5. Consider the system shown in Fig. 5, where the cylinder of radius r and mass m is subjected to an external force excitation P(t). The cylinder undergoes pure rolling and rotates freely about its axis. Also, the cylinder is constrained by a massless spring with spring constant k and a massless dashpot with damping coefficient c. The motion of the center of the cylinder is described by displacement r(t), which is measured with respect to the undeformed position of the spring. Initially, the cylinder has an initial angular velocity wo and the spring is undeformed. (Hint: The mass moment of inertia of the cylinder about the center is mr.) Given that r(t) is governed by with the initial conditions mt+c +kt=P C 2 m 8 x(0) = 0, answer the following questions: (a) Derive the differential equations (11). (b) Derive the natural frequency wn, viscous damping factor , and damped natural fre- quency wa of this system. (c) Assume that the system is underdamped. If the input load P(t) is an impulse of magni- tude I, i.e., P(t) = 18(t), where 8(t) is the Dirac delta function, what will be the velocity of the center of the cylinder immediately after the impulse is applied? m i(0) = rwo (d) Following part (c), derive the subsequent response (t). Is this an impulse response or a step response? Describe the motion of the cylinder, and when does the cylinder go as t ? (11) (e) Assume that the spring is underdamped. If the input load P(t) = Po is a constant load. Sketch how the subsequent motion (t) looks under the initial conditions (12). Describe the motion of the cylinder, and when does the cylinder go as t? x(t) (12) P(t)