An object moves in simple harmonic motion described by the equation \( d=-2 \sin \frac{2 \pi}{7} \mathrm{t} \) where t is measured in seconds and d in inches. Find the maximum displacement, the frequency, and the time required for one cycle. a. Find the maximum displacement. \( \square \mathrm{in} \). (Type an integer or a fraction.)

The maximum displacement in simple harmonic motion is determined by the coefficient in front of the sine function. Here, the equation \( d = -2 \sin \frac{2 \pi}{7} t \) indicates that the maximum displacement (amplitude) is 2 inches. Next, to find the frequency, we look at the term \(\frac{2 \pi}{7}\) in the equation. This represents the angular frequency \(\omega\), which is related to the frequency \(f\) by the formula \(f = \frac{\omega}{2\pi}\). So, substituting \(\omega\) gives us: \[ f = \frac{2 \pi/7}{2\pi} = \frac{1}{7} \text{ Hz}. \] Finally, the time required for one complete cycle (the period \(T\)) is the reciprocal of frequency: \[ T = \frac{1}{f} = \frac{1}{1/7} = 7 \text{ seconds}. \] So, to summarize the answers: - Maximum displacement: 2 inches. - Frequency: \(\frac{1}{7}\) Hz. - Time for one cycle: 7 seconds.