$(1)$ \[ f_1 = \frac{1}{2 × 10 × 10^{-2} \,\rm{m}} \sqrt \frac {100 \,\rm{kg\cdot m/s^2}}{0.01\,\rm{kg/m}} = 500 \,\rm{Hz} \] \[ \therefore500 \,\rm{Hz} \] $(2)$ \[ f_1 = \frac{1}{2 × 1 \,\rm{m}} \sqrt \frac {100 \,\rm{kg\cdot m/s^2}}{0.01\,\rm{kg/m}} = 50 \,\rm{Hz} \] \[ \therefore50 \,\rm{Hz} \] $(3)$ \[ f_1 = \frac{1}{2 × 1.5 \,\rm{m}} \sqrt \frac {20 \,\rm{N}}{0.2\,\rm{kg/m}} = 3.333 \,\rm{Hz} \] \[ \therefore3.33 \,\rm{Hz} \] $(4)$ \[ f_1 = \frac{1}{2 × 100 \,\rm{m}} \sqrt \frac {200 \,\rm{N}}{0.1\,\rm{kg/m}} = 0.2236 \,\rm{Hz} \] \[ \therefore0.224 \,\rm{Hz} \]