Problem 131 - Precalculus

Question

For Exercises 131 and \(132,\) refer to the following: Damped oscillatory motion, or damped oscillation, occurs when things in oscillatory motion experience friction or resistance. The friction causes the amplitude to decrease as a function of time. Mathematically, we can use a negative exponential function to damp the oscillations in the form of $$f(t)=e^{-t} \sin t$$ Graph the functions \(Y_{1}=e^{-t}\) \(Y_{2}=\sin t,\) and \(Y_{3}=e^{-t} \sin t\) in the same viewing window (let \(t\) range from 0 to \(2 \pi\) ). What happens as \(t\) increases?

Step-by-Step Solution

Answer

As \(t\) increases, the amplitude of \(Y_3 = e^{-t} \sin t\) decreases due to the damping effect of \(e^{-t}\).

Step 1: Understanding the Functions

We have three functions to graph: \(Y_1 = e^{-t}, Y_2 = \sin t,\) and \(Y_3 = e^{-t} \sin t.\) The first function, \(e^{-t},\) is a negative exponential function that decreases over time. The second function, \(\sin(t),\) is a sine function that oscillates between -1 and 1. The third function, \(e^{-t} \sin t,\) is the product of the first two functions and represents a damped oscillatory motion where the amplitude of the oscillation is scaled by the exponential decay of \(e^{-t}.\)

Step 2: Graphing the Functions

Graph \(Y_1 = e^{-t}\) by plotting points for \(t\) between 0 and \(2\pi\). You should notice that \(e^{-t}\) starts at 1 when \(t = 0\) and approaches 0 as \(t\) increases. Next, graph \(Y_2 = \sin t\), which will show the typical sine wave oscillating between -1 and 1. Finally, graph \(Y_3 = e^{-t} \sin t\) by multiplying the corresponding \(\sin t\) values by \(e^{-t}\), resulting in an oscillating curve that diminishes in amplitude over the same interval.

Step 3: Analyzing the Graphs

As \(t\) increases from 0 to \(2\pi\), observe how \(Y_1 = e^{-t}\) exponentially decays towards zero, indicating a decrease in size. The \(Y_2 = \sin t\) wave continues to oscillate steadily. For \(Y_3 = e^{-t} \sin t\), the amplitude of the sine wave is progressively reduced due to the effect of \(e^{-t}\). The effect of the damping is more pronounced as \(t\) gets larger. This shows how resistance gradually reduces the oscillations to negligible amounts as time passes.

Key Concepts

Exponential Decay

In the context of damped oscillations, exponential decay plays a crucial role. Exponential decay describes how a quantity decreases over time at a rate proportional to its current value. In our exercise, the decay is represented by the function \( Y_1 = e^{-t} \). As time \( t \) increases, \( e^{-t} \) decreases swiftly from 1 towards zero. This reflects a rapid reduction in magnitude, which is a defining feature of exponential decay.

In practical terms, think about how this function affects oscillations. With every increase in time, the effect of \( e^{-t} \) causes the oscillation to lose energy. The underlying mathematics show that the oscillations' amplitude diminishes continually as time continues.

Key characteristics of exponential decay in this scenario include:

• The starting value is 1 when \( t = 0 \).
• It continuously decreases as \( t \) grows.
• Approaches zero but never truly becomes zero.

Through this exponential behavior, we see how friction or resistance progressively dampens oscillations.

Sine Wave

The sine wave is a familiar concept in oscillatory motions and is represented by the function \( Y_2 = \sin t \). Sine waves are periodic, meaning they repeat at regular intervals, and they oscillate smoothly between -1 and 1. In the scope of our problem, the sine function provides the oscillatory motion that we see being damped.

Understanding the sine wave is essential:

• The wave starts at 0 when \( t = 0 \).
• It reaches its maximum at 1 when \( t = \frac{\pi}{2} \).
• It comes back to 0, then goes negative, reaching -1 at \( t = \frac{3\pi}{2} \).
• Finally, it returns to 0 to complete one full cycle by \( t = 2\pi \).

The steady, repetitive nature of the sine wave illustrates how energy oscillates back and forth. In a system without damping, this motion would continue indefinitely. However, when combined with an exponential decay factor, the sine wave's peaks decrease over time, making it a perfect example of a natural, repeating pattern affected by external forces.

Amplitude

Amplitude is a critical feature in discussing any oscillation. It refers to the maximum extent of a wave measured from its equilibrium position. In our exercise, the amplitude of the damped oscillation is affected by the function \( Y_3 = e^{-t} \sin t \).

Initially, the amplitude of the wave is determined solely by the sine function. However, due to the multiplication by \( e^{-t} \), this amplitude decreases over time.

Important points about amplitude in this damped oscillation scenario include:

• The initial amplitude is similar to undamped oscillations, with the maximum reaching 1.
• As time progresses, the amplitude decreases because the exponential decay factor \( e^{-t} \) diminishes in value.
• The wave's peaks get smaller as resistance or friction affects the motion, leading to a gradual reduction in amplitude.

This diminishing amplitude illustrates how damping impacts oscillation, showing a real-world application of how systems lose energy and settle over time.