Problem 8 - Algebra and Trigonometry

Question

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$y=1.6 \sin (t-1.8)$$

Step-by-Step Solution

Answer

Amplitude is 1.6, period is \(2\pi\), frequency is \(\frac{1}{2\pi}\).

Step 1: Identify the form of the function

The given function is in the form \( y = a \sin(b(t-c)) + d \), where \( a \) is the amplitude, \( b \) affects the period, and \( c \) is the phase shift. In this example, \( y = 1.6 \sin(t - 1.8) \).

Step 2: Determine the amplitude

The amplitude of a sine function \( y = a \sin(b(t-c)) + d \) is the absolute value of \( a \). So for \( y = 1.6 \sin(t - 1.8) \), the amplitude is \( |1.6| = 1.6 \).

Step 3: Calculate the period

The period of a sine function is determined by \( \frac{2\pi}{b} \). In this case, since \( b = 1 \), the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Determine the frequency

The frequency is the reciprocal of the period. The frequency \( f \) is given by \( f = \frac{1}{\text{Period}} = \frac{1}{2\pi} \).

Step 5: Draw the graph of the sine function

To sketch the graph over one complete period, plot the sine function for \( t \) ranging from the phase shift to the phase shift plus \( 2\pi \). For \( y = 1.6 \sin(t - 1.8) \), start at \( t = 1.8 \) through \( t = 1.8 + 2\pi \). The graph will oscillate between -1.6 and 1.6.

Key Concepts

Sine Function

The sine function plays a critical role in modeling simple harmonic motion. In mathematics, its form is typically represented as:

• \( y = a \sin(b(t-c)) + d \)

This function describes how certain patterns repeat over time, such as vibrations or waves. The sine wave smoothly oscillates between maximum and minimum values, making it ideal for representing periodic phenomena. Let's break down its components:

• \( a \) is the amplitude, which determines the height of the wave's crest from its central position.
• \( b \) affects the period of the function, dictating how quickly the wave oscillates.
• \( c \) represents the horizontal shift (or phase shift), affecting where the wave starts along the time axis.
• \( d \) is a vertical shift, which moves the entire wave up or down along the y-axis, although often \( d \) is zero, as in this case.

Understanding how these components interact allows us to predict and manipulate the behavior of systems described by simple harmonic motion.

Amplitude

Amplitude in simple harmonic motion is the maximum displacement from the equilibrium position. In the context of the sine function, such as \( y = 1.6 \sin(t - 1.8) \), amplitude is signified by \( a \). Here, \( a = 1.6 \), meaning the wave reaches 1.6 units above and below the central axis.
Amplitude tells us how "large" the oscillations are, and it’s a direct indicator of the energy in the system. Larger amplitudes represent more energy and more significant oscillations, while smaller amplitudes indicate less energy.
In practical terms, if this equation describes a pendulum, the amplitude would be the maximum angle it swings away from rest.

• Key aspect: Amplitude is always a positive value, reflecting the peak distances, regardless of whether the wave reaches a positive or negative direction.

Period and Frequency

The period of a sine function is crucial for understanding how often the cycle repeats. In the function \( y = 1.6 \sin(t - 1.8) \), the period is calculated as \( \frac{2\pi}{b} \). With \( b = 1 \), we find the period is simply \( 2\pi \).
This period represents the time it takes for a complete cycle of the motion, from start to finish, to occur. The shorter the period, the more cycles fit into a set timeframe.
On the flip side is frequency, which is the number of complete cycles per unit time. Frequency (\( f \)) can be found with the formula:

• \( f = \frac{1}{\text{Period}} = \frac{1}{2\pi} \)

This frequency measurement gives us insight into how frequently the oscillations happen. It's crucial for real-world applications like electrical currents, where you might need specific frequencies for systems to function effectively.

• Note: Period and frequency are inversely related. As one increases, the other decreases, maintaining the balance and regularity of oscillations in simple harmonic motion.