$(a)$ \[ J=m\cdot\frac{(D/2)^2}{2}=200\,\rm{kg}\times\frac{(1\,\rm{m}/2)^2}{2}\\ =25.0\,\kgsqm\hspace{20mm}\\ \underline{\therefore J=25.0\ \kgsqm} \] $(b)$ \[ \begin{align} k_{te}&=\frac{1}{\frac{1}{k_{t1}}+\frac{1}{k_{t2}}}=\frac{k_{t1}\cdot k_{t2}}{k_{t1}+k_{t2}}\\ k_{t1}&=\frac{GI_{p}}{L_1}=\frac{G\pi d_1^4} {32L_1}=\frac{81\times10^9\,\rm{Pa}\times\pi\times(0.02\,\rm{m})^4}{32\times0.1\,\rm{m}}\\ &=12.72\times10^3\,\rm{N・m/rad}\\ k_{t2}&=\frac{GI_{p}}{L_2}=\frac{G\pi d_2^4}{32L_2} =\frac{81\times10^9\,\rm{Pa}\times\pi\times(0.03\,\rm{m})^4}{32\times0.2\,\rm{m}}\\ &=32.20\times10^3\,\rm{N・m/rad}\\ k_{te}&=\frac{12.72\times10^3\times32.20\times10^3}{12.72\times10^3+32.20\times10^3}=9.118\times10^3\,\rm{N・m/rad}\\ \therefore k_{te} &=9.12\,\rm{kN・m/rad} \end{align} \] $(c)$ \[J\ddot\theta=(-k_{te}\cdot\theta)より\\ \therefore \hspace{10mm}\underline{J\ddot\theta+k_{te}\theta=0}\] $(d)$ \[\omega_n=\sqrt{\frac{k_{te}}{J}}=\sqrt \frac{9.118\times10^3\,\rm{N・m/rad}}{25.0\,\rm{kg・m}^2}=19.09\,\rm{rad/s}\hspace{10mm}\\ \underline{\therefore \omega _n=19.1\,\rm{rad/s}}\] \[f_n=\frac{\omega_n}{2\pi}=\frac{19.09\,\rm{rad/s}}{2\pi}=3.038\,\rm{Hz}\hspace{10mm}\\ \underline{\therefore f_n=3.04\,\rm{Hz}} \]