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 displacementb. the frequencyc. the time required for one cycle.d = 1/3 sin 2t

Verified step by step guidance

1. Identify the amplitude of the function, which is the coefficient of the sine function. In this case, the amplitude is \( \frac{1}{3} \). This represents the maximum displacement.
2. The general form of a simple harmonic motion equation is \( d = A \sin(Bt + C) + D \). Here, \( A = \frac{1}{3} \), \( B = 2 \), \( C = 0 \), and \( D = 0 \).
3. The frequency of the motion is given by \( \frac{B}{2\pi} \). Substitute \( B = 2 \) to find the frequency.
4. The period of the motion, which is the time required for one complete cycle, is the reciprocal of the frequency. Calculate the period using \( \text{Period} = \frac{2\pi}{B} \).
5. Summarize the findings: The maximum displacement is \( \frac{1}{3} \) inches, the frequency is \( \frac{1}{\pi} \) cycles per second, and the period is \( \pi \) seconds.

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 equation d = (1/3) sin(2t) represents the displacement of the object over time, indicating how far it moves from its central position.

Maximum Displacement

Maximum displacement, also known as amplitude, refers to the greatest distance the object moves from its equilibrium position during its oscillation. In the given equation, the coefficient of the sine function (1/3) indicates the amplitude, meaning the object reaches a maximum displacement of 1/3 inches from the center position in either direction.

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 d = (1/3) sin(2t), the coefficient of t (which is 2) relates to the frequency, allowing us to determine that the frequency is 1 Hz, and the period, which is the reciprocal of frequency, is 1 second. This means the object completes one full oscillation every second.