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lecture22 - 10.4 The Simple Pendulum A pendulum is simply a...

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10.4 The Simple Pendulum A pendulum is simply a mass attached to a string/rope (or rigid rod) that, once set to motion will swing back and forth under the influence of gravity into motion, will swing back and forth under the influence of gravity. Clearly, the motion of the pendulum is periodic, i.e. it repeats in time. But is the motion of the pendulum simple harmonic motion (SHM)??? Let’s draw the FBD for the pendulum. T x y θ L W Break the forces down into their x - and y - omponents W x W y Assume it is swinging on the way up (ccw). components. Notice, the force acting along the direction of motion is the x -component of the weight, and it acts in the – x -direction. For SHM, we need the force ( W x ) to be W W x = sin sin mg W x = proportional to , but in the opposite direction. The force is not proportional to , but to sin . Thus, the pendulum motion is strictly not SHM. But, for small angles, sin θ≈θ , so that: mg W x = So, in a small-angle approximation, pendulum motion is SHM!!!
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θ mg F = for the pendulum in the small-angle approximation. The displacement of the pendulum mass, which is the distance it travels along its arc, is given by: L s = mgs mg L F = ) ( s L = ent) (Displacem Constant) ( × = kx = Thus, for the pendulum, L mg k = , m k = ω Since we find the following result for the frequency of pendulum motion:
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lecture22 - 10.4 The Simple Pendulum A pendulum is simply a...

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