Problem 17 - 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. $$k=2, \quad c=1.5, \quad f=3$$

Step-by-Step Solution

Answer

The function is \(y = 2 e^{-1.5t} \cos(6\pi t)\). Graph it as a decaying cosine wave.

Step 1: Calculate Angular Frequency

First, we need to find the angular frequency \(\omega\). The frequency is given as \(f = 3\), and we know that \(\omega = 2\pi f\). Thus, \(\omega = 2\pi \times 3 = 6\pi\).

Step 2: Write the Damped Harmonic Motion Function

Use the cosine form of the equation \(y = k e^{-ct} \cos(\omega t)\) because it belongs to Exercises 17-20 as stated in the problem. Substitute the given values: \(k = 2\), \(c = 1.5\), and \(\omega = 6\pi\). Thus, we have the function \(y = 2 e^{-1.5t} \cos(6\pi t)\).

Step 3: Graph the Function

To graph the function \(y = 2 e^{-1.5t} \cos(6\pi t)\), notice that it is a product of an exponentially decaying envelope and a cosine wave. The amplitude of the cosine wave decreases over time due to the factor \(e^{-1.5t}\). Start by plotting the exponential decay \(2 e^{-1.5t}\) as an envelope curve. Then, superimpose the oscillating function \(\cos(6\pi t)\). The result will show decreasing oscillations within the exponentially decaying envelope.

Key Concepts

Angular Frequency

When dealing with oscillations and waves, understanding angular frequency is essential. Angular frequency, denoted by \(\omega\), measures how fast an object oscillates in a sinusoidal wave. It’s typically measured in radians per second. The relationship between angular frequency and regular frequency \(f\) is given by the formula:

• \(\omega = 2\pi f\).

This means that the angular frequency is essentially the frequency of the wave multiplied by \(2\pi\), reflecting the mathematical nature of the circle in trigonometry, as \(\pi\) radians represent half a circle. In the context of the given exercise, where \(f = 3\), we calculate:

• \(\omega = 2\pi \times 3 = 6\pi\).

Understanding angular frequency helps identify the number of oscillations a system undergoes in a given period of time, providing critical insights into the system's temporal dynamics.

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In mathematics, it is often expressed with an equation of the form \(e^{-ct}\) where \(c\) is a positive constant and \(t\) represents time. It describes how quantities such as radioactive substances diminish over time.
For damped harmonic motion, exponential decay indicates how the amplitude of oscillations reduces as time progresses. In the given exercise, this decay is represented in the function \(y = 2 e^{-1.5t} \cos(6\pi t)\), where the term \(e^{-1.5t}\) governs the decay rate. Specifically, the coefficient \(1.5\) determines how quickly the motion's amplitude decreases over time:

• Lower values of \(c\) lead to slower decay.
• Higher values of \(c\) produce rapid diminishment of amplitude.

Thus, exponential decay plays a crucial role in defining the behavior of damped oscillations, ensuring the oscillations cease after a period dictated by the decay constant \(c\).

Cosine Function

The cosine function is a fundamental concept in trigonometry and plays a vital role in understanding oscillatory motions such as waves and vibrations. It is defined as \(\cos(\theta)\), where \(\theta\) is the angle in radians.
In the context of damped harmonic motion, the cosine function models the oscillatory part of motion. The function \(y = 2 e^{-1.5t} \cos(6\pi t)\) incorporates the cosine wave \(\cos(6\pi t)\), which describes the periodic oscillations with angular frequency \(6\pi\). This means the wave completes:\vspace{-\baselineskip}

• Every cycle in \(\frac{2\pi}{6\pi} = \frac{1}{3}\) seconds.

Importantly, the presence of the cosine function explains the wave-like behavior, where:

• \(\cos(\omega t)\) cycles between -1 and 1, affecting the amplitude of the wave.

In summary, the cosine function is indispensable for representing the periodic aspect of the damped harmonic motion, providing insight into the regularity and predictability of the oscillations.