Problem 25 - 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. $$ k=0.3, \quad c=0.2, \quad f=20 $$

Step-by-Step Solution

Answer

Function: \( y = 0.3 e^{-0.2 t} \cos (40\pi t) \); Graph shows high-frequency damping.

Step 1: Determine Angular Frequency

The angular frequency \( \omega \) is determined by the frequency \( f \) using the formula \( \omega = 2\pi f \). Given \( f = 20 \), we find:\[\omega = 2\pi \times 20 = 40\pi.\]

Step 2: Choose Function Form

The problem specifies using the cosine form for exercises 19-22. Therefore, the function should be of the form:\[ y = k e^{-ct} \cos \omega t. \]

Step 3: Substitute Known Values

Substitute the given values into the function. We have \( k = 0.3 \), \( c = 0.2 \), and \( \omega = 40\pi \). Thus, the function becomes:\[ y = 0.3 e^{-0.2 t} \cos (40\pi t). \]

Step 4: Graph the Function

To graph the function \( y = 0.3 e^{-0.2 t} \cos (40\pi t) \), consider plotting in a graphing calculator or software. The graph will feature oscillations with decreasing amplitude over time due to the exponential decay \( e^{-0.2 t} \). The term \( \cos (40\pi t) \) will generate rapid oscillations due to the high angular frequency.

Key Concepts

Angular Frequency

Angular frequency, often denoted by \( \omega \), is a concept that links the frequency of an oscillation to its rotational characteristic. It tells us how fast something oscillates in a circular or repetitive manner. Angular frequency is measured in radians per second. This is different from regular frequency, which is usually measured in cycles per second, or Hertz (Hz).

• Frequency \( f \) is related to angular frequency by the formula \( \omega = 2\pi f \).
• In our example, the frequency \( f = 20 \), therefore the angular frequency is \( \omega = 40\pi \) radians per second.

This means that the oscillation cycle repeats every \( \frac{1}{20} \) seconds. Angular frequency is essential in damped harmonic motion to describe how quickly the system oscillates.

Exponential Decay

Exponential decay is a process where a quantity decreases at a rate proportional to its current value. In the context of damped harmonic motion, it describes how the amplitude of oscillations reduces over time.

• The damping factor here is represented by the term \( e^{-ct} \).
• The constant \( c \) dictates the rate at which this decay happens.

In our exercise, the damping constant \( c = 0.2 \), and it is used in the term \( e^{-0.2t} \). This makes the amplitude of the oscillation diminish as time goes on, embodying less of the initial energy of the system with each oscillation cycle. As a result, as \( t \) increases, the value of \( e^{-0.2t} \) gets smaller, showing the gradual energy loss and sound, for instance, fading into silence.

Oscillations

Oscillations refer to the repeated back-and-forth motion around an equilibrium position. In the context of damped harmonic motion, these oscillations slowly decrease in amplitude over time due to damping.

• The function \( \cos(\omega t) \) describes these oscillations in the given system.
• Here, \( \omega = 40\pi \) means there are very rapid oscillations.

Due to damping, even though the frequency remains the same initially, the oscillations gradually lose their intensity. The initial amplitude in our scenario is \( k = 0.3 \), and each oscillation loses height—eventually, the system returns to rest. It's this graceful decline of oscillations due to the exponential decay that demonstrates the energy being expened over time.

Graphing Functions

Graphing functions in damped harmonic motion provides a visual representation of the concepts discussed. The graph of our function \( y = 0.3 e^{-0.2t} \cos(40\pi t) \) shows how oscillations behave with respect to time.

• The \( e^{-0.2t} \) term causes the amplitude of oscillation to decrease exponentially, which can be seen as shrinking wave peaks over time.
• The \( \cos(40\pi t) \) term results in frequent oscillations, showing tight, rapidly-changing waves.

Using graphing tools or calculators, you can plot each part separately to see each effect: the rapid oscillation due to \( \cos(40\pi t) \), and how \( e^{-0.2t} \) diminishes the size of the peaks. Together these produce a curve where the oscillations maintain frequency but decrease in amplitude, showcasing the essence of damped harmonic motion.