Given the following position function: \begin{equation} x(t) = 4.00 \cos(3.00\pi t + \pi) \end{equation} I have to extract the following information which are the following:
- Frequency
- Amplitude
- Period
- Phase Constant
The system is said to be operating under the guidelines of radians. The goal of this post is to provide a verification of my answers, and methods.
To get an answer to my solution I decided to list all relevant equations to the information which I needed to extract from the position equation.
- Position Function \begin{equation} x(t)=A\cos(\omega t+\phi) \end{equation}
- Frequency \begin{equation} \omega = 2\pi f \end{equation}
- Period \begin{equation} T = \frac{2\pi}{\omega} \end{equation}
Using this I retrieved the extracted information by doing the following:
After I put down the formula for the position function I saw that the amplitude was $A = 4.00\ \text{meters}$. I also deduced that since one takes the cosine, and sine of angle measures I decided that the phase constant was $\phi = \pi \ \text{radians}$. I saw then that the angular frequency was the following $3.00\pi$, so then I used the relationship between frequency and angular frequency and wrote the following rearranged equation. \begin{equation} \frac{\omega}{2\pi} = f \end{equation} Using that I retrieved the frequency $f=\frac{3.00\pi}{2\pi}=\frac{3}{2} \ \text{Hz}$. Another observed relationship is the one between period and frequency leading me to say that the period is $T = \frac{2}3 \ \text{s}$