I'm not entirely sure this belongs here but it seems like a good fit. I have a question that I was pretty certain I knew how to solve but the answer that my book provides does not match up with my found answer, and I cannot understand why. I'm hoping some of the sharp minds here may be able to help me out. First a little information about finding centroids...
I've drawn a figure of the area to help understand the problem
to find the x component of the centroid of an area the following formula is utilized.. ((x1A1+x2A2+x3A3+xnAn)/(A1+A2+A3+An))
As you can see, this total area can be divided into 4 shapes(2 rectangles, a semi-circle and a circle), and we can find the area and x centroid of those shapes
for this problem, the x components (xn) of the area shapes are given in an appendix...
Rectangles are 1/2*Base
Semi-circles and circles are the center point of said circles...
(An) simply refers to the area of those shapes. Area of semi circle is 1/2pi(r^2)
For this problem, and others similar to it, I add the total areas of Parts 1, 2, and 3 then subtract 4. This method has been proven to work on previous exercises.
The following table consists of the numbers I come up with.
Xi Ai XiAi Part 1 80 12800 1024000 Part 2 120 3200 384000 Part 3 120 2513.274 301592.895 Part 4 120 1256.637 150796.447
After substituting these values into the equation ((x1A1+x2A2+x3A3)-x4A4)/((A1+A2+A3)-A4)) I get an answer of 90.33mm. However the answer in the book suggests the answer is in fact 87.3mm. I have spent the better part of 2 hours, reworking this problem, going through it step by step, doing other problems just like it in the same fashion and getting the right answers for those problems, and still cannot see how this problems answer is 87.3.
Anyone who is willing to go through this problem and either confirm my suspicions that the books answer is wrong, or find the mistake that has successfully eluded me, I would highly appreciate it because I am losing my mind trying to figure out what is going on here!!! If something is unclear please let me know and I will try to clarify it better. Thank you!!