Problem 20 - Precalculus Mathematics for Calculus

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

Step-by-Step Solution

Answer

The function is \( y(t) = 15 e^{-0.25 t} \cos(1.2 \pi t) \).

Step 1: Identify the Formula

Since the exercise falls within the range of Exercises 19-22, we will use the function form \( y = k e^{-c t} \cos(\omega t) \) to model the damped harmonic motion.

Step 2: Calculate Angular Frequency \( \omega \)

The frequency \( f \) is given as 0.6. The angular frequency \( \omega \) is related to the frequency by the formula \( \omega = 2 \pi f \). Therefore, we substitute \( f = 0.6 \) into the equation to get \[ \omega = 2 \pi \times 0.6 = 1.2 \pi. \]

Step 3: Substitute Values into the Function

Now we substitute the given values of \( k = 15 \), \( c = 0.25 \), and \( \omega = 1.2 \pi \) into the function formula: \[ y(t) = 15 e^{-0.25 t} \cos(1.2 \pi t). \] This is the function that models the damped harmonic motion.

Step 4: Graph the Function

To graph \( y(t) = 15 e^{-0.25 t} \cos(1.2 \pi t) \), plot a curve using a graphing tool with the x-axis representing time \( t \) and the y-axis representing the amplitude \( y(t) \). The graph will show oscillations that decay exponentially over time due to the damping factor \( e^{-0.25 t} \).

Key Concepts

Angular Frequency

In damped harmonic motion, angular frequency, denoted by \( \omega \), is an essential factor. It helps quantify how often oscillations occur in a certain period. The relationship between angular frequency and regular frequency \( f \) is defined by the formula \( \omega = 2 \pi f \). This means that the angular frequency is the regular frequency multiplied by \( 2\pi \), translating frequency into radians per unit time.
For our specific exercise, we are given a frequency \( f = 0.6 \). Using the formula, we find \( \omega = 2\pi \times 0.6 = 1.2\pi \). This calculation gives us the angular frequency, which indicates that the system completes 0.6 cycles per unit time, or about 1.884 radians per unit time.
Understanding this relationship helps us analyze motion affected by angular elements such as sine and cosine functions in our model.

Exponential Decay

Exponential decay plays a crucial role in modeling damped harmonic motion. The concept revolves around how an oscillating system loses energy over time, resulting in a decreasing amplitude.
In our function \( y = k e^{-ct} \cos(\omega t) \), the term \( e^{-ct} \) represents exponential decay. Here, \( c \) is the damping constant, determining how rapidly the decay happens. A larger \( c \) value means faster decay, causing the oscillations to die out more quickly. For example, in our specific exercise, the damping constant is \( c = 0.25 \).
The exponential function \( e^{-ct} \) decreases as time \( t \) advances, progressively reducing the amplitude of the oscillations. This decay reflects physical processes like friction or air resistance that hamper movement in mechanical systems.

Trigonometric Functions

Trigonometric functions, like cosine and sine, are vital when expressing periodic motion in mathematics. In damped harmonic motion, they describe the oscillations’ shape and nature over time.
In the given equation \( y = 15 e^{-0.25 t} \cos(1.2 \pi t) \), the cosine function \( \cos(1.2 \pi t) \) models the system’s periodic component. Cosine oscillates between -1 and 1, capturing the repetitive movement that characterizes oscillations.
This trigonometric function's argument, \( 1.2 \pi t \), integrates angular frequency, determining the speed and frequency of the oscillations. As time progresses and if there were no damping (i.e., \( e^{-0.25 t} \) was constant), the amplitude variations would be purely cosine in nature.
Trigonometric functions are essential for depicting the wavelike features of oscillation, such as crests and troughs, which are vital for understanding the behavior of vibrations.

Graphing Techniques

Graphing damped harmonic motion functions require an understanding of combining exponential decay and trigonometric functions to visualize the oscillations effectively.
When plotting the function \( y(t) = 15 e^{-0.25 t} \cos(1.2 \pi t) \), we take into account both components: the rapidly decaying exponential \( e^{-0.25 t} \) and the oscillatory cosine \( \cos(1.2 \pi t) \).

• The exponential decay reduces the amplitude over time, resulting in smaller oscillations as you move along the x-axis.
• The cosine function contributes to the periodic dips and peaks on the graph, suitable for representing the repeating nature of the system.

The x-axis represents time \( t \), while the y-axis represents the system's amplitude. A suitable graphing tool will help in pattern recognition and analysis. You will notice that with each subsequent period, the peaks become less pronounced, illustrating the effect of damping on the system.