Problem 24 - Algebra and Trigonometry

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

a. The maximum displacement is 8 inches. b. The frequency is \( \frac{1}{4} \) Hz. c. The time required for one complete cycle is 4 seconds.

Step 1: Finding the Maximum Displacement

The maximum displacement \(D\) is the absolute value of the coefficient of the cosine term in the given equation. Hence, \[D = |-8| = 8 \] inches.

Step 2: Finding the Frequency

The frequency \(f\) from the angular frequency can be calculated as \(f=\frac{\omega}{2 \pi}\). Here, \(\omega=\frac{\pi}{2}\). Hence, the mechanical frequency \(f\) is given as: \[f = \frac{\frac{\pi}{2}}{2\pi} = \frac{1}{4} \] Hz.

Step 3: Finding the Time Required for One Cycle

The period of oscillation \(T\) is simply the reciprocal of the frequency hence \[ T = \frac{1}{f} = \frac{1}{\frac{1}{4}} = 4 \] seconds.

Key Concepts

Understanding Maximum Displacement in Simple Harmonic Motion

When studying simple harmonic motion (SHM), a key characteristic to observe is the 'maximum displacement', often represented as the amplitude. It indicates the furthest point from the equilibrium position that the object reaches during its motion. In our equation, this is determined by the coefficient of the cosine function, which in this case is -8. It's essential to note that the maximum displacement is a positive value, as it represents distance, which means we take the absolute of -8, yielding 8 inches.

In real-life scenarios, this could represent how far a pendulum swings from the central point or the height a child might reach on a swing. Understanding maximum displacement helps us grasp the limits of the object's motion and predict how much space the motion will occupy, which is critical in engineering designs that involve oscillatory systems.

Frequency of Oscillation: The Heartbeat of Cyclic Motion

Frequency of oscillation, measured in Hertz (Hz), is another fundamental concept within the study of SHM. It tells us how often the object repeats its motion cycle in one second. To find the frequency from the given equation, we first identify the angular frequency, \(\omega\), which is related to the frequency by the formula \(f = \frac{\omega}{2\pi}\).

In our case, \(\omega = \frac{\pi}{2}\). This gives us a frequency of \(f = \frac{\frac{\pi}{2}}{2\pi} = \frac{1}{4}\) Hz, meaning the object completes one-fourth of a cycle every second. By understanding frequency, we can determine how fast processes occur in both natural phenomena and technology, such as the oscillation rate of a quartz crystal in a watch.

Period of Oscillation: Measuring the Duration of Cycles

The period of oscillation, usually denoted as \(T\), is the time it takes to complete one full cycle of motion. It's the reciprocal of frequency, as shown by the relationship \(T = \frac{1}{f}\). For the equation in our exercise, with the frequency being \(\frac{1}{4}\) Hz, the period thus becomes \(T = \frac{1}{\frac{1}{4}} = 4\) seconds.

Knowing the period of oscillation is crucial as it gives insight into the rhythm of motion. For example, in mechanics, this can be used to calculate the timing of parts in an engine, or in electronics, to determine the wave patterns in circuits. A longer period implies a slower oscillation, whereas a shorter period indicates a faster one. This concept is pivotal in designing and managing any system involving repetitive motion.