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Question

Find the center of mass of uniform semicircular ring of radius R and mass m


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Solution

Step. 1 Calculation of the center of mass at x-axis,

The mass of this element is dm=λRdθ.

R is the ring of radius.
The center of mass will be calculated along X-axis as well as Y-axis.
For this, we need to consider the distance of the massdm.

Along the X-axis the distance will be Rsinθ and the distance of this mass dm along the Y-axis will be Rcosθ .
Thus, the x-coordinate of the center of mass xcm will be:
xcm=1m(Rcosθ)λRdθ
xcm=1m0π(Rcosθ)λRdθ
As θ lies between [0,π]
xcm=λR2m0πcosθdθ
xcm=λR2m[sinθ]0π
xcm=λR2m[sinπsin0]
xcm=λR2m[0]=0

Step 2. Calculation of the center of mass at the y-axis,
The y-coordinate of the center is:
ycm=1m(Rsinθ)λRdθ
ycm=λR2m0π(sinθ)dθ
As θ lies between [0,π]
ycm=λR2m[cosθ]0π
xcm=λR2m[1(cosπcos0)]
ycm=λR2m[1(11)]
ycm=λR2m×2
But λ is mass per unit length, substitutingλ=mπR, we get
ycm=mπR×R2m×2
ycm=2Rπ
Hence the position of the center of mass of the semi-circular ring lies at (0,2Rπ).


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