An object moves in simple harmonic motion described by the equation \( d=8 \cos \frac{\pi}{2} t \), where \( t \) is measured in seconds and \( d \) in inches. Find the following. a. the maximum displacement b. the frequency c. the time required for one cycle a. \( \square \) in.

To solve the problem, we will analyze the given equation of motion for the object in simple harmonic motion: \[ d = 8 \cos \left( \frac{\pi}{2} t \right) \] ### Step 1: Maximum Displacement The maximum displacement in simple harmonic motion is given by the amplitude of the cosine function. In this case, the amplitude is the coefficient of the cosine term. - **Maximum Displacement**: \( A = 8 \) inches ### Step 2: Frequency The frequency \( f \) of simple harmonic motion can be determined from the angular frequency \( \omega \), which is the coefficient of \( t \) in the cosine function. The relationship between angular frequency and frequency is given by: \[ \omega = 2\pi f \] From the equation, we have: \[ \omega = \frac{\pi}{2} \] Now, we can solve for \( f \): \[ f = \frac{\omega}{2\pi} = \frac{\frac{\pi}{2}}{2\pi} = \frac{1}{4} \text{ Hz} \] ### Step 3: Time Required for One Cycle The time period \( T \) is the reciprocal of the frequency: \[ T = \frac{1}{f} = \frac{1}{\frac{1}{4}} = 4 \text{ seconds} \] ### Summary of Results - a. Maximum Displacement: \( 8 \) inches - b. Frequency: \( \frac{1}{4} \) Hz - c. Time Required for One Cycle: \( 4 \) seconds Thus, the maximum displacement is \( 8 \) inches.