$(1)$ \[r=\sqrt{R^2-z^2}\\ 微小体積: dV=\pi r^2 \cdot dz=\pi(R^2-z^2)dz\\ \] \[ 重心位置: Z_G=\frac{1}{V}\int{ZdV} = \frac{\int{ZdV}}{\int{dV}} \] \[\begin{align} 分子=\int{z dV} =\int^{R}_{0}z\pi\left(R^2-z^2 \right)dz &=\pi\int^{R}_{0}\left(zR^2-z^3 \right)dz\\ &=\pi\left[R^2\frac{z^2}{2}-\frac{z^4}{4} \right]^R_0\\ &=\pi\left[\frac{R^4}{2}-\frac{R^4}{4} \right]\\ &=\frac{\pi R^4}{4} \end{align}\] \[\begin{align} 分母=\int {dV} &=\int^{R}_{0}\pi\left(R^2-z^2 \right)dz\\ &=\pi\left[R^2z-\frac{z^3}{3} \right]^R_0\\ &=\pi\left[R^3-\frac{R^3}{3} \right]\\ &=\frac{2}{3} \pi R^3 \end{align}\] \[\begin{align} z_G &=\frac{\frac{1}{4}\pi R^4}{\frac{2}{3}\pi R^3}\\ &=\frac{3}{4\times 2}R\\ &=\frac{3}{8}R \end{align}\] $(2)$ \[\begin{align} R&=50\ \rm{mm}のとき、\\ Z_G &=\frac{3}{8}\times 50\,\rm{mm}\\ &=18.75\, \rm{mm} \end{align}\]