$(a)$ \[ \begin{align} 軸のばね定数 : k &= \frac{48EI}{l^3} \\ &=\frac{48 \times 206 \times 10^9 \,\rm{Pa} \times \frac{\pi}{64} \times (0.025\,\rm{m})^4}{(0.6\,\rm{m})^3} \\ &=8.777 \times 10^5 \,\rm{N/m} \end{align} \] \[ \therefore k=8.78 \times 10^5 \,\rm{N/m} \] $(b)$ \[ \begin{align} 軸の質量 : m_b &=\rho A l \\ &=7.8 \times 10^3 \,\rm{kg/m^3} \times \frac{\pi}{4} \times (0.025\,\rm{m})^2 \times 0.6 \,\rm{m} \\ &=2.297 \,\rm{kg} \\ \end{align} \] \[ \begin{align} w_n&=\sqrt{\frac{k}{m+0.5m_b}} \\ &=\sqrt{\frac{8.777 \times 10^5 \,\rm{N/m} }{30\,\rm{kg} +0.5\times 2.297\,\rm{kg}}} \\ &=167.8 \,\rm{rad/s} \end{align} \] \[ \therefore w_n=168 \ \,\rm{rad/s} \] \[ \begin{align} f_n&=\frac{w_n}{2 \pi} \\ &=\frac{167.8 \,\rm{rad/s} }{2 \pi} \\ &=26.70 \,\rm{Hz} \end{align} \] \[ \therefore f_n=26.7 \ \,\rm{Hz} \] $(c)$ \[\begin{align} N_c &= \frac{60}{2\pi}w_n \\ &=\frac{60}{2 \pi}\times 167.8\,\rm{rad/s} \\ &=1602\,\rm{rpm}\\ \\ または、\\ N_c&=60 f_n \\ &=60 \times 26.70 \,\rm{Hz}\\ &=1602\,\rm{rpm} \end{align}\] \[ \therefore N_c=1600\,\rm{rpm} \]