Center of Mass of a Semi-Circle using Cartesian Coordinates

Question

So I'm currently trying to figure this out but am not sure where to start. I know that you can figure the center of mass using polar coordinates, but I know that it's possible to do it using Cartesian system. Any suggestions. The radius of the semi-circle is 9 cm.

Answer 1

You know by symmetry that the center of mass is going to be along the center line, so simplify the computation by centering the semicircle on, say, the $x$-axis. Now, slice it up into thin vertical strips and integrate:

$$\bar x = {2 \over \pi R^2}\int_0^R2x\sqrt{R^2-x^2}\,dx$$

Answer 2

$${\iint _S\ y\ dxdy\over {\iint_S\ dxdy}}$$ will give you the Y co - ordinate, where $$S = {(x, y)|-9\le x\le 9, 0\le y \le \sqrt{81-x^2}}$$ Then by fubini's theorem, you can write $${{\int_{-9}^9 \left ( \int_0^{{(81-x^2)}^{1/2}}\ ydy \right )\ dx}\over {{81\ \pi}\over 2}}$$ which will give you the Y - coordinate of center of disk.