Center of mass of a semi circle using polar coordinates

[xdrgnh]

Homework Statement

Semi circle of Radius R given. Find center of mass using polar coordinates, not double integrals.

Homework Equations

.5 intergral(r^2dpheta)

(1/M) integral y dm

r=R

The Attempt at a Solution

.5(2/piR^2) integral(R^3sinpheta do pheta) from 0 to pi, when I evaluate it I don't get 4R/3pi

 

[SammyS]

The centroid of a circular sector (circle of radius, r) subtending angle dθ at angle θ from the positive x-axis is given by:

$$(\bar{x},\,\bar{y})=\left(\frac{2r}{3}\cos(\theta),\,\frac{2r}{3}\sin(\theta)\right)$$

The following gives the y coordinate for a semi-circle of radius r, centered at the origin.

$$\frac{\displaystyle (0.5)\int_{0}^{\pi}\left(\frac{2r}{3}\sin(\theta)\right)r^2\,d\theta }{\displaystyle (0.5)\int_{0}^{\pi}r^2\,d\theta}$$

Get the x coordinate from symmetry.