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Recall that an annulus is the region between two concentric circles. Prove that the centroid of an annulus is the common center of the two circles.

Short Answer

Expert verified

the centroid of an annulus is the common center of the two concentric circles.

The Center of two concentric circles x2+y2=a2&x2+y2=b2is (0,0)and the centroid of the annulus is(x,y)=(0,0).

Step by step solution

01

Step 1. Given information.   

The given statement state that the centroid of an annulus is the common center of the two concentric circles.

02

Step 2. x coordinate of the centroid 

Consider two concentric circles x2+y2=a2&x2+y2=b2of constant density with a center at (0,0)where a<b.

xcoordinate of the centroid of the annulus is following.

x¯=xρ(x,y)dxdyρ(x,y)dxdy

change the system into a polar system by substituting x=rcosθ,y=rsinθ&dxdy=rdrdθand limits are role="math" localid="1650440508501" arb&0θ2π.

x¯=θ=02πθ=ab(rcosθ)rdrdθθ=02πθ=abrdrdθx¯=θ=abr2dr×θ=02πcosθdθθ=abr2dr×θ=02πdθx¯=(b3-a33)×(0)(a33)×(2π)x¯=0

03

Step 3. y coordinate of the centroid 

ycoordinate of the centroid of the annulus is following.

y¯=yρ(x,y)dxdyρ(x,y)dxdy

change the system into a polar system by substituting x=rcosθ,y=rsinθ&dxdy=rdrdθand limits are arb&0θ2π.

y¯=θ=02πr=ab(rsinθ)rdrdθθ=02πr=abrdrdθy¯=r=abr2dr×θ=02πsinθdθr=abr2dr×θ=02πdθy¯=(b3-a33)×(0)(b3-a33)×(2π)y¯=0

The centroid of annulus is (x,y)=(0,0).

the centroid of an annulus is the common center of the two circles.

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