Problem 132 - 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 \(Y_{1}=e^{-t} \sin t, Y_{2}=e^{-2 t} \sin t\) and \(Y_{3}=e^{-4 t} \sin t\) in the same viewing window. What happens to \(Y=e^{-k t} \sin t\) as \(k\) increases?

Step-by-Step Solution

Answer

As \(k\) increases, \(Y = e^{-kt} \sin t\) exhibits faster amplitude decrease.

Step 1: Understanding the Function

The function given is a product of an exponential decay function and a sine function: \[ f(t) = e^{-kt} \sin(t) \].This represents a damped oscillation, where the term \(e^{-kt}\) decreases with time, causing the amplitude of the sine function to reduce.

Step 2: Plotting the Graphs

We need to graph three functions in the same window: \( Y_1 = e^{-t} \sin(t), \ Y_2 = e^{-2t} \sin(t), \ Y_3 = e^{-4t} \sin(t) \). These graphs show how the sine wave is affected by different damping factors \(k\).

Step 3: Observation of Damping Effect

As \(k\) increases from 1 to 4, the rate at which the amplitude decreases is faster. This means that: \( Y_1 \) with \( k = 1 \) is damped less than \( Y_2 \) with \( k = 2 \), and \( Y_3 \) with \( k = 4 \) is damped the most quickly.

Step 4: Conclusion on Increasing k

From observing the plotted graphs, as the constant \( k \) increases, the graphical representation of \( Y = e^{-kt} \sin(t) \) shows a more rapid decrease in amplitude of the sine wave over time. This means a stronger damping effect.

Key Concepts

Exponential Decay

Exponential decay refers to the process by which a quantity decreases at a rate proportional to its current value. In the context of damped oscillations, the exponential decay is represented by the term \[ e^{-kt} \]in the equation \[ f(t) = e^{-kt} \sin(t) \]. This term causes the amplitude of the sine wave to decrease over time.

The key characteristics of exponential decay include:

• A rapid decrease at the beginning, which slows down over time.
• The decay rate is determined by the constant \( k \), where a larger \( k \) results in a faster decrease.
• Ultimately, the quantity approaches zero, but never truly reaches it.

In the exercise, the exponential decay essentially 'tames' the oscillations of the sine wave, causing them to diminish and eventually level out to a negligible point as time passes. This illustrates the effect of friction or resistance on oscillatory motion.

Sine Wave

A sine wave is a smooth periodic oscillation often used to describe regular wave-like motion. It is typically characterized by its amplitude, frequency, and phase. In the formula \[ f(t) = e^{-kt} \sin(t) \], the sine wave is represented by \( \sin(t) \).

A sine wave naturally oscillates between a maximum positive and minimum negative value, producing a recognizably wave-like shape. However, when combined with an exponential decay function, as seen in this damped oscillation, its amplitude decreases over time. This means the peaks and troughs become less pronounced as time progresses.

In the exercise example, the sine wave's natural oscillation is slowed and reduced in intensity by the accompanying exponential decay. This blend results in what is known as a damped oscillation, where the wave continues its regular pattern but with progressively smaller amplitudes due to damping. This reflects real-world phenomena such as a swinging pendulum that gradually comes to a stop due to air resistance.

Graphing Functions

Graphing functions like \( Y_1 = e^{-t} \sin(t), \, Y_2 = e^{-2t} \sin(t), \, Y_3 = e^{-4t} \sin(t) \) gives visual insight into how varying the decay factor \( k \) affects damped oscillations.

When graphing these functions, notice:

• Each \( Y \) function begins with similar initial oscillations due to the sine component.
• As \( k \) increases, the exponential decay intensifies, causing the amplitude to drop more sharply.
• \( Y_1 \) (smallest \( k \)) retains a recognizable wave form longer than \( Y_2 \) and \( Y_3 \), demonstrating softer damping.
• \( Y_3 \) (largest \( k \)) rapidly approaches zero, showcasing the aggressive damping effect.

Visual representation helps in comparing the effects each different damping constant has on the sine wave. With graphing, students observe firsthand the direct correlation between the size of \( k \) and the speed at which oscillations diminish, providing a clear picture of how real-world damping works.