Suppose that the mass in a mass-spring-dashpot system with m = 20, c = 17 and k = 3 is set in motion with x(0) = 0 and x'(0) = 8. (a) Find the position function x(t) and show that its graph looks as indicated in the figure. (b) Find how far the mass moves to the right before starting back toward the origin. 160 5 (a) x(t) = (Type an exact answer.) e + 160 M 7 ... (b) The mass moves 7.1344 units to the right before turning back. (Round to four decimal places as needed.) 8- 10 20 30 40 50

+ b 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent…

A body weighing 5kg is attached to a very light bar as depicted below. The set-up is free to rotate in the vertical plane about pin support O. The force applied at point C makes the bar oscillate(via small rotation) about the position of equilibrium illustrated below a. Derive the differential equation of motion for the steady state angular displacement (theta) of the bar

please define the system and provide the free-body diagram with x-y-axis Please answer these two questions together cause there is a follow-up question.