Problem 7 - 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. $$y=5 \cos \left(\frac{2}{3} t+\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(3\pi\), Frequency: \(\frac{1}{3\pi}\).

Step 1: Identify the Amplitude

The amplitude of a cosine function in the form \( y = a \cos(bx + c) \) is given by the absolute value of the coefficient \( a \). Here, \( y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right) \), so the amplitude is \( |5| = 5 \).

Step 2: Determine the Period

The period of a cosine function is given by \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( t \). Here, \( b = \frac{2}{3} \), so the period is \( \frac{2\pi}{\frac{2}{3}} = 3\pi \).

Step 3: Calculate the Frequency

The frequency is the reciprocal of the period. Given the period \( 3\pi \), the frequency is \( \frac{1}{3\pi} \).

Step 4: Sketch the Graph of the Displacement Over One Complete Period

To graph the function over one period, start by plotting the points from \( t = 0 \) to \( t = 3\pi \). The function is \( y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right) \). At \( t = 0 \), the value is \( 5 \cos\left(\frac{3}{4}\right) \). At \( t = \frac{3\pi}{2} \), it should have a maximum or minimum, depending on the phase shift.

Key Concepts

Amplitude

The amplitude in a trigonometric function, like our cosine example, represents the maximum displacement from its central position. Think of it as the measurement of how far the object moves from the equilibrium point. In the standard equation form, such as \(y = a \cos(bx + c)\), the amplitude is given by the absolute value of the coefficient \(a\).

In our equation, \(y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right)\), the amplitude is \(|5| = 5\). This means that the object will oscillate 5 units above and below the central axis, demonstrating the extent of its movement.

Period

The period of a trigonometric function tells you how much input is needed for the function to complete one full cycle of motion. For a cosine function, this is determined by the coefficient \(b\), as seen in the equation \(y = a \cos(bx + c)\).

The formula to calculate the period is \(\frac{2\pi}{|b|}\). In our function \(y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right)\), \(b\) is \(\frac{2}{3}\).

• Calculating the period yields \(\frac{2\pi}{\frac{2}{3}} = 3\pi\).

This indicates that the function completes a full cycle every \(3\pi\) units of \(t\).

Frequency

Frequency in simple harmonic motion describes how often the cycle repeats over a unit of time. It is inversely related to the period, meaning as the period increases, frequency decreases, and vice versa.

• The formula to find the frequency is \(\frac{1}{\text{Period}}\).

Given our period was calculated as \(3\pi\), the frequency is \(\frac{1}{3\pi}\).

This means the motion repeats its cycle approximately \(\frac{1}{3\pi}\) times per unit of time, indicating a slower oscillation when compared to functions with smaller periods.

Cosine Function

The cosine function is one of the key trigonometric functions and is often used to model periodic phenomena like simple harmonic motion. It has the basic form \(y = a\cos(bx + c)\), where:

• \(a\) is the amplitude, affecting the vertical stretch.
• \(b\) influences the period of the function.
• \(c\) causes a horizontal phase shift.

In our equation, \(y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right)\),

• Amplitude \(a = 5\)
• Period is determined by \(b = \frac{2}{3}\)
• Phase shift due to \(c = \frac{3}{4}\)

The cosine function starts at its maximum value when \(c = 0\). However, in our function, the phase shift moves its starting point.

Graphing Trigonometric Functions

Graphing trigonometric functions like cosine involves plotting their behavior over one full period. It helps visualize how variables like amplitude, period, and phase shift alter the curve's shape.

To graph \(y = 5 \cos\left(\frac{2}{3} t + \frac{3}{4}\right)\):

• Start by identifying key points: maximum, minimum, and zeros based on one period \(t = 0\) to \(t = 3\pi\).
• Calculate values of \(y\) at these points, considering phase shift \(\frac{3}{4}\).
• Sketch the curve beginning from the cosine point at \(t = 0\).
• Continue until completing a single cycle at \(t = 3\pi\).

The graph will exhibit a wave pattern, reaching a maximum at \(5\) and a minimum at \(-5\), symmetrically oscillating around the axis.