Problem 16 - Algebra and Trigonometry

Question

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude 6.25 in., frequency 60 Hz

Step-by-Step Solution

Answer

The function is \( x(t) = 6.25 \cos(120\pi t) \).

Step 1: Understanding Simple Harmonic Motion

Simple harmonic motion can be described using a cosine or sine function. The general formula for the displacement of an object in simple harmonic motion is \( x(t) = A \cos(2\pi f t + \phi) \) or \( x(t) = A \sin(2\pi f t + \phi) \), where \( A \) is the amplitude, \( f \) is the frequency, and \( \phi \) is the phase shift.

Step 2: Identify Given Values

From the problem, we know the amplitude \( A = 6.25 \) inches, the frequency \( f = 60 \) Hz, and that the displacement is at its maximum at time \( t = 0 \), indicating a cosine function with no phase shift \( \phi = 0 \).

Step 3: Substitute Values into the Formula

Based on the information and the choice of a cosine function (since maximum displacement occurs at \( t = 0 \)), we substitute \( A = 6.25 \), \( f = 60 \), and \( \phi = 0 \) into the formula: \[ x(t) = 6.25 \cos(2\pi \times 60 \times t) \]

Step 4: Finalize the Harmonic Motion Function

The function describing the simple harmonic motion is therefore \[ x(t) = 6.25 \cos(120\pi t) \]

Key Concepts

Amplitude

Amplitude is one of the key components of simple harmonic motion and represents the maximum displacement of the object from its equilibrium position. In simple terms, it is how far the object moves from its starting point.
The amplitude's value is always a positive number and is noted as "A" in the mathematical formulas for harmonic motion. This value determines how "big" the motion is, meaning the larger the amplitude, the farther the object swings or oscillates.
For example, in the given exercise, the amplitude is specified as 6.25 inches. This means that the object moves 6.25 inches away from its central or resting position. Without enough amplitude, the motion would be barely visible or flat. When thinking about sound, a larger amplitude equates to a louder sound, while in light, it means a brighter light.

Frequency

Frequency is related to how often the oscillation repeats itself in one second and is an essential factor in understanding simple harmonic motion. Measured in Hertz (Hz), frequency defines the number of complete cycles or oscillations that occur in one second.
In simple harmonic motion, frequency is denoted by "f" in the mathematical equation. If the frequency is higher, the oscillations happen more rapidly. Conversely, a lower frequency means fewer oscillations per second.
For example, the exercise specifies that the frequency is 60 Hz. This means that the object completes 60 full oscillations every second. In many applications, such as tuning musical instruments or evaluating machine vibrations, precisely determining the frequency is crucial.

Phase Shift

Phase shift (\(\phi\)) in simple harmonic motion describes how the starting point of the motion is advanced or delayed concerning the central cycle. In simpler terms, it tells us if the motion is shifted in time.
In cosine or sine equations, the phase shift allows adjustments to the wave's starting point. When \(\phi\) is zero, there is no shift—meaning the displacement is greatest at time \(t = 0\), as in the exercise.
If the phase shift is a positive value, the motion starts later, moved to the right on a time graph. Conversely, a negative phase shift means the motion starts earlier. Although often unnoticed, understanding phase shift can be critical when fine-tuning models of oscillating systems such as antennas or seismic waves.