Problem 21 - 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. $$k=7, \quad c=10, \quad p=\pi / 6$$

Step-by-Step Solution

Answer

The damped motion model is \( y = 7 e^{-10t} \cos 12t \). It shows a decaying amplitude cosine wave.

Step 1: Calculate Frequency

We start by calculating the frequency \( f \) using the period \( p \). The relationship is given by \( f = \frac{1}{p} \). Substituting in the given period: \[ p = \frac{\pi}{6} \] Thus, \[ f = \frac{1}{\pi / 6} = \frac{6}{\pi} \]

Step 2: Determine Angular Frequency

The angular frequency \( \omega \) is related to the frequency \( f \) by the equation \( \omega = 2\pi f \). Using the calculated frequency: \[ \omega = 2\pi \times \frac{6}{\pi} = 12 \]

Step 3: Formulate the Damped Harmonic Motion Function

Based on the problem, we use the form \( y = k e^{-c t} \cos \omega t \) for exercises 17-20. Substituting the values \( k = 7 \), \( c = 10 \), and \( \omega = 12 \), the function becomes: \[ y = 7 e^{-10t} \cos 12t \]

Step 4: Graph the Function

The function \( y = 7 e^{-10t} \cos 12t \) should be graphed against time \( t \). The graph will show a cosine wave with amplitude diminishing exponentially over time due to the damping factor \( e^{-10t} \). The period of the cosine function on this graph aligns with \( \omega = 12 \), which means the graph oscillates rapidly with each cycle having a duration \( \frac{2\pi}{12} = \frac{\pi}{6}\).

Key Concepts

Understanding Angular Frequency

When dealing with oscillatory systems, angular frequency is a crucial concept. It captures how quickly something is oscillating or rotating. The angular frequency \( \omega \) is closely linked to the regular frequency \( f \). Recall that \( f \) is the number of oscillations per unit time. The relationship between angular frequency and frequency is given by the formula:

• \( \omega = 2\pi f \)

Angular frequency is measured in radians per second, a standard unit in physics that aligns with the nature of oscillations involving trigonometric functions. In our problem, we calculated the angular frequency from the given period \( p = \frac{\pi}{6} \). If \( p \) is the time for one complete cycle, then \( f = \frac{1}{p} = \frac{6}{\pi} \), and thus, \( \omega = 2\pi \times \frac{6}{\pi} = 12 \). This means that in our damped harmonic motion, the system completes 12 radians of oscillation per second, dictating how fast the cosine wave cycles through its peaks and troughs.

Exploring the Cosine Function in Damped Harmonic Motion

The cosine function is a pivotal trigonometric function used to describe periodic motion. In damped harmonic motion, \( \cos(\omega t) \) represents the periodic oscillations of the system, where \( \omega \) is the angular frequency. Cosine functions are known for their smooth wave shape that oscillates between -1 and 1.
For our particular damped motion, the function is defined as:

• \( y = k e^{-ct} \cos \omega t \)

Here, the amplitude of the cosine wave is modulated by the exponential decay \( e^{-ct} \). This modification ensures the oscillations decrease over time, reflecting the real-world effects of energy loss in systems like swinging pendulums or vibrating strings. The cosine wave in the function \( y = 7 e^{-10t} \cos 12t \) contributes to a pattern of repeating peaks and valleys, with each cycle being \( \frac{2\pi}{12} = \frac{\pi}{6} \) seconds long. This quantitative measure confirms the rapid rate of oscillation we expect from the calculated angular frequency of 12 radians per second.

The Role of Exponential Decay

In the expression for damped harmonic motion, the exponential decay term \( e^{-ct} \) plays an essential role. Exponential decay describes how the amplitude of the oscillations diminishes over time. In our given function, the value \( c = 10 \) indicates how quickly this decay occurs. The function

• \( y = 7 e^{-10t} \cos 12t \)

shows that the initial amplitude of 7 decreases exponentially as time \( t \) increases. Each increase in \( t \) causes a multiplicative decrease by a factor of \( e^{-10} \).This means that the oscillations will quickly become smaller and smaller. This decay simulates energy loss due to factors like friction or air resistance in real systems, making it vital for modeling realistic physical phenomena. Over time, the predominant effect is this exponential decay, which tames the vigorous oscillations of the cosine function, leading to a smooth fading of the waveform.