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Consider a semicircular ring of radius R. Its linear mass density varies as λ=λ0sinθ. Locate its centre of mass.
769793_d8798eb3af884ad9bb86f167ce6a7e34.png

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Solution

λ=λ0sinθdm=(Rdθ)λ=λ0RsinθdθXcom=xdmdm=(Rcosθ)(λ0Rsinθdθ)λ0RsinθdθXcom=λ0R22π0sin2θdθλ0Rπ0sinθdθ=R2×[cos2θ2]π00[cosθ]π0=0Ycom=ydmdm=(Rsinθ)(λ0Rsinθdθ)λ0Rπ0sinθdθ=R2=[π0sin2θdθπ0sinθdθ]=R2[π0(1+cos2θ]dθ[cosθ]π0]=R2[θsin2θ2]π02=πR4
952124_769793_ans_ceb61417070d4f1481cd246166ca63fd.JPG

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