Undergrad Surface Integrals: Finding Centroid & Inertia of Circle

[CaptainX]

How to find the centroid of circle whose surface-density varies as the nth power of the distance from a point O on the circumference. Also it's moments of inertia about the diameter through O.

I'm getting x'=2a(n-2)/(n+2)
And about diameter
-4(a×a)M{something}

 

[Infrared]

Working in coordinates, we can assume that $O$ is the origin, and the circle is $(x-1)^2+y^2=1$. In polar coordinates, this circle is $r=2\cos(\theta)$. The $y$-coordinate of the center of mass is $0$ by symmetry, and the $x$-coordinate is the integral of $x r^n dx dy=\cos\theta \ r^{n+2} dr d\theta$ over the disk, divided by the mass, which should be straightforward to calculate (bounds are $-\pi/2\leq\theta\leq\pi/2$ and $0\leq r\leq 2\cos\theta$).

The moment of inertia calculations are similar.

 

[CaptainX]

Working in coordinates, we can assume that $O$ is the origin, and the circle is $(x-1)^2+y^2=1$. In polar coordinates, this circle is $r=2\cos(\theta)$. The $y$-coordinate of the center of mass is $0$ by symmetry, and the $x$-coordinate is the integral of $x r^n dx dy=\cos\theta \ r^{n+2} dr d\theta$ over the disk, divided by the mass, which should be straightforward to calculate (bounds are $-\pi/2\leq\theta\leq\pi/2$ and $0\leq r\leq 2\cos\theta$).

The moment of inertia calculations are similar.

Thanks a lot!