Area and Center of Gravity for a Crescent
Date: 9/5/95 at 16:16:7 From: Anonymous Subject: area and center of gravity of crescent Dear Dr. Math, I need to calculate the area and center of gravity of a crescent. For example take a 10.00" dia circle center at x,y 0,0 add a 9.00" dia circle center at x,y 2,0. The crescent is formed in 2 areas that are not overlapping.
Date: 9/22/95 at 20:52:39 From: Doctor Andrew Subject: Re: area and center of gravity of crescent Calculus is the best way I know of to solve this problem. The equation for a circle is: r^2 = (x-a)^2 + (y-b)^2, where (a,b) is the center of the circle and r is the radius of the circle. You can write separate equations for each circle in terms of x and in terms y. Once you've got these equations you'll need to find the x and y coordinates of where the circles intersect. If you draw a picture of the two circles and label the points where they intersect and then shade the crescent this might help you solve the problem. On the picture, draw lines from the intersection points perpendicular to the x and y axes. These will be the boundaries for your integration later. If you already know how to find the area and center of mass of a shape using calculus, then you can use those techniques to find the area and moment of one circle in a range and subtract the area and moment from another circle at each dx and dy. If you are integrating along the x axis, for instance, integrate between x-coordinates of the points of intersection and subtract the contribution of one circle from the other at each dx. If this doesn't make sense to you yet, write back and I can go a little more into how to take the integrals. Good luck! Hope this isn't too late to help. -Doctor Andrew, The Geometry Forum
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