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solution

So we break up the integration into x and y directions.

By symmetry if we get the answer along the x direction, we can easily figure out what it should be along the y direction. In fact it's intuitively clear that the center of mass lies along a line that goes 45 degrees through the right angled corner, as seen below.

{

  figure199

So let's get the answer for the x direction.

equation203

This means the following. We've divided up the integration into vertical strips. We first integrate over y to find out the answer in one strip. Then we integrate over all x to find the final answer of all integrated over all strips.

{

  figure207

Doing the top and bottom integrals over dy we have

equation211

By complete chance, we just did this calculation for the one dimensional rod, see 1.27. The answer was tex2html_wrap_inline1237 . So we know the center of mass in the x direction. The y-direction is almost the same problem, but slightly different. In the x-direction, the triangle is becoming thicker with increasing x, in the y direction it is becoming thinner with increasing y. So it makes sense just to say the center of mass in the y direction is tex2html_wrap_inline1237 from the thinner end. This means its tex2html_wrap_inline1241 from the thicker end, so it has coordinate tex2html_wrap_inline1243 .

{

  figure219

Tricks like these save you from having to do tedious calculations. You can do the integral the more complicated way if you're unsure that this reasoning is correct.



Joshua Deutsch
Fri Jan 17 12:19:41 PST 1997