Problem 25 - Algebra and Trigonometry

Question

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: 10 cycles/min, (b) Graph: cosine wave, (c) Max displacement: 8.2 m.

Step 1: Identify the formula structure

The equation for simple harmonic motion is generally given by \(y = A \cos(Bt + C) + D\). Here, \(A\) is the amplitude, \(B\) is related to the frequency, \(t\) is time, and \(D\) is the vertical shift. In the given equation \(y = 0.2 \cos(20\pi t) + 8\), we identify \(A = 0.2\), \(B = 20\pi\), and \(D = 8\). The term \(C\) is missing, which indicates there is no horizontal phase shift.

Step 2: Determine the frequency of the motion

The angular frequency \(\omega\) is given by the coefficient of \(t\) in the cosine function, which is \(20\pi\). The relationship between angular frequency and frequency \(f\) is \(\omega = 2\pi f\). Therefore, \(20\pi = 2\pi f\). Solving for \(f\), we get \(f = \frac{20\pi}{2\pi} = 10\). So, the frequency of the motion is 10 cycles per minute.

Step 3: Sketching the graph of the motion

The function \(y = 0.2 \cos(20\pi t) + 8\) represents a cosine wave oscillating between 7.8 and 8.2 with its midline at 8. The amplitude of 0.2 means that the cork moves 0.2 meters above and below the midline (vertically shifted by 8 meters). Completing one full cycle in 0.1 minutes (due to a frequency of 10 cycles/minute), the graph will be compact, showing repetitive peaks spaced narrowly within each minute.

Step 4: Find the maximum displacement

In a cosine function \(y = A \cos(Bt) + D\), the maximum value of \(y\) happens when the cosine term is +1, giving \(y = A + D\). For \(y = 0.2 \cos(20\pi t) + 8\), the maximum displacement is \(0.2 + 8 = 8.2\). Thus, the maximum displacement of the cork above the lake bottom is 8.2 meters.

Key Concepts

Frequency Calculation

In the realm of simple harmonic motion, frequency is a key concept that refers to how often an event occurs per unit time. For the bobbing cork in our exercise, the formula for its motion is given as \(y = 0.2 \cos(20\pi t) + 8\). The term \(20\pi\) attached to \(t\) is essential for frequency calculations. This part of the formula represents the angular frequency \(\omega\), which is linked to the actual frequency \(f\) through the equation \(\omega = 2\pi f\).

To find the frequency \(f\), we equate and solve the angular frequency for \(f\):

• Start with \(20\pi = 2\pi f\)
• Divide both sides by \(2\pi\) to isolate \(f\)
• Which yields \(f = \frac{20\pi}{2\pi} = 10\)

This calculation tells us that the cork completes 10 cycles every minute. Frequency is measured in cycles per minute, enlightening us about the tempo at which simple harmonic motion recurs.

Graph Sketching

Graphing the simple harmonic motion of the cork helps visualize how it moves over time. The function \(y = 0.2 \cos(20\pi t) + 8\) represents the height of the cork above the lake bottom as a cosine wave. The key elements of this graph include the amplitude, midline, and wave frequency.

The amplitude of the wave is 0.2, which tells us how far the cork deviates above and below a central position, known as the midline. In this case, the midline is at \(y = 8\), setting a consistent height baseline.

With a calculated frequency of 10 cycles per minute, the graph will include multiple tightly packed oscillations over each minute, indicating the cork completes a full up-and-down movement 10 times in one minute.

• The crests (highest points) of the wave will reach \(8.2\)
• The troughs (lowest points) will descend to \(7.8\)

Drawing the graph, one would see a recurring pattern of peaks and valleys centered around 8, offering a clear representation of how the cork's displacement changes with time.

Maximum Displacement

The maximum displacement of an object in simple harmonic motion is the greatest distance it travels from its central position. For the cork bobbing in the lake, this maximum is determined by the cosine function: \(y = A \cos(Bt) + D\). In this context, \(A\) represents amplitude and \(D\) the midline shift, both influencing displacement.

The formula implies that the maximum height occurs when the cosine component equals 1,

• This results in the expression \(y = A + D \)

Plugging in the values from our given function \(y = 0.2 \cos(20\pi t) + 8\), the calculation is straightforward:

• Maximum displacement \(= 0.2 + 8 = 8.2\) meters

Thus, the furthest point from the lake's bottom the cork can reach is 8.2 meters. This measure of displacement is crucial, as it defines the extent of the motion produced by forces acting on the cork.