Propeller blades totaling \(1200 \mathrm{~kg}\) with a total mass moment of inertia of \(155 \mathrm{~kg} \cdot \mathrm{m}^{2}\) are attached to a solid circular shaft ( \(ho=5000 \mathrm{~kg} / \mathrm{m}^{3}\), \(G=60 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, E=140 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) ) of radius \(40 \mathrm{~cm}\) and length \(20 \mathrm{~m}\). The other end of the shaft is fixed in an ocean liner. Determine

(a) The lowest natural frequency for torsional oscillations of the propeller.

(b) The lowest natural frequency for longitudinal motion of the propeller.