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 = 10 cos 2πt

Verified step by step guidance

1. <b>Step 1:</b> Identify the amplitude of the motion from the equation \(d = 10 \cos 2\pi t\). The amplitude is the coefficient of the cosine function, which represents the maximum displacement. In this case, it is 10 inches.
2. <b>Step 2:</b> Determine the frequency of the motion. The frequency is related to the coefficient of \(t\) inside the cosine function. Here, the coefficient is \(2\pi\), which indicates that the frequency \(f\) is 1 cycle per second (since \(2\pi\) is the angular frequency and \(f = \frac{\text{angular frequency}}{2\pi}\)).
3. <b>Step 3:</b> Calculate the time required for one cycle, also known as the period \(T\). The period is the reciprocal of the frequency. Since the frequency is 1 cycle per second, the period \(T = \frac{1}{f} = 1\) second.
4. <b>Step 4:</b> Review the characteristics of simple harmonic motion. The maximum displacement (amplitude) is the farthest distance the object moves from its equilibrium position, and the period is the time it takes to complete one full cycle of motion.
5. <b>Step 5:</b> Verify the understanding of the cosine function in the context of simple harmonic motion. The function \(d = 10 \cos 2\pi t\) describes a motion that starts at the maximum displacement when \(t = 0\), oscillates back and forth, and completes a full cycle in 1 second.

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 sinusoidal function, such as sine or cosine, which captures the oscillatory nature of the movement. In the given equation, the cosine function indicates that the object starts at its maximum displacement.

Maximum Displacement

Maximum displacement, also known as amplitude, refers to the furthest distance the object moves from its equilibrium position during its oscillation. In the equation d = 10 cos 2πt, the coefficient '10' represents the maximum displacement in inches, indicating that the object oscillates between +10 and -10 inches from the center position.

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, the term '2π' in the cosine function indicates the angular frequency, which can be used to calculate the frequency as 1 cycle per second. The period, which is the reciprocal of frequency, tells us how long it takes for the object to return to its starting position.