# sg16 - Chapter 16 Study Guide for Periodic Motion 16.1...

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Unformatted text preview: Chapter 16 Study Guide for Periodic Motion 16.1 Periodic motion and energy Skill 16.1 Understand the fundamentals of periodic motion. Periodic motion is any motion that repeats itself. It is due to a continuous back and forth conversion between potential and kinetic energy in an isolated system. Note: In reality, each of these conversions is accompanied by some energy dissipation, causing the periodic motion to die out. This process is called damping . To begin with, we will act as if damping does not occur. The time duration of a full cycle of periodic motion is called the period , T . Example: For an object in circular motion, the period is the amount of time it takes to trace out one circle. The number of cycles completed in one second is called the frequency , f . It is the inverse of the period: f = 1 T . (16.1) The amplitude of periodic motion is the maximum displacement from the equilibrium position. 16.2 Simple harmonic motion Skill 16.2 Understand the basic properties that define simple harmonic motion. Any oscillating object whose position vs. time curve is a sinusoidal function (sine or cosine) is called a simple harmonic oscillator , and its motion is called simple harmonic motion . The period of a simple harmonic oscillator is independent of the amplitude and determined solely by the properties of the system of which the object is a part. Any system whose period exhibits this property is called isochronous (See Figure 16.3 for examples). Every simple harmonic oscillator is subject to a linear restoring force (remember, a linear restoring force tends to return the object to its equilibrium position and is linearly proportional to the object’s displacement from equilibrium). 16.3 Fourier’s theorem Skill 16.3 Understand how Fourier’s theorem can be used to construct arbitrary periodic functions from simple harmonic functions. Fourier’s theorem states that any periodic function with period T can be written as a sum of simple harmonic functions of frequency f n = n T , where n is a positive integer. This means that any periodic motion can be represented by the sum of a bunch of simple harmonic motions. 1 The lowest frequency harmonic component, called the fundamental frequency , has the same frequency f 1 = 1 T as the original periodic function. By adding other components, called higher harmonics , whose frequencies are integer multiples of the funda- mental frequency, the shape of the resulting periodic function changes, but the period remains the same . The reason the period remains the same is simple. Since the frequencies of the higher harmonics are integer multiples of the fundamental frequency, their periods are rational fractions of the original period. Any compo- nent that repeats every T n seconds repeats for the n th time at T seconds. Thus, all the components repeat simultaneously at T seconds....
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sg16 - Chapter 16 Study Guide for Periodic Motion 16.1...

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