Problem 26 - 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. $$ k=12, \quad c=0.01, \quad f=8 $$

Step-by-Step Solution

Answer

The function is \( y = 12 e^{-0.01 t} \cos(16\pi t) \). It describes damped oscillations.

Step 1: Determine Angular Frequency

To find the angular frequency \( \omega \), we use the given frequency \( f = 8 \) Hz. The relationship between frequency and angular frequency is given by \( \omega = 2\pi f \). Plug in the value of \( f \):\[ \omega = 2\pi \times 8 = 16\pi. \]

Step 2: Choose the Function Model

Since we are working within exercises 19-22, we use the cosine-based model for damped harmonic motion:\[ y = k e^{-c t} \cos(\omega t). \]

Step 3: Substitute Known Values

Substitute the known parameters into the equation. We are given \( k = 12 \), \( c = 0.01 \), and \( \omega = 16\pi \). Thus, the function becomes:\[ y = 12 e^{-0.01 t} \cos(16\pi t). \]

Step 4: Graph the Function

To graph the function \( y = 12 e^{-0.01 t} \cos(16\pi t) \), note:1. The amplitude decreases over time due to the damping factor \( e^{-0.01t} \).2. The oscillatory behavior is governed by the \( \cos(16\pi t) \) part, with a frequency of 8 oscillations per time unit.3. Use plotting software or graphing calculator to visualize these elements concurrently, indicating a wave that loses amplitude as \( t \) increases.

Key Concepts

Angular Frequency

In damped harmonic motion, angular frequency is an essential concept. Angular frequency, denoted by \( \omega \), describes how quickly the system oscillates in radians per unit of time. It is connected to the frequency \( f \), which is measured in cycles per second or Hertz. For calculation, the relationship is expressed as \( \omega = 2\pi f \).
Let's consider our specific example where the given frequency \( f \) is 8 Hz. By substituting into the equation, we find \( \omega = 2\pi \times 8 = 16\pi \) radians per unit time. This result shows how fast the system completes its oscillatory cycle in terms of radians, which is essential for constructing the function that models the motion.
Understanding angular frequency helps clarify how often the system returns to the same point in its cycle, making it fundamental in understanding oscillatory systems.

Damping Factor

The damping factor plays a crucial role in damped harmonic motion. It represents the gradual reduction in amplitude over time, mitigating the system's oscillations. Mathematically, it's expressed as \( e^{-ct} \), where \( c \) is the damping constant, and \( t \) is time.
In our exercise, the damping constant \( c \) is given as 0.01. This indicates that, over time, the oscillations will become progressively smaller, as the factor \( e^{-0.01t} \) effectively decreases the amplitude. This damping leads to the system eventually coming to a rest.
Understanding how the damping factor influences the motion is key to analyzing how the energy in a system is lost over time. It explains why real-world systems do not oscillate indefinitely, as energies like friction or resistance are at play.

Cosine Function

The cosine function is often used to model oscillations due to its wave-like oscillatory properties. In the context of damped harmonic motion, it represents the periodic aspect of the motion.
Our function in this exercise is modeled as \( y = k e^{-ct} \cos(\omega t) \). Here, the cosine component \( \cos(\omega t) \) dictates the wave's shape, determining the pacing of the oscillations. The cosine function’s important properties include its periodic nature and its ability to start at a maximum when \( t = 0 \), aligning well with real-world wave scenarios where a system begins in a state of maximum energy or displacement.
Simply put, the cosine function is vital for illustrating the time-based variation of the position within harmonic motion.

Frequency-Period Relationship

The frequency-period relationship is a fundamental principle in harmonic motion, simplifying the analysis of oscillations. Frequency \( f \) refers to how many cycles occur in one second, measured in Hertz. Conversely, the period \( p \) is the time required to complete one full cycle of the motion, measured in seconds.
This relationship is expressed by \( f = \frac{1}{p} \). In our exercise, the frequency is given as 8 Hz, allowing for immediate calculation of the period, \( p = \frac{1}{8} \) s. This succinct relation provides insights into timing and motion characteristics, essential for both theoretical and practical applications.
Understanding this relationship helps decode any harmonic system's temporal behavior, illustrating that period and frequency are inversely related, thus allowing adjustments based on requirements of time or cycles needed.