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notes13a - The Simple Pendulum An application of Simple...

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The Simple Pendulum An application of Simple Harmonic Motion A mass m at the end of a massless rod of length L There is a restoring force which acts to restore the mass to θ =0 Compare to the spring F s =-kx The pendulum does not display SHM m θ T mg θ sin mg F = mgsin θ L
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But for very small θ (rad), we can make the approximation ( θ <0.5 rad or about 25°) simple pendulum approximation L mg k kx F s L mg L s mg F L mg F s = = = = = = = ) ( r s since sin θ θ θ θ θ Arc length Looks like spring force Like the spring constant This is SHM Now, consider the angular frequency of the spring
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L g f L g m L mg m k π ω ω 2 1 / = = = = = Simple pendulum angular frequency Simple pendulum frequency With this ω , the same equations expressing the displacement x , v , and a for the spring can be used for the simple pendulum, as long as θ is small For θ large, the SHM equations (in terms of sin and cos) are no longer valid more complicated functions are needed (which we will not consider) A pendulum does not have to be a point-particle
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