Question

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)

Accepted Answer

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. 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 $$\left| -\frac{1}{2} \right|. To $$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 $$T = \frac{2\pi}{B}. $$The frequency $$f is $$the reciprocal of the period, so calculate $$f = \frac{1}{T}$$ once you have found $$T. $$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.

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Key Concepts

Simple Harmonic Motion (SHM)

Amplitude, Frequency, and Period

Phase Shift in Trigonometric Functions