Derive the equation of motion for the homogeneous circular cylinder, which rolls without slipping. If the cylinder mass is \(50 \mathrm{~kg}\), the cylinder radius \(0.5 \mathrm{~m}\), the spring constant \(75 \mathrm{~N} / \mathrm{m}\), and the damping coefficient \(10 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\), determine

(a) The undamped natural frequency

(b) The damping ratio

(c) The damped natural frequency

(d) The period of the damped system.

In addition, determine \(x\) as a function of time if the cylinder is released from rest at the position \(x=-0.2 \mathrm{~m}\) when \(t=0\).

Transcribed Image Text:

m x+