In Exercises 37–40, 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, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = − 1/2 sin(πt/4 − π/2)

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1. Identify the general form of the simple harmonic motion equation, which is usually written as \(d = A \sin(Bt - C)\) or \(d = A \cos(Bt - C)\), where \(A\) is the amplitude (maximum displacement), \(B\) affects the frequency and period, and \(C\) is the phase shift.
2. From the given equation \(d = -\frac{1}{2} \sin\left(\frac{\pi t}{4} - \frac{\pi}{2}\right)\), recognize that the amplitude \(A\) is the absolute value of the coefficient in front of the sine function, which is \(\left| -\frac{1}{2} \right|\).
3. To find the frequency, use the relationship between \(B\) and frequency: \(B = \frac{2\pi}{T}\), where \(T\) is the period (time for one cycle). Here, \(B = \frac{\pi}{4}\), so solve for \(T\) using \(T = \frac{2\pi}{B}\).
4. The frequency \(f\) is the reciprocal of the period, so calculate \(f = \frac{1}{T}\) once you have found \(T\).
5. Determine the phase shift by solving the equation inside the sine function for zero: set \(\frac{\pi t}{4} - \frac{\pi}{2} = 0\) and solve for \(t\). This value of \(t\) gives the horizontal shift of the graph.

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. It is typically modeled by sine or cosine functions, representing periodic movement such as vibrations or waves. Understanding SHM helps interpret the given equation and its physical meaning.

Amplitude, Frequency, and Period

Amplitude is the maximum displacement from the equilibrium position, frequency is the number of cycles per unit time, and period is the time for one complete cycle. These parameters are derived from the equation's coefficients and arguments, essential for analyzing the motion's characteristics.

Phase Shift in Trigonometric Functions

Phase shift refers to the horizontal displacement of the sine or cosine wave along the time axis, caused by added or subtracted constants inside the function's argument. It affects the starting point of the motion and is crucial for accurately graphing and interpreting the equation.