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 = −4 sin 3π/2 t

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object moves back and forth around an equilibrium position. The motion can be described by a sine or cosine function, which captures the oscillatory nature of the movement. In this context, the displacement equation indicates that the object oscillates with a specific amplitude and frequency, which are key to understanding its motion.

Amplitude

Amplitude refers to the maximum displacement of an object from its equilibrium position in simple harmonic motion. In the given equation, the amplitude can be determined from the coefficient of the sine function. It represents how far the object moves from the center point during its oscillation, which is crucial for identifying the maximum displacement.

Frequency and Period

Frequency is the number of cycles an object completes in one second, while the period is the time taken to complete one full cycle. In the equation provided, the frequency can be derived from the coefficient of 't' in the sine function. Understanding the relationship between frequency and period (where period = 1/frequency) is essential for determining how quickly the object oscillates.