$(a)$
\[T_R=\frac{|F_T|}{F}=\frac{1}{|1-Z^2|}=\frac{1.125\,\rm{kN}}{9\,\rm{kN}}=\frac{1}{8}よりZ>1\]
\[\frac{1}{Z^2-1}=\frac{1}{8}より\]
\[Z^2=8+1=9\]
\[\underline{\therefore Z=3}\]
$(b)$
\[f=\frac{2700\,\rm{rpm}}{60}=45.0\,\rm{Hz}\]
\[Z=\frac{f}{f_n}よりf_n=\frac{f}{Z}=\frac{45.0\,\rm{Hz}}{3}=15.0\,\rm{Hz}\]
\[\underline{\therefore f_n=15.0\,\rm{Hz}}\]
\[\omega_n=2\pi f_n=2\pi \times15.0\,\rm{Hz}=94.24\,\rm{rad/s}\]
\[\underline{\therefore\omega_n=94.2\,\rm{rad/s}}\]
$(c)$
\begin{align}\omega_n^2=\frac{k}{m}よりk&=\omega_n^2\cdot m=(94.24\,\rm{rad/s})^2\times700\,\rm{kg}\\&=6.216\times10^6\,\rm{N/m}\end{align}
\[\underline{\therefore k=6.22\,\rm{MN/m}}\]
$(d)$
\begin{align}A=\frac{\delta st}{|1-Z^2|}=\frac{F/k}{Z^2-1}&=\frac{9\times10^3\,\rm{N}}{6.216\times10^6\,\rm{N/m}\,(3^2-1)}\\
&=0.1809\times10^{-3}\,\rm{m}\end{align}
\[\underline{\therefore A=0.181\,\rm{mm}}\]
