Problem 90 - Precalculus with Limits

Question

The displacement from equilibrium of a weight oscillating on the end of a spring is given by \( y = 1.56e^{-0.22t} \cos 4.9t \), where \( y \) is the displacement (in feet) and \( t \) is the time (in seconds).Use a graphing utility to graph the displacement function for \( 0 \le t \le 10 \). Find the time beyond which the displacement does not exceed \( 1 \) foot from equilibrium.

Step-by-Step Solution

Answer

To find the exact time, use the graphing utility accurately. As the question requires approximate time, it might not be the same for different graphing utilities due to rounding off errors and resolution. However, the approximate time could be around 8 seconds.

Step 1: Graphing the function

Begin by inputting the function \( y = 1.56e^{-0.22t} \cos 4.9t \) into the graphing utility. Set the domain for the time \( t \) as 0 to 10 seconds. The graph will represent the oscillating motion of the weight on the spring over time.

Step 2: Interpret the graph

Observe the graph. It should show an oscillating function that gradually decreases in amplitude due to the damping term \( e^{-0.22t} \). The displacement oscillates between positive and negative values, indicating the back-and-forth motion of the weight.

Step 3: Determine the time

Find the point on the graph where the amplitude first becomes less than or equal to 1 foot, and does not exceed this value again. The correspond time \( t \) is when the displacement does not exceed 1 foot from equilibrium.

Key Concepts

Damping

Damping is a crucial concept when studying oscillations. It refers to the process by which the amplitude of an oscillating system, like a spring, gradually decreases over time. Imagine you have a weight bouncing on a spring. Initially, the bounces are quite high, but as time progresses, they get smaller. This decrease in bounce height is due to damping.

• Energy loss: Damping happens because energy is gradually lost from the system. This energy could be lost through friction, air resistance, or internal material resistance.
• Amplitude reduction: In our equation, the term \(e^{-0.22t}\) represents damping. It ensures the oscillations become smaller as time goes on.
  
• Practical significance: Damping is beneficial in real-world applications for reducing unwanted vibrations in structures like bridges and buildings.

Understanding damping helps in predicting how quickly an oscillation will die down. In the graph you've plotted, the damping effect is visible as the decreasing height of the oscillations over time.

Exponential Decay

Exponential decay is a mathematical concept that describes processes where quantities decrease at a rate proportional to their current value.
In the context of our oscillating spring, it describes how the amplitude of oscillation gets smaller with time.

• The Exponential factor: In the displacement equation \(1.56e^{-0.22t}\), \(e^{-0.22t}\) is the exponential decay factor. It dictates the rate at which the amplitude decreases.
• Understanding \(e\): The base of the natural logarithm, \(e\), is approximately equal to 2.718. It is used extensively in growth and decay processes.
• Decay rate: The constant \(-0.22\) in the exponent indicates the rate of decay. A larger magnitude would mean a faster decrease in amplitude.

By recognizing exponential decay, you can determine how long it will take for oscillations to reduce significantly, which is crucial for applications needing specific damping levels.

Trigonometric Functions

Trigonometric functions, like cosine and sine, are fundamental in describing oscillations. For the problem at hand, the motion of the spring uses the cosine function. Here's why it's used and its implications:

• Periodic nature: The cosine function is periodic, meaning it repeats its pattern over definite intervals. This makes it perfect to model oscillations which repeat after a regular time period.
• Cosine in the equation: In the displacement function \(1.56e^{-0.22t} \cos 4.9t\), \(\cos 4.9t\) describes the oscillatory behavior. The term \(4.9t\) inside the cosine function controls the frequency of the oscillation.
• Oscillation frequency: The frequency is related to how fast the oscillations occur. A higher frequency means more oscillations in a given time.

Trigonometric functions like cosine make it possible to accurately express periodic phenomena through mathematics. They capture the essence of the back-and-forth motion typical of springs or pendulums.