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Bounding curves in harmonic motion
Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying
amplitude , such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.
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Key equations
Standard form of sinusoidal equation |
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Simple harmonic motion |
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Damped harmonic motion |
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Key concepts
- Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See
[link] and
[link] .
- Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See
[link] .
- Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See
[link] ,
[link] ,
[link] and
[link] .
- Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See
[link] .
- Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See
[link] ,
[link] ,
[link] ,
[link] , and
[link] .
- Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See
[link] .
Section exercises
Verbal
Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?
Physical behavior should be periodic, or cyclical.
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What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.
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If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?
Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.
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Algebraic
For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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