$(20$ pts $)$ Use the potentiometer shown in Fig. $3$ to answer the following. The input $x_{in}$ is the displacement of the potentiometer wiper, $R_x$ is the resistance between the potentiometer output leads, $V_{\text{o}\text{u}\text{t}}$ is the output voltage, $R_p$ is the maximum resistance of the potentiometer, and $L$ is the maximum amount of wiper travel.

$R_p$ : Potendinate

Figure $3$: Schematic for Problem $5.$

$2.$ Derive an expression for input$-$output relationship $(\frac{V_{\text{o}\text{u}\text{t}}}{x_{in}}).$

b$.$ Is the system a zero$-$order system? Why or why not?

b$.$ What is the gain or sensitive of the system?

d$.$ What is the bandwidth of the system?

$.$ Does the system have amplitude linearity?

f$.$ Does the system have phase linearity?

$5.$ Suppose the wiper position was attached to a linear actuator that could move the wiper up and down. If

$x_{in}=$Asin$(t+)$

find an expression for the $V_{\text{o}\text{u}\text{t}}.$