Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R1 and R2 (figure 9-E5).
ycm=∫y dmM
or dm=Mπrdrπ(R22−R21)2
=M(R22−R21)2dr
ycm=∫R2R1(2rπ).M(R22−R21)2.rdrM
=2π(R22−R21)2∫R2R1r2dr
=2π(R22−R21)2[R32−R313]
=4(R21+R1R2+R22)3π(R1+R2)