2. Figure 2 shows a roller rolling and slipping at the same time on a slippery...

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2. Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The center of the roller moves with a displacement x(t) and the roller rotates with angle (t). The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is pulled by a spring under a given (i.c., prescribed) displacement u(t), whereas the spring has spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows the free-body diagram of the roller. Answer the following questions. (a) Apply F = ma to derive an equation of motion governing the translation x(t). (b) Apply M = Ia to derive an equation of motion governing the rotation (t). (c) Eliminate the variable (t) from the two equations of motion derived in parts (a) and (b). You should obtain a third-order differential equation in x(t). (d) Initially (i.c., at t = 0), the roller has no displacement but translates with a velocity vo. The roller also has an initial spin rate wo. Derive the initial conditions for the third-order different equation governing x(t) in part (c). x(t) mg 8(t) u(t) k(u-x) k M c(x-r) m, I M Figure 2: A slipping roller with a pre- scribed displacement Figure 3: Free-body diagram of the slipping roller 2. Figure 2 shows a roller rolling and slipping at the same time on a slippery surface. The center of the roller moves with a displacement x(t) and the roller rotates with angle (t). The roller has mass m, mass moment of inertia I, and radius r. In addition, a layer of fluid with viscous damping coefficient c is present to lubricate the roller. Moreover, the roller is pulled by a spring under a given (i.c., prescribed) displacement u(t), whereas the spring has spring constant k. Also, the roller is subjected to an applied torque M(t). Figure 3 shows the free-body diagram of the roller. Answer the following questions. (a) Apply F = ma to derive an equation of motion governing the translation x(t). (b) Apply M = Ia to derive an equation of motion governing the rotation (t). (c) Eliminate the variable (t) from the two equations of motion derived in parts (a) and (b). You should obtain a third-order differential equation in x(t). (d) Initially (i.c., at t = 0), the roller has no displacement but translates with a velocity vo. The roller also has an initial spin rate wo. Derive the initial conditions for the third-order different equation governing x(t) in part (c). x(t) mg 8(t) u(t) k(u-x) k M c(x-r) m, I M Figure 2: A slipping roller with a pre- scribed displacement Figure 3: Free-body diagram of the slipping roller