In Exercises 21–28, 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 π/2 t

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1. Identify the general form of the simple harmonic motion equation, which is \(d = A \cos(\omega t)\) or \(d = A \sin(\omega t)\), where \(A\) is the amplitude (maximum displacement), and \(\omega\) is the angular frequency in radians per second.
2. From the given equation \(d = -8 \cos \left( \frac{\pi}{2} t \right)\), determine the amplitude \(A\) by taking the absolute value of the coefficient in front of the cosine function. This gives the maximum displacement.
3. Find the angular frequency \(\omega\) by identifying the coefficient of \(t\) inside the cosine function, which is \(\frac{\pi}{2}\) in this case.
4. Calculate the frequency \(f\) using the relationship between angular frequency and frequency: \(f = \frac{\omega}{2\pi}\). Substitute \(\omega = \frac{\pi}{2}\) to find \(f\).
5. Determine the period \(T\), which is the time required for one complete cycle, using the formula \(T = \frac{1}{f}\) or equivalently \(T = \frac{2\pi}{\omega}\). Substitute the value of \(\omega\) to find \(T\).

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion describes oscillatory motion where the restoring force is proportional to displacement and acts in the opposite direction. The motion can be modeled by sinusoidal functions like sine or cosine, representing displacement over time. Understanding SHM helps interpret the given equation and analyze the object's movement.

Amplitude and Maximum Displacement

Amplitude is the maximum distance an object moves from its equilibrium position in SHM. It corresponds to the coefficient in front of the cosine or sine function in the displacement equation. In this problem, the amplitude indicates the maximum displacement the object reaches during its motion.

Frequency and Period of Oscillation

Frequency is the number of complete oscillations per unit time, while the period is the time taken for one full cycle. They are inversely related: frequency = 1/period. The angular frequency (coefficient of t inside the cosine) helps calculate both frequency and period, essential for answering parts b and c of the question.