TEAM PROJECT. Harmonic Motions in Different Physical

Chapter 2, Problem 2.4

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TEAM PROJECT. Harmonic Motions in Different Physical Systems. Different physical or other systems may have the same or similar models, thus showing the ullifyillg power of mathematical methods. Illustrate this for the systems in Figs. 43-45. (a) Flat spring (Fig. 43). The spring is horizontally clamped at one end, and a body of weight 25 nt (about 5.6Ib) is attached at the other end. Find the motion of the system, assuming that its static equilibrium is 2 cm below the horizontal line, we let the system start from this position with initial velocity 15 cm/sec, and damping is negligible. (b) Torsional vibrations (Fig. 44). Undamped torsional vibrations (rotations back and forth) of a wheel attached to an elastic thin rod are modeled bv the ODE " . loe + Ke = 0, where e is the angle measured from the state of equilibrium, 10 is the polar moment of inertia of the wheel about its center, and K is the torsional stiffness of the rod. Solve this ODE for Kilo = 17.64 sec-2, initial angle 45, and initial angular velocity 15 sec-I. (c) Water in a tube (Fig. 45). What is the frequency of vibration of 5 liters of water (about 1.3 gal) in a U-shaped tube of diameter 4 cm, neglecting friction?