy plane with its center at the origin and diameter lying along the x axis. Find the position of the Center of Mass using polar coordinates.

Answer 1

T is angle

area of small radial slice subtending dA = (1/2)R^2 dT
distance of that cg from x axis = (2R/3)sin T
moment about x axis = (R/3)R^2 sin T dT
integrate 0 to pi/2 and divide by area of quarter circle, (1/4) pi R^2

(R^3/3) (-cos pi/2 + cos 0) /.25 pi R^2

(4/3pi)R = .425 R

moment of that slice = (1/2)