Problem 19 - 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. $$k=100, \quad c=0.05, \quad p=4$$

Step-by-Step Solution

Answer

The function is \( y = 100 e^{-0.05 t} \cos \left(\frac{\pi}{2} t\right) \).

Step 1: Calculate Angular Frequency

First, we need to determine the angular frequency \( \omega \). We know that the frequency \( f \) is related to the period \( p \) by the equation \( f = \frac{1}{p} \). For period \( p = 4 \), the frequency \( f \) is \( \frac{1}{4} = 0.25 \). The angular frequency \( \omega \) is given by \( \omega = 2\pi f \). Therefore, \( \omega = 2\pi \times 0.25 = \frac{\pi}{2} \).

Step 2: Choose the Correct Function Form

Since this problem involves using the form \( y = k e^{-c t} \cos \omega t \) for exercises 17-20, we will utilize this cosine function form to represent the damped harmonic motion.

Step 3: Substitute Given Values

Substitute the given values into the chosen function form. The values are \( k = 100 \), \( c = 0.05 \), and \( \omega = \frac{\pi}{2} \). Thus, the function becomes \( y = 100 e^{-0.05 t} \cos \left(\frac{\pi}{2} t\right) \).

Step 4: Graph the Function

To graph the function \( y = 100 e^{-0.05 t} \cos \left(\frac{\pi}{2} t\right) \), start by plotting the exponential decay \( 100 e^{-0.05 t} \) which indicates how the amplitude decreases over time. Overlay the cosine wave \( \cos \left(\frac{\pi}{2} t\right) \) which oscillates between -1 and 1. The complete graph will show an oscillating cosine wave with amplitude decreasing exponentially.

Key Concepts

Angular Frequency in Damped Harmonic Motion

Angular frequency, often represented as \( \omega \), plays a crucial role in damped harmonic motion. It helps determine how rapidly the system oscillates. This can be pivotal in understanding the behavior of systems like vibrating springs or even electrical circuits.

To find angular frequency, we need the frequency \( f \), which we derive from the period \( p \). The relationship is straightforward: \( f = \frac{1}{p} \). Once we've calculated \( f \), we employ the equation \( \omega = 2\pi f \) to find \( \omega \).

• For a given period \( p \), like \( p = 4 \), the frequency is \( f = \frac{1}{4} = 0.25 \).
• Multiplying by \( 2\pi \), we find \( \omega = \frac{\pi}{2} \).

With \( \omega \) in hand, the formula of motion can be precisely described, showcasing how it affects the oscillation speed of harmonic motion.

Understanding Exponential Decay

Exponential decay is an essential feature in the context of damped harmonic motion, indicating how the system loses energy over time. In the equation \( y = k e^{-c t} \cos \omega t \), the term \( e^{-c t} \) represents exponential decay.

Let's break it down:

• The constant \( c \) is the damping constant, which measures how quickly the system's amplitude decreases.
• \( e^{-c t} \) implies that as time \( t \) increases, the amplitude diminishes, approaching zero.

This is akin to watching a swinging pendulum slowly come to a stop. The larger the damping constant, the quicker the system loses its motion. Understanding exponential decay showcases not only the transient nature of the oscillation but also the stability of systems over prolonged periods.

Role of the Cosine Function in Oscillations

In damped harmonic motion, the cosine function reflects the oscillatory movement of the system. The equation \( y = k e^{-c t} \cos \omega t \) uses \( \cos \omega t \) to define the periodic oscillation.

The cosine function is a fundamental function in trigonometry signifying the wave-like pattern of oscillations. Consider:

• The cosine component moves between -1 and 1, ensuring the motion oscillates smoothly.
• It contributes to the "+" and "-" swings of the harmonic behavior without affecting the exponential decay.

Overlaying exponential decay and the cosine function provides a comprehensive picture of the oscillation: damped peaks and troughs as the system slows down. The cosine function, characterized by its wave shape, captures the essence of the natural rhythmic motion found in many physical systems.