Waves and oscillations.

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timer Asked: Apr 21st, 2016

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damped oscillator! (a) A mass spring damper system is driven by a sinusoidal force with amplitude Fo and frequency w. (i) Does the system's trajectory (i.c., the position of the mass as a function of time) depend on Fo, the driving amplitude? Maybe (ii) Does the system's trajectory (i.c., the position of the mass as a function of time) depend on w, the driving frequency? Yes Maybe Yes No No (iii) Does the impedance depend on For the driving amplitude? Yes No Maybe Yes (iv) Does the impedance depend on w, the driving frequency? Yes No Maybe (v) Does the phase angle depend on Fo, the driving amplitude? Yes No Maybe (vi) Does the phase angle depend on w, the driving frequency? No Maybe (vii) Does the quality factor depend on Fo, the driving amplitude? Yes No Maybe (viii) Does the quality factor depend on w, the driving frequency? Yes No Maybe (b) Two sets of mass spring damper systems are comprised of two ideal springs, with spring constants 5 N/m and 20 N/m, respectively, along with a damper of strength bi 12 N.s/m, together supporting a massless platform, on which a block of mass M = 2 kg rests. In one system, the springs are arranged in series (as in II) while in the other system, they are in parallel (as in III). A rather long lasting tempest subjects the two mass spring damper systems to a thorough sinusoidal shaking with amplitude Fo = 4 n, and angular frequency w = 2 V3rad/s. (i) Ascertain the period of these tempestuous vibrations. (ii) Determine the average rate at which the series system studied in Problem II absorbs and re-emits energy. (iii) Determine the average rate at which the parallel system studied in Problem III absorbs and re-emits energy. Page 4 of 7 (VI) In problems (VI) and (VII) we shall analyse another proposed design for a campus thermometer. Consider a thin copper rod with square cross section and face area mm? (i... , edge length 0.5 mm), and length 2.000m at T = 25C. The rod is stretched between two posts set in a concrete pad which are 2.006 m apart at 25c. The Young's modulus for this type of copper is 120 GPa, and its density is 9000 kg/m copper rod post concrete pad (a) (i) Determine the lineal strain that the rod endures (at T = 25C). (ii) Determine the lineal stress experienced by the rod (at T = 25C). (iii) Compute the force of tension within the rod, ET (b) (i) Determine the (average) mass per unit length of the unstretched rod. (ii) Argue that this is a good approximation to the lineal density of the stretched rod. (c) Determine the speed of transverse waves on the copper rod (at t = 25C). (d) Determine the frequency of the fundamental (first harmonic) standing wave supported on the stretched rod. Page 6 of 7 (VII) Consider the same copper rod stretched between two posts, but now at temperatures differing from 25c. The thermal expansion coefficients for concrete and copper are 1.0 * 10-5c-1 and 1.7 x 10-5C), respectively. (a) Suppose that the temperature increases to 35 C. (i) Deterniine the change in the length of the unstretched bar arising from the change in temperature (ii) Determine the change in the distance of separation of the posts embedded in the concrete pad (iii) Ascertain by how much the copper rod must be stretched to span the distance between the posts (at 35C). (b) But the rod is not free to move; rather it is attached to the posts. (i) Determine the lineal strain that the rod now endures (at T = 35C). (ii) Determine the lineal stress experienced by the rod (at T = 35C). (iii) Compute the force of tension within the rod, Fr. (c) Using the same mass per unit length as before, determine the speed of transverse wave on the copper rod (at t = 35C). (d) Determine the frequency of the fundamental (first harmonic) standing wave support on the stretched rod. HIGHER THAN (e) Without doing any computation, one might expect that the fundamental frequenc the rod at T = 5C is CHOOSE ONE: THE SAME AS LOWER THAN fundamental frequency at T = 25 C. (f) Comment on the design of this thermometer. Page 7 of 7
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