Originally Posted by

**Mei Liu** Question:

A wire takes the shape of the semicircle y= $\displaystyle x^2 + y^2, y \geq 0 $. Find the center of mass of the wire if the linear density at any point is proportional to its distance from the line $\displaystyle y=1$.

So far I have this:

$\displaystyle p(x,y) = K(sqrt(1-y^2)); polarized = K(1-rsin\theta)$

The Double Integral I have is:

$\displaystyle \int\int(K(1-rsin\theta))rdrd\theta$ with $\displaystyle 0\leq r\leq 1; 0\leq \theta \leq \pi$

Which works out to $\displaystyle K\pi/2$

I then attempt to solve the $\displaystyle \overline{y} $ expression:

$\displaystyle

2/K\pi \int\int (rsin\theta)(K)(1-sin\theta)rdrd\theta$

I end up with the answer of:

$\displaystyle (4-\pi)/(3\pi)$ whereas the correct answer is: $\displaystyle (4-\pi)/(2(2\pi-2))$

Any ideas where I went wrong?

Thanks,

Mike