Problem 5 - 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. $$y=-0.25 \cos \left(1.5 t-\frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

Amplitude: 0.25, Period: \(\frac{4\pi}{3}\), Frequency: \(\frac{3}{4\pi}\).

Step 1: Identify the Amplitude

The coefficient in front of the cosine function represents the amplitude of the motion. In the given function \( y = -0.25 \cos(1.5t - \frac{\pi}{3}) \), the amplitude is \(-0.25\). Since amplitude is always positive, we take the absolute value, so the amplitude is \(0.25\).

Step 2: Determine the Period

The period of a cosine function is given by \( \frac{2\pi}{|b|} \), where \(b\) is the coefficient of \(t\) in the cosine function. Here, \(b = 1.5\). Therefore, the period is \( \frac{2\pi}{1.5} = \frac{4\pi}{3} \).

Step 3: Calculate the Frequency

The frequency of the motion is the reciprocal of the period. We find the frequency by computing \( \frac{1}{\text{Period}} \), which is \( \frac{1}{\frac{4\pi}{3}} = \frac{3}{4\pi} \).

Step 4: Sketch the Graph

The graph of \( y = -0.25 \cos(1.5t - \frac{\pi}{3}) \) should be a cosine wave shifted to the right by \( \frac{\pi}{3} \) and inverted due to the negative amplitude. Starting at \(t = 0\), the wave reaches -0.25, and it completes a cycle at \(t = \frac{4\pi}{3}\). The graph should show one full wave over this period with peaks and troughs aligned with fractional periods \(\frac{\pi}{3}\), \(\frac{2\pi}{3}\), etc.

Key Concepts

Amplitude

The amplitude of a harmonic motion describes the maximum displacement of an object from its equilibrium position. This tells how far the object moves from its starting point. In the function given, the amplitude is determined by the coefficient of the cosine function, which is -0.25. Amplitude is always a positive value because it represents a distance. Therefore, we take the absolute value, which results in an amplitude of 0.25.
This means the object moves 0.25 units away in either direction from its origin. If you're dealing with a negative coefficient, don't worry — just focus on its size while ignoring the sign. This characteristic of amplitude plays a crucial role in defining the "height" of the wave in the graph.

• The function's highest point is 0.25 above the x-axis.
• The lowest point is 0.25 below the x-axis.
• No matter what transformation is applied, amplitude remains constant.

Understanding amplitude helps in predicting the object's range of movement in harmonic motion.

Period

The period of a harmonic motion is the time it takes for the object to complete one full cycle of motion. It's like the "length" of one wave repetition. For the given function, we calculate the period using the formula \( \frac{2\pi}{|b|} \), where \(b\) is the coefficient of the term \(t\).
In this case, \(b = 1.5\). When we plug it into the formula, we get \( \frac{2\pi}{1.5} = \frac{4\pi}{3} \). Thus, the period is \(\frac{4\pi}{3}\). This means that, after this interval, the wave pattern begins to repeat itself.

• The time interval \(\frac{4\pi}{3}\) represents the completion of a full cycle.
• The object's movement from its highest, through to the lowest point, and back again, fits within this framework.
• Knowing the period allows us to predict the behavior of the motion over time.

Tracking the period shows us how frequently the object retraces its path.

Frequency

Frequency tells us how often the object repeats its cycle in a unit of time. Simply put, it's the number of complete cycles per unit time. If the period is the length of one cycle, then frequency is the number of those cycles you can fit in a certain time frame.
For our function, frequency is the reciprocal of the period. This is calculated as \( \frac{1}{\text{Period}} = \frac{1}{\frac{4\pi}{3}} = \frac{3}{4\pi} \). Thus, the frequency is \( \frac{3}{4\pi} \), indicating how many cycles happen per unit time.

• Frequency gives us insight into the relative speed of the harmonic motion.
• Higher frequency means more cycles in a given time, whereas lower frequency means fewer cycles.
• Understanding frequency is crucial for applications where timing and cycles are essential, like in musical instruments or oscillating systems.

In essence, frequency helps us understand the rhythm and timing of the object's motion.