A hemisphere $($yar$$k$$re$)$ or radius RR and mass MM is placed on the x$$zx$$z plane with the center $($of the full sphere$)$ at the origin and yy axis is the axis of symmetry. We want to find the height of its center of mass $($cm$)$ from its base. We divide it into thin slices of thickness dydy and integrate.

Part A $-$ Mass dMdM of the slice between yy and y$+$dyy$+$dy

What is the mass$$dMdM$$of the slice between$$yy$$and$$y$+$dyy$+$dy$?$

$3$M$2$R$3($R$2+$y$2)$dy$3$M$2$R$3($R$2+$y$2)$dy$$Not enough information in the problem$3$M$2$R$2($R$2+$y$2)$dy$3$M$2$R$2($R$2+$y$2)$dy$3$M$2$R$3($R$2$y$2)$dy$3$M$2$R$3($R$2$y$2)$dy$3$M$2$R$2($R$2$y$2)$dy$3$M$2$R$2($R$2$y$2)$dy$$

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Part B $-$ Contribution of the slice to cm

What is the contribution of this thin slice to the center of mass position?

$3$M$2$R$3($R$2+$y$2)$dy$3$M$2$R$3($R$2+$y$2)$dy$$There is not enough information in the problem$3$M$2$R$3($R$2$y$2)$Rdy$3$M$2$R$3($R$2$y$2)$Rdy$3$M$2$R$3($R$2+$y$2)$Rdy$3$M$2$R$3($R$2+$y$2)$Rdy$3$M$2$R$3($R$2$y$2)$ydy$3$M$2$R$3($R$2$y$2)$ydy$$

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Part C $-$ The integral

The y coordinate of the center of mass represented as an integral is given as;

$1$MMR$3$R$0($R$2$y$2)$ydy$1$MMR$30$R$($R$2$y$2)$ydy$3$M$2$R$3$R$1/$R$($R$2$y$2)$ydy$3$M$2$R$31/$RR$($R$2$y$2)$ydy$1$M$2$M$3$R$3$R$0($R$2$y$2)$ydy$1$M$2$M$3$R$30$R$($R$2$y$2)$ydy$1$M$3$M$2$R$3$R$0($R$2$y$2)$ydy$1$M$3$M$2$R$30$R$($R$2$y$2)$ydy$1$M$3$M$2$R$2$R$$R$($R$2$y$2)$ydy$1$M$3$M$2$R$2$RR$($R$2$y$2)$ydy$$

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Part D $-$ The cm

The y coordinate of the center of mass is:

MR$2/3$MR$2/34$R$/34$R$/33$R$/43$R$/43$R$/83$R$/8$RR