Consider a mass and spring system with mass m = 3 and spring constant k = 75, and damping constant c. What would the frequency of oscillation be if there were no damping at all (c = 0)? What is the critical value of the damping constant c that separates oscillatory from non-oscillatory motion? fc=25, is the system underdamped, overdamped, or neither?
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- For Parts I and II, use the pendulum equation and solve to predict the length needed for a pendulum with a period of 1 s and 2 s respectively. Use a piece of string (or thread, dental floss, shoe lace, etc.) and a steel nut (or washer or something small but with enough mass to weigh down the string) to build each pendulum. Time each pendulum for 30 periods and then find the average time for one period. Remember, a period is the time to complete one cycle of motion, out and back. Tape each pendulum up in perhaps a doorway where it is stationary and has room to swing. Be precise with your measuring and timing. Show your work. Part I Given: T = 1.00 s l =? Part II Given: T = 2.00 s l =?3 b) A mass weighing 4 pounds is attached to a spring whose constant is 2 Ib/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 14 ft/s. Find the time (in s) at which the mass attains its extreme displacement from the equilibrium position (the extreme distance after passing the equilibrium position.) Round your answer to two digits after the decimal sign.damped harmanic oscillator, haS damping constant a = 2 We, that is acted upon by a driving force F = Fo sin wt, The system Starts from rest With an and initial displacement of Xo lie,xLO) = Xo 10)=0), Find the equation of motion and its corres panding salution xlt), determine all of the coefficients le.g., Ai,A2, B,, Bzretc) %3D Be,Bz,et c)
- s" + bs' +5s = 0, find the values of b that make the general solution overdamped, underdamped, or critically damped. (For each, give an interval or intervals for b for which the equation is as indicated. Thus if the the equation is overdamped for all b in the range -1Problem 2 (Estimating the Damping Constant). Recall that we can experimentally mea- sure a spring constant using Hooke's law-we measure the force F required to stretch the spring by a certain y from its natural length, and then we solve the equation F = ky for the spring constant k. Presumably we would have to determine the damping coefficient of a dashpot empirically as well, but how would we do so? As a warm-up, suppose we have a underdamped, unforced spring-mass system with mass 0.8 kg, spring constant 18 N/m, and damping coefficient 5 kg/s. We pull the mass 0.3 m from its rest position and let it go while imparting an initial velocity of 0.7 m/s. %3D (a) Set up and solve the initial value problem for this spring-mass system. (b) Write your answer from part (a) in phase-amplitude form, i.e. as y(t) = Aeºt sin(ßt – 4) and graph the result. Compare with a graph of your answer from (a) to check that you have the correct amplitude and phase shift. (c) Find the values of t at which y(t)…one-dimensional crystal assembled from three identical atoms connected to one another and to the walls by identical springs as shown.find the normal modes of given system.plot relative displacements versus time for all three normal modes.