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 = 3 cos(πt + π/2)

Accepted Answer

Identify the general form of the simple harmonic motion equation: $$d = A \cos(\omega t + \phi)$$, where $$A is $$the amplitude (maximum displacement), $$\omega is $$the angular frequency, and $$\phi is $$the phase shift. From the given equation $$d = 3 \cos(\pi t + \frac{\pi}{2})$$, determine the amplitude $$A by $$looking at the coefficient of the cosine function. This gives the maximum displacement. Find the angular frequency $$\omega by $$identifying the coefficient of $$t$$ inside the cosine function, which is $$\pi in $$this case. Use the relationship between angular frequency and frequency: $$f = \frac{\omega}{2\pi} to $$find the frequency. Calculate the time period $$T ($$time required for one cycle) using the formula $$T = \frac{1}{f} or $$equivalently $$T = \frac{2\pi}{\omega}. $$Determine the phase shift by analyzing the term $$\phi = \frac{\pi}{2}$$ inside the cosine function. The phase shift in time units is given by $$-\frac{\phi}{\omega}$$, which tells how much the graph is shifted horizontally.

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Concetti chiave

Simple Harmonic Motion (SHM)

Amplitude, Frequency, and Period

Phase Shift in Trigonometric Functions