Problem 22 - Precalculus Mathematics for Calculus

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 \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t}\) sin \(\omega t\) in Exercises \(23-26\) (b) Graph the function.

Step-by-Step Solution

Answer

Model using \(y = k e^{-ct} \cos(\omega t)\) or \(y = k e^{-ct} \sin(\omega t)\) and plot it.

Step 1: Understand the Given Information

We are given the initial amplitude \(k\), the damping constant \(c\), and either the frequency \(f\) or the period \(p\). These are critical in modeling the damped harmonic motion.

Step 2: Relate Frequency and Period

Use the equation \(f = \frac{1}{p}\) to convert between frequency and period if necessary. This helps us to determine the angular frequency \(\omega\).

Step 3: Calculate Angular Frequency

For damped harmonic motion, the angular frequency \(\omega\) is related to the frequency by the equation \(\omega = 2\pi f = \frac{2\pi}{p}\). Use this formula to calculate \(\omega\) based on \(f\) or \(p\).

Step 4: Model the Damped Harmonic Motion

If the exercise specifies using cosine (Exercises 19-22), the function will be \(y = k e^{-ct} \cos(\omega t)\). If it specifies sine (Exercises 23-26), then \(y = k e^{-ct} \sin(\omega t)\). Use the appropriate form based on the exercise number.

Step 5: Graph the Function

Plot the function \(y = k e^{-ct} \cos(\omega t)\) or \(y = k e^{-ct} \sin(\omega t)\) on graph paper or using graphing software. Ensure to label axes and indicate the amplitude damping over time.

Key Concepts

Initial Amplitude

In the study of damped harmonic motion, the initial amplitude plays a fundamental role. It is symbolized by the letter \( k \). This term refers to the maximum displacement from the equilibrium position at the start of the motion. Think of it as the starting "size" of the wave. Initially, if you were to stretch or compress a spring, the furthest point away from its rest position before releasing it would be the initial amplitude.

The initial amplitude gives us insight into how vigorous or large the oscillation is when it begins. Over time, because of damping, this amplitude will decrease. However, knowing the starting point is crucial for modeling and graphical accuracy. Just imagine throwing a ball into a valley; how high it goes initially (initial amplitude) affects how it bounces down the valley initially. This concept is essential for calculating the resulting damped harmonic motion using the given formulae.

Damping Constant

The damping constant, represented as \( c \), serves to quantify how quickly the oscillations lose energy over time. In other words, it determines how fast the amplitude decreases. This parameter is crucial as it impacts how long the oscillation remains significant before it fades away.

When \( c \) is large, the damping effect is also strong, leading to a rapid decrease in amplitude. This means that the system will quickly settle back into equilibrium without much oscillating. Conversely, a smaller \( c \) results in a gentler damping effect, allowing the oscillations to continue for a longer time.

For example:

• If you are modeling the damped motion of a car suspension system, the damping constant ensures the car doesn't bounce excessively over rough terrain.
• In electrical circuits, it could mean preventing circuits from oscillating indefinitely.

Understanding the damping constant will enable you to comprehend and predict the duration and behavior of damped oscillations effectively.

Angular Frequency

Angular frequency, commonly denoted as \( \omega \), is essential in describing how fast the oscillations occur over time. It is connected with the rate at which the system oscillates through one complete cycle in terms of radians per unit time. Unlike standard frequency that tells us how many cycles occur per second, angular frequency deals with how much angular displacement occurs per unit time.

The relationship between angular frequency and frequency is given by \( \omega = 2\pi f \) or equivalently, when using the period \( p \), \( \omega = \frac{2\pi}{p} \). These relationships show that there's a direct conversion between linear and angular descriptions of frequency.

Understanding angular frequency helps in plotting the damped harmonic motion graph precisely and ensuring each oscillation represents the real-time behavior of the system. Consider a clock's pendulum, where knowing \( \omega \) helps predict how smoothly and regularly it swings back and forth despite energy loss.