Question

A thin wire is bent into the shape of a semicircle x^2 + y^2 = 144, x ≥ 0. If the linear density is a constant k, find the mass and center of mass of the wire.

Answer 1

finding the mass is trivial - take the density k and multiple by the perimeter 12*pi. so the mass is 12 pi k.
to find the center of mass, first it can be noted that the y coordinate is 0, since the semicircle is symmetric about the x axis. to find the x coordinate of the center of mass, integrate this coordinate with respect to theta : xbar=1/pi * integral(12cos(x) dt,{t,-pi/2,pi/2}).