A \(200 \mathrm{~kg}\) machine is placed on a massless simply supported beam as shown in Figure P6.59. The machine has a rotating unbalance of \(1.41 \mathrm{~kg} \cdot \mathrm{m}\) and operates at \(3000 \mathrm{rpm}\). The steady-state vibrations of the machine are to be absorbed by hanging a mass attached to a \(40 \mathrm{~cm}\) steel cable from the location on the beam where the machine is attached. What is the required diameter of the cable such that machine vibrations are eliminated at \(3000 \mathrm{rpm}\) and the amplitude of the absorber mass is less than \(50 \mathrm{~mm}\) ?

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FIGURE P6.59 2 m 200 kg 1 m E 200 x 109 N/m I = 1.8 104 m4 m