$(1)$
\[\begin{align}
k_1=\frac{T}{\theta}
&=\frac{5000\,\rm{N\cdot m}}{\left\{3\,\rm{deg}\cdot \frac{\pi\,\rm{rad}}{180\,\rm{deg}} \right\}}\\
&=\frac{5000\,\rm{N\cdot m}}{0.05235\,\rm{rad}}\\
&=95490\,\rm{N\cdot m/rad}\\
&＝ 95.5\,\rm{kN\cdot m/rad}
\end{align}\]
$(2)$
\[\begin{align}
\omega_n＝\sqrt{\frac{k_t}{J}}
&=\sqrt{\frac{95490\,\rm{N\cdot m/rad}}{2\,\rm{kg\cdot m^2}}}\\
&=\sqrt{\frac{95490\,\rm{kg\cdot m\cdot m/s^2\cdot rad}}{2\,\rm{kgm^2}}}\\
&=218.5 \,\rm{rad/s}\\
&＝ 219 \,\rm{rad/s}\\\end{align}\]
\[
f_n ＝\frac{\omega_n}{2\pi} ＝\frac{218.5 \,\rm{rad/s}}{2\pi\,\rm{rad}}＝ 34.77\,\rm{Hz} ＝ 34.8\,\rm{Hz}
\]
