Centroid of an ArcDate: 03/20/2002 at 07:18:00 From: derrick weedon Subject: Centroid of an arc How do I find the centroid (or centre of gravity) of an arc of a circle? Date: 03/20/2002 at 08:07:07 From: Doctor Jerry Subject: Re: Centroid of an arc Hi Derrick, Suppose a curve C is described parametrically by x = x(t) y = y(t) where t varies over the interval [a,b]; also suppose that the density of the curve at a point (x,y) is g(x,y). The total mass m of the curve is given by the integral m = int(t=a,t=b,g(x(t),y(t))*sqrt(x'(t)^2+y'(t)^2)*dt). If (X,Y) is the center of gravity, then X = (1/m)*int(t=a,t=b,x(t)*g(x(t),y(t))*sqrt(x'(t)^2+y'(t)^2)*dt) Y = (1/m)*int(t=a,t=b,y(t)*g(x(t),y(t))*sqrt(x'(t)^2+y'(t)^2)*dt). If C is the arc circle of radius a described by x = a*cos(t) y = a*sin(t) where t varies over the interval [0,T] (radian measure), and the density is constant, say, g(x,y) = k, then m = k*a*T, X = a*sin(T)/T Y = a*(1-cos(T))/T If a = 1 and T = pi/2, then X = 2/pi Y = 2/pi. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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