Problem 21 - 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. $$k=100, \quad c=0.05, \quad p=4$$

Step-by-Step Solution

Answer

The function is \( y = 100 e^{-0.05 t} \cos\left(\frac{\pi}{2}t\right) \).

Step 1: Compute the Angular Frequency

Since the period \(p\) is given, we can calculate the angular frequency \(\omega\) using the relation \(f=1/p\). The angular frequency is given by \(\omega = 2\pi f\). Substitute \(f=1/4\) into the equation: \[ \omega = 2\pi \times \frac{1}{4} = \frac{\pi}{2} \]

Step 2: Identify the Function Form

The problem requires us to identify the appropriate form of the damped harmonic motion function. Since the problem asks for a cosine function in exercises 19-22, we need to use the form:\[ y = k e^{-c t} \cos(\omega t) \]

Step 3: Substitute the Values into the Function

Now, we substitute the given values \(k=100\), \(c=0.05\), and \(\omega = \frac{\pi}{2}\) into the function form identified in Step 2:\[ y = 100 e^{-0.05 t} \cos\left(\frac{\pi}{2}t\right) \]

Step 4: Graph the Function

Using graphing tools or software, graph the function \( y = 100 e^{-0.05 t} \cos\left(\frac{\pi}{2}t\right) \). The graph should show an exponentially damped cosine wave, starting with an amplitude of 100, gradually reducing its amplitude over time.You would notice the amplitude decreasing exponentially with respect to time, indicative of the damping effect.

Key Concepts

Amplitude

In damped harmonic motion, the amplitude refers to the maximum extent of a vibration or oscillation. Initially, the oscillation in our example has an amplitude of 100, represented by the constant \( k \) in the equation.
- Amplitude is crucial because it indicates the level of energy contained within a system. - In the provided formula, \( y = k e^{-ct} \cos(\omega t) \), the initial amplitude is \( k = 100 \).
As damping takes effect due to the term \( e^{-ct} \), this initial amplitude decreases over time, which reflects the gradual loss of energy in the system. In essence, each oscillation effectively 'shrinks' in size when considering the graph, where peaks progressively diminish. This decrease continues until the oscillating object comes practically to rest.
Understanding amplitude in damped motion helps us visualize how vigorous or pronounced the motion initially is and provides insight into how quickly it will diminish due to damping factors.

Angular Frequency

Angular frequency in oscillating systems like damped harmonic motion determines how rapidly an object oscillates. It's denoted by \( \omega \) and calculated based on the relation between frequency and period.
- The formula to find angular frequency is \( \omega = 2\pi f \), where \( f \) is the frequency. - For systems with a period \( p \), you determine frequency using \( f = \frac{1}{p} \).
In our problem, the period is given as 4 seconds, indicating that a full cycle takes this amount of time. Thus, frequency \( f \) becomes \( \frac{1}{4} \) and the angular frequency is calculated as \( \omega = 2\pi \times \frac{1}{4} = \frac{\pi}{2} \).
Angular frequency communicates how fast the motion oscillates per unit time, providing crucial information on how quickly the waveform progresses along the time axis of a graph. When shown on a graph, this affects how 'compressed' or 'spread out' the wave appears. In practical terms, if the angular frequency is high, cycles occur rapidly, leading to a tightly packed wave.

Exponential Decay

Exponential decay in the context of damped harmonic motion refers to the steady reduction in amplitude as time progresses. This decay is represented by the term \( e^{-ct} \) within the function.
- \( c \) is the damping constant, dictating the decay rate. - As \( t \) increases, \( e^{-ct} \) exponentially reduces, leading to a reduction in amplitude over time.
With \( c = 0.05 \) in this case, the damping effect is mild, causing a gradual decrease in amplitude. If \( c \) were larger, the decay would be more rapid, leading to a quicker diminishment of motion. The term \( e^{-ct} \) ensures that each oscillation's peak height decreases, which is visually represented on a graph as a shrinking envelope that surrounds the wave pattern.
The process of exponential decay ensures that in systems such as a pendulum, the energy dissipates over time, eventually causing the motion to cease. This concept is essential in explaining why mechanical and radiant systems rarely vibrate indefinitely or with constant amplitude.