Midterm 2 Practice - N me le rin ~ PID Practice Midterm...

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Unformatted text preview: N me le rin ~ PID: Practice Midterm Exam #2 Total points = 25. Show all of your work! 1. [12 points] A damped oscillator consists of a spring (with a spring constant k = 8.00 N/m), a block of mass m = 1.50 kg, and a damping force given by —bi (where b = 0.230 kg/s). Suppose that the block is initially pulled so that the spring is extended a distance of x = 0.12 m and then released from rest. (a) [3 points] What is the equation of motion for the block? (b) [3 points] What is the solution to the equation of motion i.e. what is x(t)? (c) [3 points] Calculate the time required for the amplitude of the oscillations to fall to one-third of its initial value. ((1) [3 points] How many oscillations are made by the block in this time? Note: There is another question on the next page! 2. [13 points] Consider a particle of mass m constrained by a weightless, extensionless rod to move in a vertical circle of radius l (i.e. a plane pendulum). The particle can be thought of as moving in a periodic potential U(6) = mgl(1-cos6). (a) [2 points] If the total energy of the particle is E1 = mgl, what is the maximum value of the angular velocity, 0? (b) [1 point] At what angle does this maximum occur? (0) [1 point] At what angle is 6 = 0? (d) [4 points] If the total energy = 132 = 3mgl, what are the maximum m minimum values of the angular velocity, 5? (e) [2 points] At what am angles do these extreme values occur? 0 (f) [3 points] Sketch the phase paths (9 vs. 0) for both the energies E1 and E. W Oscillations: SHM 3: + (002x = 0 x(t) = Asin(mot — a) Damped SHM 3: +213)? + (002x = 0 The general solution is:- X(t) = MIX-[30[I’xlt‘J‘PU/(l32 - who + AzeXP(-\/(l32 ' (1)050] depending on the relative value of (002 and [32. ...
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