21-28= Damped Harmonic Motion An initial amplitude k ,...

$21-28=$ Damped Harmonic Motion 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 $f=1 / p$ . $$ \begin{array}{l}{\text { (a) Find a function that models the damped harmonic }} \\ {\text { motion. Use a function of the form } y=k e^{-c t} \cos \omega t \text { in }} \\ {\text { Exercises } 21-24 \text { and of the form } y=k e^{-c t} \sin \omega t \text { in }} \\ {\text { Exercises } 25-28 \text { . }} \\ {\text { (b) Graph the function. }}\end{array} $$ $$ k=7, \quad c=10, \quad p=\pi / 6 $$

$21-28=$ Damped Harmonic Motion 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 $f=1 / p$ .

$$
\begin{array}{l}{\text { (a) Find a function that models the damped harmonic }} \\ {\text { motion. Use a function of the form } y=k e^{-c t} \cos \omega t \text { in }} \\ {\text { Exercises } 21-24 \text { and of the form } y=k e^{-c t} \sin \omega t \text { in }} \\ {\text { Exercises } 25-28 \text { . }} \\ {\text { (b) Graph the function. }}\end{array}
$$

$$
k=7, \quad c=10, \quad p=\pi / 6
$$

Key Concepts

Graphing of Damped Oscillations

Graphing damped oscillations provides a visual representation of both the decay of amplitude over time and the periodic nature of the oscillation. It helps in understanding the combined effects of exponential decay and sinusoidal oscillation, illustrating how the motion evolves from higher to lower energy states.

Angular Frequency

Angular frequency is a measure of how quickly the oscillation cycles occur, expressed in radians per unit time. It is related to the period and frequency of the motion by formulas such as ? = 2?f or ? = 2?/p. This parameter is crucial for accurately modeling the timing and phase of the oscillations.

Relationship Between Frequency and Period

Frequency and period are reciprocals of one another, with frequency being the number of cycles per unit time and period being the duration of one cycle. This relationship, expressed as f = 1/p, is essential when converting between these two quantities in the analysis of periodic or oscillatory systems.

Exponential Decay

Exponential decay refers to a process where the magnitude of a quantity decreases at a rate proportional to its current value. In the context of damped harmonic motion, the exponential decay factor models how the amplitude of the oscillation diminishes over time, characterizing the energy loss in the system.

Damped Harmonic Motion

Damped harmonic motion describes a type of oscillatory behavior in which the amplitude of the oscillations decreases over time due to the presence of a damping force such as friction or resistance. This concept is central to understanding systems where energy is gradually dissipated, resulting in a decay of the oscillatory behavior.

Sinusoidal Functions

Sinusoidal functions, including sine and cosine, are fundamental in modeling periodic phenomena. They capture the oscillatory component of the motion, and the choice between sine and cosine depends on the initial conditions of the system, such as whether the motion starts at a maximum displacement or from the equilibrium position.

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