Problem 18 - 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. $$k=15, \quad c=0.25, \quad f=0.6$$

Step-by-Step Solution

Answer

The function is \( y = 15 e^{-0.25t} \cos(3.77 t) \). Graph it to visualize its damped oscillation.

Step 1: Determine Angular Frequency

First, we need to determine the angular frequency \( \omega \). The frequency \( f \) is given as 0.6, and the relationship between frequency and angular frequency is \( \omega = 2 \pi f \). So, compute \( \omega = 2 \pi \times 0.6 = 1.2 \pi \approx 3.77 \).

Step 2: Choose the Right Function Form

Since this exercise does not specify if we should use the cosine or sine form directly, we can choose either. For this explanation, we'll select the cosine form, which means we'll use the function \( y = k e^{-ct} \cos(\omega t) \).

Step 3: Plug Values into the Function

Now substitute the values of \( k \), \( c \), and \( \omega \) into the function. Our expression will be:\[ y = 15 e^{-0.25t} \cos(3.77 t) \]

Step 4: Graph the Function

To graph the function, you plot \( y \) against \( t \). The graph will exhibit oscillatory behavior with an exponentially decreasing amplitude due to the term \( e^{-0.25t} \). The peaks of the waves will decrease as \( t \) increases, reflecting the damping effect. The period of oscillation can be calculated as \( \frac{2\pi}{3.77} \approx 1.67 \). Ensure that the amplitude starts at 15 and diminishes over time.

Key Concepts

Angular Frequency

Angular frequency, often denoted by \( \omega \), plays a central role in understanding harmonic motion, including the damped variety. It describes how quickly the system oscillates over time. In this context, when we refer to frequency \( f \), it tells us how many cycles occur per second, while \( \omega \) takes the form \( \omega = 2\pi f \). This relationship highlights that angular frequency is measured in radians per second.

For our exercise, with a frequency \( f = 0.6 \), we calculate the angular frequency as \( \omega = 2\pi \times 0.6 = 1.2\pi \). This numerical value reflects how the oscillation progresses over time, specifically in a time span marked in radians. Understanding this concept is crucial for diving deeper into oscillations and ensuring accurate modeling of the harmonic motion.

Damping Constant

The damping constant \( c \) is a key factor in determining how the amplitude of oscillation decreases over time. Unlike a pure harmonic oscillator that continues indefinitely with a constant amplitude, a damped harmonic motion gradually loses energy due to the damping effect, often caused by friction or resistance.

In our function, the damping constant is \( c = 0.25 \). This value is part of the exponential decay term \( e^{-ct} \), which modifies the behavior of the oscillations by reducing their amplitude as time \( t \) increases. The higher the damping constant, the quicker the amplitude shrinks, causing the oscillations to fade faster. The presence of this factor makes the motion more realistic and applicable to scenarios like automotive suspensions or pendulums in real-world contexts, where energy dissipation is a natural occurrence.

Exponential Decay

Exponential decay is a mathematical concept that describes how quantities reduce over time at a rate proportional to their current value. In damped harmonic motion, it's represented by the term \( e^{-ct} \), where \( c \) is the damping constant.

This term dictates that the amplitude of the oscillation decreases exponentially, showcasing a gradual reduction. Exponential decay is ubiquitous in nature and physics, from radioactive decay to cooling processes. Its role in harmonic motion indicates an energy dissipation mechanism that impacts the system's dynamics. As time progresses, this term ensures that the oscillations dwindle, graphing lower peaks and reflecting how a system transitions toward rest. Recognizing this effect is pivotal in mastering concepts related to energy loss and transient states in oscillatory systems.

Oscillatory Behavior

Oscillatory behavior refers to the motion that repeats itself in cycles over time, showing wave-like patterns. It is a quintessential aspect of systems under harmonic motion, where the object oscillates back and forth in a regular manner.

In the problem at hand, the oscillatory behavior is captured by the cosine component \( \cos(\omega t) \) in the function \( y = ke^{-ct}\cos(\omega t) \). This term signifies the repetitive up and down swings that characterize oscillations. As time advances, the damping component \( e^{-ct} \) modifies this repetitive motion by reducing its amplitude, illustrating how oscillations in real-world systems lose energy and intensity.

Understanding oscillatory behavior is crucial for analyzing systems from mechanical springs to electrical circuits, where such patterns dictate performance and design requirements.