A one degree-of-freedom model of a 9 -m steel flagpole ( \(ho=7400 \mathrm{~kg} / \mathrm{m}^{3}\), \(\left.E=200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) is that of a beam fixed at one end and free at one end. The flagpole has an inner diameter of \(4 \mathrm{~cm}\) and an outer diameter of \(5 \mathrm{~cm}\).

(a) Approximate the natural frequency of transverse vibration.

(b) Approximate the natural frequency of torsional oscillation.