51. If the torque exerted by the spring on the rod is proportional to the angle of rotation of the rod and if the torque tends to pull the rod toward its equilibrium orientation, then the rod will oscillate in simple harmonic motion. If τ= –Cθ, where is the torque, is the angle of rotation, and Cis a constant of proportionality, then the angular frequency of oscillation is ω=CI/and the period isTIC==22ππ//,whereIis the rotational inertia of the rod. The plan is to find the torque as a function of and identify the constant Cin terms of given quantities. This immediately gives the period in terms of given quantities. Let A0be the distance from the pivot point to the wall. This is also the equilibrium length of the spring. Suppose the rod turns through the angle , with the left end moving away from the wall. This end is now (L/2) sin further from the wall and has moved a distance (L/2)(1 – cos ) to the right. The length of the spring is
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.