A solid disk of diameter \(30 \mathrm{~cm}\) and mass \(10 \mathrm{~kg}\) is attached to the end of a solid 3-cm-diameter, 1-m-long steel shaft \(\left(G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\). A torsional vibration absorber consists of a disk attached to a shaft that is then attached to the primary system. If the absorber disk has a mass of \(3 \mathrm{~kg}\) and a diameter of \(10 \mathrm{~cm}\), what is the required diameter of a \(50-\mathrm{cm}\)-long absorber shaft to eliminate steady-state vibrations of the original system when excited at \(500 \mathrm{rad} / \mathrm{s}\) ?