Problem 27 - Precalculus Mathematics for Calculus

Question

A Bobbing Cork A cork floating in a lake is bobbing in simple harmonic motion. Its displacement above the bottom of the lake is modeled by $$ y=0.2 \cos 20 \pi t+8 $$ where \(y\) is measured in meters and \(t\) is measured in minutes. (a) Find the frequency of the motion of the cork. (b) Sketch a graph of \(y .\) (c) Find the maximum displacement of the cork above the lake bottom.

Step-by-Step Solution

Answer

(a) Frequency is 10 cycles per minute. (b) Sketch a cosine wave oscillating between 7.8 and 8.2 with a period of 0.1 minutes. (c) Maximum displacement is 8.2 meters.

Step 1: Identify Angular Frequency

The model of the cork's displacement is given by \( y = 0.2 \cos(20 \pi t) + 8 \). In this equation, the term \( 20\pi \) represents the angular frequency \( \omega \). Recall that the formula to find the frequency \( f \) from angular frequency is \( \omega = 2\pi f \).

Step 2: Solve for Frequency

From Step 1, we have \( \omega = 20\pi \). Substitute into the formula \( 2\pi f = 20\pi \). Divide both sides by \( 2\pi \) to find \( f = \frac{20\pi}{2\pi} = 10 \). So, the frequency of the cork's motion is 10 cycles per minute.

Step 3: Understand the Graph

The function \( y = 0.2 \cos(20\pi t) + 8 \) is a cosine wave with an amplitude of 0.2, a period of \( \frac{1}{10} \) minutes, and a vertical shift of 8 units. The graph will oscillate above and below the line \( y=8 \), with a maximum height of 8.2 meters and a minimum height of 7.8 meters. The period of the wave is the reciprocal of the frequency, which is \( \frac{1}{10} \approx 0.1 \) minute.

Step 4: Sketch the Graph

When sketching the graph of \( y \), plot the vertical axis for displacement and the horizontal axis for time. The graph starts at maximum displacement at \( t=0 \), which is 8.2 meters. It reaches the minimum of 7.8 meters at \( t = 0.05 \) minutes, back to 8.2 meters at \( t = 0.1 \) minutes. This cycle repeats every 0.1 minutes.

Step 5: Calculate Maximum Displacement

The maximum displacement occurs when the cosine function is at its maximum value, which is 1. At this point, \( y = 0.2 \cdot 1 + 8 = 8.2 \). Thus, the maximum displacement above the lake bottom is 8.2 meters.

Key Concepts

Frequency Calculation

In simple harmonic motion, understanding the frequency is key to describing how often the motion repeats. The frequency ( f ), measured in cycles per minute, tells us how many oscillations occur in one minute. To find the frequency from the given equation, we first look at the angular frequency ( \omega ). In our equation, y = 0.2 \cos(20 \pi t) + 8 , the term 20\pi is the angular frequency.The formula to convert angular frequency to frequency is:

• \( \omega = 2\pi f \)

To find f , we rearrange the formula to f = \frac{\omega}{2\pi} . By substituting 20\pi for \omega , we get:

• \( f = \frac{20\pi}{2\pi} = 10 \)

This means the cork completes 10 cycles in one minute, showing that it has a frequency of 10 cycles per minute. The higher the frequency, the faster the cork bobs up and down on the lake.

Graph of Trigonometric Function

Creating a graph of a trigonometric function helps visualize the behavior of the cork's motion. The equation y = 0.2 \cos(20\pi t) + 8 represents a cosine wave that adds a constant to model the cork's vertical movement in the lake.In this function:

• The amplitude is 0.2 , indicating the extent of the bobbing above and below a central value.
• The vertical shift is 8 , moving the entire wave up by 8 meters, indicating a baseline above the lake bed.
• The period is derived from the frequency, calculated as \frac{1}{10} , or 0.1 minute, since frequency ( f ) and period ( T ) are inverses: \( T = \frac{1}{f} \).

To sketch this graph, place time ( t ) on the horizontal axis and displacement ( y ) on the vertical axis. At t = 0 , the displacement starts at its maximum of 8.2 meters. It then descends to a minimum of 7.8 meters around t = 0.05 minutes and rises back to 8.2 meters by t = 0.1 minutes, repeating this cycle.Such visual graphs are vital as they allow one to immediately understand the operation and see the cork's motion over time. The alternating peaks and troughs symbolize the bobbing motion characteristic of simple harmonic motion.

Maximum Displacement

In simple harmonic motion, the maximum displacement occurs when the function value is at its peak. For the cork bobbing on the lake, its displacement is modeled by the equation y = 0.2 \cos(20 \pi t) + 8 .To find the maximum displacement:

• Identify when the cosine function is at a maximum, which is when \cos(\theta) = 1 . For a cosine function, this happens at the start of the cycle where \theta corresponds to t=0 and other points separated by full periods.
• Substitute this maximum into the displacement equation:\[ y = 0.2 \times 1 + 8 = 8.2 \]
• The result, 8.2 meters, is the maximum height above the lake bottom.

Understanding maximum displacement is important as it gives us the furthest point the cork reaches during its motion, providing insights into the energy and extent of the oscillation each time it bobs up.