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Question

An initial amplitude k, damping constant c, and frequency f or period p are given. (Recall that frequency and period are related by the equation f=1 / p.)(a) Find a function that models the damped harmonic motion. Use a function of the form $y=k e^{-c t} \sin \omega t$ in Exercises given below.(b) Graph the function.$k=1, \quad c=1, \quad p=1$

Solution

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If the equation describing the displacement $y$ of an object at time $t$ is

$y=ke^{-ct}\sin\omega t$         or         $y=ke^{-ct}\cos\omega t$ ,     $(c>0)$

then the object is in $\fbox{damped harmonic motion}$.

The constant $c$ is the damping constant, $k$ is the initial amplitude, and $2\pi/\omega$ is the period.

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