Problem 6 - 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. $$y=-\frac{3}{2} \sin (0.2 t+1.4)$$

Step-by-Step Solution

Answer

Amplitude: \( \frac{3}{2} \), Period: \( 10\pi \), Frequency: \( \frac{1}{10\pi} \). Sketch shows one complete sine wave cycle.

Step 1: Understanding the Amplitude

The amplitude of a function in the form \( y = A \sin( B t + C ) \) is the absolute value of \( A \). In the given function, \( y = -\frac{3}{2} \sin (0.2 t + 1.4) \), the coefficient \( A = -\frac{3}{2} \). To find the amplitude, we take the absolute value: \[ \text{Amplitude} = |-\frac{3}{2}| = \frac{3}{2} \].

Step 2: Calculating the Period

The period of a sine function is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \) in the argument of the sine function. Here, \( B = 0.2 \). The period is calculated as:\[\text{Period} = \frac{2\pi}{0.2} = \frac{2\pi}{\frac{1}{5}} = 10\pi\].

Step 3: Determining the Frequency

The frequency is the reciprocal of the period. With the period being \( 10\pi \), the frequency is calculated as:\[\text{Frequency} = \frac{1}{10\pi}\].

Step 4: Sketching the Graph

To sketch the graph over one complete period, plot the function \( y = -\frac{3}{2} \sin (0.2 t + 1.4) \) over the interval from \( t = 0 \) to \( t = 10\pi \). The graph should show a sine wave oscillating between \( \frac{3}{2} \) and \( -\frac{3}{2} \), starting at an angle of \( 1.4 \) radians due to the phase shift. Ensure the wave completes exactly one full cycle within this range.

Key Concepts

Amplitude

The amplitude in a simple harmonic motion refers to the maximum extent of the vibration or wave, measured from the position of equilibrium. It tells us how far the object moves from the central position. Think of amplitude as the height of a wave or vibration peak.
In mathematical terms, for a sine function of the form \( y = A \sin(Bt + C) \), the amplitude \( A \) is the coefficient directly in front of the sine function. In our example of \( y = -\frac{3}{2} \sin(0.2t + 1.4) \), the coefficient \( A = -\frac{3}{2} \).

• The amplitude is found by taking the absolute value of \( A \): \( |A| = |-\frac{3}{2}| = \frac{3}{2} \).
• A higher amplitude means a taller wave, while a lower amplitude leads to a shorter wave.

This means our object moves \( \frac{3}{2} \) units away from the center in either direction.

Period

The period of a simple harmonic motion helps determine how long it takes to complete one full cycle of oscillation. It's often associated with the time interval for the wave to repeat itself. Understanding the period is crucial, as it allows you to anticipate the motion of the object at any given time.
For a sine function \( y = A \sin(Bt + C) \), the period is calculated using the formula:

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

In the example \( y = -\frac{3}{2} \sin(0.2t + 1.4) \):

• Here, \( B = 0.2 \).
• So, the period becomes \( \frac{2\pi}{0.2} = 10\pi \).

Hence, every \( 10\pi \) units of time, the wave repeats itself. This cycle dictates how frequently the wave pattern occurs over a set duration.

Frequency

The frequency of a wave describes how often the wave occurs in a specific time frame. It is a measure of how many complete wavelengths or cycles pass a specific point per unit of time.
Frequency is the reciprocal of the period. Therefore, knowing the period allows us to easily find frequency using the relationship:

• \( \text{Frequency} = \frac{1}{\text{Period}} \)

For the function \( y = -\frac{3}{2} \sin(0.2t + 1.4) \), the period is \( 10\pi \). So the frequency is calculated as:

• \( \text{Frequency} = \frac{1}{10\pi} \).

A lower frequency means slower oscillations, and a higher frequency denotes faster oscillations. In simple terms, the lower the frequency, the fewer cycles of the wave occur in the same time frame.

Sine Function

The sine function is a fundamental part of simple harmonic motion, modeling the oscillatory motion found in various physical systems like springs and pendulums. It represents a smooth, periodic oscillation that can accurately depict many types of wave-like motions.
The general form of the sine function is \( y = A \sin(Bt + C) \). In our example, \( y = -\frac{3}{2} \sin(0.2t + 1.4) \), several components work together:

• \( A \) determines the amplitude or the peak height of the oscillation.
• \( B \) affects the period, influencing how quickly the cycle repeats.
• \( C \) is the phase shift, altering where the wave begins its cycle on the time-axis.

Utilizing the sine function helps us not just visualize but also predict the future behavior of oscillating systems. This mathematical tool simplifies the complex study of motions into manageable and understandable components.