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

Step-by-Step Solution

Answer

The function is \(y = 0.3 e^{-0.2t} \cos(40\pi t)\). Graph it to observe rapid oscillations with damping.

Step 1: Identify Given Values

We are given the initial amplitude \(k = 0.3\), the damping constant \(c = 0.2\), and the frequency \(f = 20\). We will use these values to find our damped harmonic motion function.

Step 2: Calculate Angular Frequency

The frequency is given by \(f = 20\), and the angular frequency \(\omega\) can be calculated using the formula \(\omega = 2\pi f\). Thus, \(\omega = 2\pi \times 20 = 40\pi\).

Step 3: Formulate the Damped Harmonic Motion Function

Using the given format for exercises 19-22, the function is \(y = k e^{-c t} \cos(\omega t)\). Substituting the values, we get \(y = 0.3 e^{-0.2t} \cos(40\pi t)\).

Step 4: Graph the Function

Plot the function \(y = 0.3 e^{-0.2t} \cos(40\pi t)\) using graphing software or graph paper. The function will exhibit a rapidly oscillating cosine curve with an exponential decay envelope due to the damping factor. The oscillations will have a high frequency, repeating 20 times per second.

Key Concepts

Amplitude

In the context of damped harmonic motion, amplitude refers to the initial size or "peak" of the oscillations in the system. Here, the initial amplitude, denoted as \(k\), is 0.3. This means that, prior to any damping influences, the maximum displacement from the equilibrium position is 0.3 units.

• The amplitude plays a vital role in determining how high or low the oscillations reach from the center point (equilibrium).
• As time progresses, the amplitude will decrease due to damping, which we'll explore in more detail in the next section.

It is important to remember that amplitude directly affects the initial "strength" of the wave, but it does not influence the frequency or the damping rate of the oscillation.
When graphing the function, the peak of the initial wave can be observed at this amplitude, thereafter gradually diminishing over time due to the damping effect.

Damping Constant

The damping constant, represented by \(c\), is a crucial factor in damped harmonic motion, as it quantifies the rate at which the oscillation amplitude decreases. In this particular exercise, the damping constant is 0.2.

• The damping constant causes an exponential decay in the system.
• It slows down the oscillatory motion over time, leading to progressively smaller oscillations.

The function incorporates the damping constant in the term \(e^{-ct}\), which dominates how quickly the oscillations decrease. A larger damping constant would mean a quicker reduction in amplitude, resulting in faster cessation of oscillations.
Understanding the damping constant is crucial for predicting how long the motion continues and how quickly it reaches rest. The nature of damping constantly affects the smoothness and speed at which the system returns to equilibrium.

Angular Frequency

Angular frequency, denoted by \(\omega\), is an essential concept that describes how fast the oscillation cycles through its motion. In the given problem, we calculated \(\omega\) using the relationship \(\omega = 2\pi f\), resulting in \(\omega = 40\pi\).

• Angular frequency relates to the rapidity of the oscillations within the system.
• It is measured in radians per second.

The angular frequency is directly proportional to the linear frequency \(f\), which is given as 20 Hz (meaning 20 cycles per second).
In the damped harmonic motion function, the angular frequency determines how tightly packed the peaks and troughs of the oscillating wave will be. A high angular frequency leads to rapid oscillations within a short time frame.

Cosine Function

The cosine function, \(\cos(\omega t)\), is used as the basis for this particular damped harmonic motion model. The role of the cosine function in the equation is to provide the periodic oscillations around the equilibrium point.

• The cosine function produces the wave pattern of the motion.
• It oscillates between -1 and 1, affecting the overall shape and symmetry of the wave.

In the equation, the term \(\cos(40\pi t)\) is responsible for the wave’s repeatable nature, oscillating at a constant frequency dictated by \(\omega\). This aspect of the model depicts the inherent cyclic nature of the harmonic motion, where the damping constant and amplitude adjust the size and sustainability of the oscillation.