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Homework Statement
Find the center of mass of the next plate if:
A) Is homogeneous
B) Its density per unit mass is ##\sigma=Axy##, where ##A## is a constant.
Homework Equations
##X_{cm}=\frac{\int\sigma x dA}{\int\sigma dA}##
##Y_{cm}=\frac{\int\sigma y dA}{\int\sigma dA}##
The Attempt at a Solution
I did part A) as is shown in the picture below:
Am I correct?
My main problem is with part B), I tried to do it as I did part A), I found ##X_{cm} without any problems, like this:
But when I tried to find ##Y_{cm}## I got stucked with this:
##Y_{cm}=\frac{\int_{0}^7 \sigma ydA_1+ \int_{1}^7 \sigma ydA_2}{\int_{0}^7 \sigma dA_1 + \int_{1}^7 \sigma dA_2}=\frac{A\left[ \int_{0}^7 y^2\sqrt{y1}dy + \int_{1}^7 y^2(y1)dy\right]}{A\left[ \int_{0}^7 y\sqrt{y1}dy + \int_{1}^7 y(y1)dy\right]}##, but when I tried to integrate ##\int_{0}^7 y^2\sqrt{y1}dy## and ##\int_{0}^7 y\sqrt{y1}dy##, I ended up with imaginary numbers, what am I doing wrong? Hope you can help me.
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