Center of Mass of a SemicircleDate: 06/14/99 at 18:43:50 From: Mayyada Abdou Subject: Center of mass of a semicircle I'm studying Mechanics (IGCSE - AS level), and need to get an equation or a standard formula for the center of mass of a semicircle. I'm sure that it will be on the symmetry line of the semicircle, but where? Thanking you Dr. Math in advance for your help. Date: 06/14/99 at 19:07:28 From: Doctor Pat Subject: Re: Center of mass of a semicircle I will assume the semi-circle in question is centered at the origin, located in the first two quadrants, and its circular periphery can be given by y = sqrt(r^2-x^2). If these things are not so, a translation and rotation will produce the given answer. Draw the semi-circle in question with a radius of r... The center of gravity, as you note, will be on the y-axis. To find out where, we need to find y-bar, the center of rotation about the x-axis. The value is given by Mx (the moment about x-axis) divided by the area. The moment can be found by drawing a single representative segment and integrating this across all values of x. Draw a thin rectangular region reaching from the x-axis up to the curve. The area (weight, force) of this section is its height times its width. Its height is f(x) = sqrt(r^2-x^2) and its width will be the infinitesimal dx. Now we need to find the center of gravity of this thin section. Since it is infinitely thin, its center will be halfway up, or f(x)/2. Now the force x distance formula for the moment clears the square root for us, leaving a moment for each tiny section of (r^2-x^2)dx/2 (remember r is a constant, x is the variable). We can integrate this from -r to r to sum up the total moment due to ALL the little sections like the one we drew. (1/2) Int (-r,r) [r^2-x^2] dx When I did this I got 2/3 r^3, but check my work. Now to find y-bar we divide this by the area of the semi-circle .5 pi r^2 and come up with (4r)/(3 pi). This should put the center of gravity about 42% of a radius out (or up) the y-axis. I hope this helps. Good luck. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/ |
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