Show the orthogonality of the two lowest mode shapes of the system in Chapter Problem 10.5.

Data From Chapter Problem 10.5:

A motor of mass moment of inertia \(85 \mathrm{~kg} \cdot \mathrm{m}^{2}\) is attached to the end of the shaft of Chapter Problem 10.1. 

Data From Chapter Problem 10.1:

A \(5000 \mathrm{~N} \cdot \mathrm{m}\) torque is statically applied to the free end of a solid \(20-\mathrm{cm}\) radius steel shaft ( \(\left.G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, ho=7500 \mathrm{~kg} / \mathrm{m}^{3}\right)\) with a length of \(1.5 \mathrm{~m}\) that is fixed at one end and free at its other end. The torque is suddenly removed, and torsional oscillations begin.