**Harmonic Motion Analysis**The position of a particle in cm is given by the equation \( x = (8) \cos{6\pi t} \), where \( t \) is in seconds.### Questions:**(a)** What is the frequency?\[ \text{Frequency:} \_\_\_\_ \text{ Hz} \]**(b)** What is the period?\[ \text{Period:} \_\_\_\_ \text{ s} \]**(c)** What is the amplitude of the particle's motion?\[ \text{Amplitude:} \_\_\_\_ \text{ cm} \]**(d)** What is the first time after \( t = 0 \) that the particle is at its equilibrium position?\[ \text{Time:} \_\_\_\_ \text{ s} \]#### In what direction is it moving at that time?- \( \bigcirc \) in the positive direction- \( \bigcirc \) in the negative direction### Explanation of Terms:1. **Frequency (f)**: The number of oscillations per second.2. **Period (T)**: The time taken for one complete cycle of oscillation.3. **Amplitude (A)**: The maximum displacement from the equilibrium position.4. **Equilibrium position**: The position where the net force on the particle is zero, typically at \( x = 0 \).### Steps to Solve:**(a) Calculating Frequency:**\[ x = 8 \cos{6\pi t} \]The term inside the cosine function \((6\pi t)\) has the form \( \omega t \), where \( \omega \) is the angular frequency. Here,\[ \omega = 6\pi \]The relationship between angular frequency (\( \omega \)) and frequency (f) is:\[ \omega = 2\pi f \]Hence,\[ 6\pi = 2\pi f \]\[ f = \frac{6\pi}{2\pi} \]\[ f = 3 \text{ Hz} \]**(b) Calculating Period:**The period \( T \) is the reciprocal of the frequency (\( f \)).\[ T = \frac{1}{f} \]Given \( f = 3 \text{ Hz} \):

Name: **Harmonic Motion Analysis**The position of a particle in cm is given
Uploaded: 2024-03-20T06:58:04.000Z
Channel: Bartleby
Description: **Harmonic Motion Analysis**The position of a particle in cm is given by the equation \( x = (8) \cos{6\pi t} \), where \( t \) is in seconds.### Questions:**(a)** What is the frequency?\[ \text{Frequency:} \_\_\_\_ \text{ Hz} \]**(b)** What is the