A rigid pendulum of mass m swings about point A on a horizon
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A rigid pendulum of mass m swings about point A on a horizontal axis. Its moment of inertia about that axis is I . The c
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A rigid pendulum of mass m swings about point A on a horizontal axis. Its moment of inertia about that axis is I . The centre of mass C of the pendulum is at distance a from A. When the pendulum hangs at rest, C is directly under A. (Why?) Suppose the pendulum is swinging. Let D .t / measure the angular displacement of the line AC from the vertical at time t. ( D 0 when the pendulum is in its rest position.) Use a conservation of energy argument similar to that in Example 4 to show that 1 2 I d dt 2 ? mga cos D constant and, hence, differentiating with respect to t, that d 2 dt2 C mga I sin D 0: This is a nonlinear differential equation, and it is not easily solved. However, for small oscillations (j j small) we can use the approximation sin . In this case the differential equation is that of simple harmonic motion. What is the period?
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