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 \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=1, \quad c=1, \quad p=1 $$

Step-by-Step Solution

Answer

The function that models the motion is \( y = e^{-t} \cos(2\pi t) \).

Step 1: Determine the Angular Frequency

To find the angular frequency \( \omega \), we first need to find the frequency \( f \) using the given period \( p \). Since \( f = \frac{1}{p} \), substituting \( p = 1 \) gives \( f = 1 \). The relation between angular frequency \( \omega \) and frequency \( f \) is \( \omega = 2\pi f \). Therefore, \( \omega = 2\pi \times 1 = 2\pi \).

Step 2: Write the Damped Harmonic Motion Equation

Since the task refers to exercises 19-22, we use the cosine form of the damped harmonic motion function: \( y = k e^{-ct} \cos(\omega t) \). Substituting \( k = 1 \), \( c = 1 \), and \( \omega = 2\pi \), the function becomes: \[ y = e^{-t} \cos(2\pi t) \].

Step 3: Graph the Function

To graph \( y = e^{-t} \cos(2\pi t) \), recognize that it represents an exponentially decaying envelope \( e^{-t} \) modulating a cosine wave with frequency corresponding to \( 1 \) (as one period is over \( 0 \) to \( 1 \)). The graph should show oscillations that decay as \( t \) increases, with peaks reducing in height, moving closer to zero.

Key Concepts

Understanding Angular Frequency

Angular frequency is an important concept when studying oscillatory systems like damped harmonic motion. It describes the rate at which an object travels through its oscillatory path. You can think of it as how quickly something goes through its cycles. The relationship between the angular frequency, denoted as \( \omega \), and the regular frequency \( f \) is given by the formula \( \omega = 2\pi f \).

For example, if the period \( p \) of an oscillation is 1, then the frequency \( f \) is obtained by \( f = \frac{1}{p} = 1 \). Therefore, the angular frequency is \( \omega = 2\pi \times 1 = 2\pi \).

This angular measurement is essential for determining the speed of oscillations in the motion, given that \( \omega \) is in radians per second. Knowing this helps us build a function that captures the motion accurately, ensuring we account for both the time factor and the oscillation frequency.

Exploring Exponential Decay

Exponential decay plays a crucial role in damped harmonic motion. It describes how, over time, the amplitude of the oscillations decreases. In the equation \( y = k e^{-ct} \cos(\omega t) \), the term \( e^{-ct} \) represents this decay.

Here, \( c \) is a constant that affects how fast the decay happens. If \( c = 1 \), it means the amplitude decreases steadily with time due to multiplying by the exponential part.

This exponential factor causes the motion to "damp," which means the wave loses energy and the motion dies out faster as \( t \) increases. Over time, the oscillations' peaks will shrink, demonstrating how exponential decay controls the dissipation of energy from the system.

In practical terms, exponential decay can model anything from the fading of sound waves in the air to the dying away of a bouncy ball’s motion. The concept captures how real-world systems often exhibit decreasing oscillations due to friction or other resistive forces.

Understanding the Role of a Cosine Wave

The cosine wave is one of the essential parts of modeling oscillatory behavior. In the damped harmonic motion function \( y = e^{-t} \cos(2\pi t) \), the \( \cos(\omega t) \) component represents the oscillatory or wave-like nature of the function.

Cosine waves are periodic and continuous, which means they repeat their pattern at regular intervals—which in this case occurs once every full cycle, due to the frequency and angular frequency we calculated earlier.

Cosine waves start at their maximum value when the time \( t = 0 \). As time progresses, the wave fluctuates between maximum and minimum values in a smooth, cyclic manner. This oscillating behavior is crucial in describing phenomena like waves in sound, light, or even the alternating current in electrical circuits.

When combined with exponential decay, as in damped harmonic motion, the cosine wave dictates the direction of movement while the envelope set by the exponential part shrinks the extent of movement over time.

Graphing Functions in Damped Harmonic Motion

Graphing the function \( y = e^{-t} \cos(2\pi t) \) provides a visual representation of how damped harmonic motion manifests over time. This function integrates both the decaying and oscillatory components, offering a rich visualization of the phenomenon.

To construct the graph, start by acknowledging that the cosine wave oscillates up and down symmetrically about the horizontal axis. Meanwhile, the value of \( e^{-t} \) decays, acting as an envelope that both limits and shapes the height of the wave.

The peaks of the cosine wave will get lower and lower as time advances due to the \( e^{-t} \) factor. This decay effect is why, on a graph, you will see the amplitude of the wave gradually decrease. This results in a wave that appears constrained and pulled toward the axis as time goes on.

By graphing the function correctly with these considerations, we can effectively visualize the process of energy damping in mechanical systems or signal attenuation in communications, offering a vivid insight into the world of oscillatory motion.