Question

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 = −8 cos π/2 t

Accepted Answer

Identify the general form of the simple harmonic motion equation, which is $$d = A \cos(\omega t) or$$ $$d = A \sin(\omega t)$$, where $$A is $$the amplitude (maximum displacement), and $$\omega is $$the angular frequency in radians per second. From the given equation $$d = -8 \cos \left( \frac{\pi}{2} t \right)$$, determine the amplitude $$A by $$taking the absolute value of the coefficient in front 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 $$\frac{\pi}{2} in $$this case. Calculate the frequency $$f$$ using the relationship between angular frequency and frequency: $$f = \frac{\omega}{2\pi}. $$Substitute $$\omega = \frac{\pi}{2} to $$find $$f. $$Determine the period $$T$$, which is the time required for one complete cycle, using the formula $$T = \frac{1}{f} or $$equivalently $$T = \frac{2\pi}{\omega}. $$Substitute the value of $$\omega to $$find $$T.$$

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 = −8 cos π/2 t

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Key Concepts

Simple Harmonic Motion (SHM)

Amplitude and Maximum Displacement

Frequency and Period of Oscillation