A \(35 \mathrm{~kg}\) machine is attached to the end of a cantilever beam of length \(2 \mathrm{~m}\), elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\), and moment of inertia \(1.3 \times 10^{-7} \mathrm{~m}^{4}\). The machine operates at \(180 \mathrm{rpm}\) and has a rotating unbalance of \(0.3 \mathrm{~kg} \cdot \mathrm{m}\).

(a) What is the required stiffness of an undamped absorber of mass \(5 \mathrm{~kg}\) such that steady-state vibrations are eliminated at \(180 \mathrm{rpm}\) ?

(b) With the absorber in place, what are the natural frequencies of the system?

(c) For what range of operating speeds will the steady-state amplitude of the machine be less than \(8 \mathrm{~mm}\) ?