Problem 24 - Algebra and Trigonometry

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 \(17-20,\) and of the form \(y=k e^{-c t}\) sin \(\omega t\) in Exercises \(21-24\). (b) Graph the function. $$k=12, \quad c=0.01, \quad f=8$$

Step-by-Step Solution

Answer

The function is \( y = 12 e^{-0.01t} \cos 16\pi t \), and it should be graphed with decaying oscillations.

Step 1: Determine Angular Frequency

Since the frequency \( f \) is given as 8, we can calculate the angular frequency \( \omega \) using the formula \( \omega = 2\pi f \). Thus, \( \omega = 2\pi \times 8 = 16\pi \).

Step 2: Construct the Damped Harmonic Motion Function

Given that we need to use the function form \( y = k e^{-ct} \cos \omega t \), we substitute the known values \( k = 12\), \( c = 0.01 \), and \( \omega = 16\pi \) into the equation. Thus, the function becomes \( y = 12 e^{-0.01t} \cos 16\pi t \).

Step 3: Graph the Function

To graph the function \( y = 12 e^{-0.01t} \cos 16\pi t \), note that the cosine component has a period of \( \frac{1}{8} \), due to \( f = 8 \). The exponential component causes the amplitude to decay over time. The graph will initially oscillate with a maximum amplitude of 12, decreasing over time as determined by the exponential decay factor \( e^{-0.01t} \).

Key Concepts

Angular Frequency

Angular frequency is a fundamental concept when dealing with periodic motion, including damped harmonic motion. It represents how quickly the system oscillates in terms of radians per unit time. When the frequency, denoted as \( f \), is known, the angular frequency \( \omega \) can be determined using the formula \( \omega = 2\pi f \).

For example, given a frequency of 8, the angular frequency \( \omega \) would be \( 2\pi \times 8 = 16\pi \). This simply means that the system completes 16\( \pi \) radians per unit of time, illustrating how fast the oscillation occurs. Understanding angular frequency helps us visualize the speed and dynamics of the motion over time.

Exponential Decay

Exponential decay is another crucial element in damped harmonic motion. It reflects the gradual decrease in amplitude as time progresses. This decay is characterized mathematically by the term \( e^{-ct} \), where \( c \) is the damping constant and \( t \) is time.

The damping constant \( c \) dictates how quickly the amplitude diminishes. In our original problem, with \( c = 0.01 \), this constant creates a slow decay, allowing the oscillation to decrease gently over time.

The inclusion of exponential decay in the function ensures that the oscillations do not continue indefinitely but instead fade at a rate controlled by the damping constant. This concept is vital to systems where energy is lost over time due to factors like friction or resistance.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are integral to modeling oscillatory behavior in physics, especially in the context of damped harmonic motion. These functions help represent the oscillating part of the motion.

In the given function, the cosine function \( \cos \omega t \) is used, where \( \omega \) represents the angular frequency. This function allows the system to oscillate back and forth in a predictable way.

Cosine and sine functions are periodic, meaning they repeat values in a cyclical manner, which is ideal for capturing the essence of waves and oscillations. When combined with exponential decay, the trigonometric component ensures that the system exhibits a realistic model of real-world phenomena, such as the swaying of a pendulum gradually coming to a stop.

Graphing Functions

Graphing functions is an essential tool for visualizing the behavior of damped harmonic motion over time. Creating a graph of the function \( y = 12 e^{-0.01t} \cos 16\pi t \) provides a clear depiction of how the oscillations and amplitude change.

Initially, the graph displays oscillations with a high amplitude of 12, corresponding to the initial energy in the system. As time progresses, the exponential decay factor \( e^{-0.01t} \) reduces the amplitude, tracing a decreasing envelope over the oscillation.

The graph also shows the influence of the angular frequency \( \omega = 16\pi \), dictating how rapidly the oscillations occur within each period. Seeing the function graphed helps students and practitioners observe the interaction between the damping and trigonometric elements more vividly. Utilizing a graph facilitates a complete understanding of the underlying dynamics involved in damped harmonic motion.