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Prove that the centroid of a circle is the center of the circle.

Short Answer

Expert verified

the centroid of a circle is the center of the circle.

The centroid of a circlex2+y2=a2is at(x,y)=(0,0)and the center is also at(0,0).

Step by step solution

01

Step 1. Given information.  

The given statement is the centroid of a circle is the center of the circle.

02

Step 2. x coordinate of the centroid

Consider a circle x2+y2=a2of constant density with a center at (0,0).

xcoordinate of the centroid of the circle is following.

x¯=xρ(x,y)dxdyρ(x,y)dxdyx¯=x=aay=a2x2a2x2xdxdyx=aay=a2x2a2x2dxdy

change the system into a polar system by substituting x=rcosθ,y=rsinθ&dxdy=rdrdθ.

localid="1650439208659" x¯=θ=02πθ=0a(rcosθ)rdrdθθ=02πr=0ardrdθx¯=r=0ar2dr×θ=02πcosθdθr=0ar2dr×θ=02πdθx¯=(a33)×(0)(a33)×(2π)x¯=0

03

Step 3. y coordinate of the centroid

ycoordinate of the centroid of the circle is following.

y¯=yρ(x,y)dxdyρ(x,y)dxdyy¯=x=aay=a2x2a2x2ydxdyx=aay=a2x2a2x2dxdy

change the system into a polar system by substitutingx=rcosθ,y=rsinθ&dxdy=rdrdθ

y¯=θ=02πr=0a(rsinθ)rdrdθθ=02πr=0ardrdθy¯=r=0ar2dr×θ=02πsinθdθr=0ar2dr×θ=02πdθy¯=(a33)×(0)(a33)×(2π)y¯=0

the centroid of the circle is p(x,y)=(0,0).

So the centroid of a circle is the center of the circle.

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