Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5 feet from its high point to its low point (see figure), and it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy, where the high point corresponds to the time \(t=0.\)

Step-by-Step Solution

Answer

The equation that describes the motion of the buoy is \( y = 1.75 \cos\left(\frac{\pi}{5}t\right) \).

Step 1: Understand Basic Concepts

Simple harmonic motion describes the motion of an object where the restoring force is directly proportional to the displacement of the object. The equation for simple harmonic motion is of the form \( y = A \cos(B(t - C)) + D \) where, \n- \( y \) represents the height at time \( t \)\n- \( A \) represents the amplitude (the maximum distance from the equilibrium position)\n- \( B \) represents the frequency (the number of oscillations per unit time)\n- \( (t - C) \) represents the phase shift (how much the wave is shifted horizontally from the usual position)\n- \( D \) represents the vertical shift (how much the wave is shifted vertically from the usual position)

Step 1: Determine the Amplitude

The buoy's maximum distance from the central position or equilibrium (halfway between the high point and low point) is defined as the amplitude. The buoy moves a total of 3.5 feet from its high point to its low point, so the amplitude would be half of 3.5 feet which is 1.75 feet.

Step 2: Determine the Period

The period is the time taken for one complete cycle of the motion. It's given that the buoy returns to its high point every 10 seconds, so the period is 10 seconds. The frequency B is given by \( B = \frac{2\pi}{T} \), where \( T \) is the period. So, \( B = \frac{2\pi}{10} = \frac{\pi}{5} \).

Step 3: Write the Equation

Since the buoy starts at the high point, there is no phase shift or vertical shift. This indicates that \( C = D = 0 \). Substituting for \( A \), \( B \), \( C \), and \( D \) into the equation, the function describing the motion of the buoy is \( y = 1.75 \cos\left(\frac{\pi}{5}t\right) \).

Key Concepts

Amplitude in Simple Harmonic Motion

The concept of amplitude is crucial in understanding simple harmonic motion (SHM). In the context of SHM, amplitude, represented by the symbol 'A', refers to the maximum extent of displacement from the equilibrium position — the central or the mean position of motion. For an object in SHM, this displacement is the furthest it moves away from this midpoint on either side before changing direction.For instance, with the buoy oscillating in the water, the amplitude measures how high or low the buoy moves relative to its resting position. In the exercise provided, the buoy travels a total distance of 3.5 feet from its highest to its lowest point. Therefore, to calculate the amplitude, we take half of this distance, resulting in an amplitude of 1.75 feet. This value is significant as it quantifies the extent of the buoy's motion and is a central variable in the equation that models the SHM.

Period and Frequency in Simple Harmonic Motion

The period (T) and frequency (f) of simple harmonic motion are two interrelated concepts describing how motion repeats over time. The period is the time it takes to complete one full cycle of motion — from one extreme back to the same point. The exercise mentions that the buoy takes 10 seconds to return to its high point, which is one complete oscillation, hence its period is 10 seconds.Frequency, on the other hand, is the number of complete cycles per unit time and is often measured in hertz (Hz), equivalent to cycles per second. Frequency and period are inversely related, as shown by the formula \( f = \frac{1}{T} \). Consequently, if the period is known, the frequency can be easily found. With our buoy's period of 10 seconds, its frequency would be \( \frac{1}{10} \text{ Hz} \) or 0.1 Hz. Understanding the concepts of period and frequency is vital because they characterize the 'rhythm' of the motion in SHM.

Equation of Simple Harmonic Motion

The equation of simple harmonic motion (SHM) is a mathematical representation of the cyclical movement observed in SHM systems. This equation takes the form \( y = A \text{cos}(B(t - C)) + D \), where each variable has a distinct meaning:

• 'y' represents the displacement of the object at any given time 't'.
• 'A' is the amplitude, or maximum displacement.
• 'B' is related to the frequency, and thereby the period of the motion.
• 'C' accounts for any phase shift — if the motion does not start at t=0.
• 'D' stands for any vertical displacement from zero.

For the buoy's motion, since it starts at the high point at time \( t=0 \), there is no phase shift; thus \( C \) is zero. There's also no vertical displacement from zero, so \( D \) is also zero.Substituting the values from the problem into the SHM equation, we arrive at \( y = 1.75 \text{cos}\big(\frac{\text{π}}{5}t\big) \), which gives us the specific model for the buoy's motion. This equation not only provides us with a blueprint to understand the buoy's current position at any given time but also allows us to predict its future positions as it continues to oscillate with the waves.