Question

The motion of a pendulum can be modeled by the function \(d=4 \cos 8 \pi t\), where \(d\) is the horizontal displacement (in inches) of the pendulum relative to its position at rest and \(t\) is the time (in seconds). Find and interpret the period and amplitude in the context of this situation. Then graph the function.

Short answer

The amplitude of the pendulum motion, represented by the function \(d=4 \cos 8\pi t\), is 4 inches. This means the pendulum moves 4 inches on either side of its rest position. The period of the pendulum motion is \(\frac{1}{4}\) seconds, meaning the pendulum completes a full swing back and forth in \(\frac{1}{4}\) of a second.

Step-by-step solution

Identify the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient in front of the function. In this case, the coefficient in front of the cosine function is 4, so the amplitude is \(|4|\) which is 4. This is the maximum horizontal displacement on either side of the pendulum’s rest position and it is measured in inches.

Calculate the Period

The period of a cosine function is given by \( \frac{2\pi}{|B|} \) where \(B\) is the coefficient of \(t\). Here, \(B\) is \(8\pi\), so the period is \( \frac{2\pi}{8\pi} = \frac{1}{4} \) seconds. This is the time it takes for the pendulum to make a complete swing back and forth.

Graph the Function

To graph the function \(d=4\cos(8\pi t)\), plot time \(t\) on the X-axis and displacement \(d\) on the Y-axis. Start at the origin (0,0). A complete period happens from 0 to \(\frac{1}{4}\) on the time axis. The maximum displacement is 4, so the graph should oscillate between 4 and -4 on the Y-axis. The main points are (0,4), middle of the period the pendulum at rest position (\(\frac{1}{8}\), 0), end of the period the pendulum is back to its maximum displacement (\(\frac{1}{4}\), 4). The graph has the shape of a 'wave' that peaks at every quarter increment starting from 0.

Key concepts

Amplitude of a Cosine Function

When studying pendulums or other oscillatory motions, understanding the amplitude of a cosine function is crucial. In trigonometry, the amplitude represents how far the function rises above and drops below its midline, which is essential for determining the maximum or minimum values of a wave. Consider a general cosine function of the form
\[ d = A \times \text{cos}(Bt) \]
In the context of a pendulum, the amplitude, represented by the coefficient \( A \), corresponds to the maximum horizontal displacement from the rest position. For the given function \( d=4 \text{cos}(8 \text{\(\text{\pi}\)} t) \), the amplitude is \( |4| \) or simply 4 inches. This tells us that the pendulum swings 4 inches to the left and right from its rest position, capturing the extent of its motion. Despite the complexity of motion, the amplitude presents a straightforward measurement; it's the pendulum's reach in its back-and-forth journey.

Period of a Cosine Function

The period is another fundamental component in the study of trigonometric functions like the cosine. For a pendulum or any cyclic process, the period is the duration required to complete one full cycle of motion.
For a cosine function modeled by \( d = A \text{cos}(Bt) \), the period \( T \) can be calculated using the formula:
\[ T = \frac{2\text{\(\text{\pi}\)}}{|B|} \]
where \( B \) is the frequency of the function. In our pendulum example, with \( B = 8\text{\(\text{\pi}\)} \), the period is \( \frac{1}{4} \) seconds. This means that it takes one-quarter of a second for the pendulum to swing from its farthest point on one side, through the rest position, to the farthest point on the opposite side, and back again. Recognizing how the coefficient \( B \) affects the period helps students predict the behavior of the pendulum over time.

Graphing Trigonometric Functions

Visualizing trigonometric functions like the cosine through graphing can significantly enhance a student's comprehension of wave motion. To graph the cosine function \( d=4\text{cos}(8\text{\(\text{\pi}\)} t) \), appropriate scales for both the time (\( t \)) axis and displacement (\( d \)) axis must be chosen. With the period being \( \frac{1}{4} \) seconds and the amplitude 4 inches, we can start graphing.
At \( t = 0 \), the pendulum is at its maximum displacement of 4 inches. This represents the peak of the cosine wave on the graph. As time proceeds to \( \frac{1}{8} \) seconds, the pendulum passes through the rest position (\( d = 0 \)), which corresponds to the zero crossing of the wave. Upon reaching \( \frac{1}{4} \) seconds, the pendulum completes a full cycle, returning to its maximum displacement, and the wave peaks once again. Charting these primary points and connecting them with a smooth curve unveils the oscillating 'wave' pattern of the pendulum's displacement over time. Graphing not only aids in visual learning but also supports the interpretation of the pendulum's physical behavior through abstract mathematical representation.