Problem 24 - Precalculus Essentials

Question

An object moves in simple harmonic motion described by the given equation, where \(t\) is measured in seconds and \(d\) in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. $$d=-8 \cos \frac{\pi}{2} t$$

Step-by-Step Solution

Answer

The maximum displacement is 8 inches. The frequency is 0.25 Hz. The time required for one cycle is 4 seconds.

Step 1: Extract the Maximum Displacement

The maximum displacement or the amplitude of a cosine function is simply the absolute value of the coefficient in front of the cosine. In this case, the amplitude is 8 inches.

Step 2: Extract the Frequency

The frequency of the motion must be derived from the term within the cosine function. The frequency \(f\) is computed as \(f=\frac{\omega}{2\pi}\), where \(\omega\) is the coefficient in the cosine expression. Here, \(\omega = \frac{\pi}{2}\) which equates to a frequency of \(f = \frac{1}{4}\) cycles per second or 0.25 Hz.

Step 3: Compute the Time Required for One Cycle

The period \(T\) of the motion is the reciprocal of the frequency. So, \(T = \frac{1}{f}\). Given the frequency to be 0.25 Hz, the period of the motion is then \(T = 4\) seconds. This indicates that one full cycle of the motion takes 4 seconds to complete.

Key Concepts

Maximum Displacement

When studying simple harmonic motion (SHM), one key characteristic to understand is the maximum displacement, often referred to as the amplitude. Imagine a child swinging back and forth; the highest point the swing reaches on either side is akin to the maximum displacement in SHM. It determines how far the object moves from its equilibrium position (the central point) before reversing its direction.

In the given exercise, the object's motion is described by the equation \(d=-8 \cos \frac{\pi}{2} t\). Here, the coefficient in front of the cosine, which is 8, represents the maximum displacement in inches. This means that the object moves 8 inches away from the central point before coming back towards it. This concept is pivotal because it sets the limits within which the object oscillates.

Frequency of Motion

The frequency of motion in SHM indicates how often the oscillations occur over a set period of time. It's like counting the number of times a pendulum swings back and forth each second. Frequency is measured in Hertz (Hz), which translates to cycles per second. It's an important factor that helps us understand the rapidity of the oscillations.

From the exercise equation, we find the term inside the cosine function, \(\frac{\pi}{2}\), which is instrumental when calculating the angular frequency \(\omega\). By using the formula \(f = \frac{\omega}{2\pi}\), we can determine the frequency. With \(\omega = \frac{\pi}{2}\), the resulting frequency is \(f = \frac{1}{4}\) Hz or 0.25 Hz. Thus, the object completes a quarter of a cycle every second.

Period of Oscillation

The period of oscillation, often symbolized as \(T\), is the time it takes to complete one full cycle of motion. If you look at a grandfather clock, the period is the time taken for the pendulum to swing to the far left and back to the far right.

In our exercise, to find the period, one would take the reciprocal of the frequency. Given that the frequency is 0.25 Hz, the period can be calculated as \(T = \frac{1}{f} = 4\) seconds. This denotes that it takes 4 seconds for the object to swing back and forth once, fully embodying one complete oscillation. Understanding the period is essential for grasping the timing of oscillatory systems, predicting their movements, and exploring their properties in various applications like engineering and physics.