Center of Mass

Date: 02/22/99 at 14:06:54
From: Genevieve Gallagher
Subject: Center of Mass

My question is:
Find the center of mass of a thin plate of constant density covering 
the region bounded by the parabolas y = 2x^2-4x and y = 2x-x^2.

I'm really close to getting this problem but I can't ask my teacher 
because I'm homeschooled, so please help! I've already found the x 
coordinate of the center of mass by dividing the Moment about the 
y-axis by the Mass. I know that you find the Moment about the x-axis in 
a similar way, but I cannot figure out what way that is. My textbook 
only gives examples of finding the moment about the y-axis.

Thank you.
Genevieve

Date: 02/23/99 at 12:08:38
From: Doctor Peterson
Subject: Re: Center of Mass

I'm not sure which method you are currently learning for moments; 
the most general method is to integrate y dx dy over the region 
to get the moment about the x axis, and integrate x dx dy to get 
the moment about the y axis. If you're doing it with a single integral, 
you'd be integrating x times the area of a vertical strip; that is, 
x (y_2 - y_1) dx, to get the latter. So the "similar way" is to use x 
instead of y multiplying the area.

But in this case, symmetry makes it unnecessary to bother with that, 
and I'm sure that is what you are expected to see. The region is 
symmetrical about the line x = 1, so the center of gravity must be on 
that line.

Symmetry is a wonderful tool for math and science - it saves a lot of 
work in cases like this.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/