Consider 2 wires, A and B, tied together at both ends, and call the point they are tied together point X. Wire B does not bend (it stays fixed) and only wire A bends. You need to measure the angle of wire A relative to wire B. When the wires are fully bent around point X, and are side by side, the angle is 0. When wire A is extended outward in the same plane as the top of wire B, wire A would be at 90 degrees. Assume that you have a large single–turn potentiometer (figure 1, maximum resistance = 100 kilo-ohms) where the wiper can rotate freely from a center on point X . Assume the maximum angle to be 120 degrees.
1) Derive an equation to describe the resistance measured across the output voltage (see figure 1) as a function of the angle.
2) Design an op-amp based circuit to output a voltage corresponding to the angle of wire A relative to wire B. Choose an input voltage and resistance values such that the maximum output of the circuit is either +5V or -5V.
3) Derive an equation that relates the voltage output to the angle of wire A relative to wire B.
This is part of my problem...I'm not really sure what the relevant equations are. I know that for a basic potentiometer, it can be thought of as a voltage divider. One end of the resistor where the output is placed is R1 and the other end is R2, so RL = R1 + R2. And then you have a fraction setting x (Im not really sure what this is, but its in my notes..). So R1 = x*RL and R2 = (1-x)*RL. So the voltage divider equation comes out as Vo = Vi(R2/RL) = (1-x)*Vi. This is all I know.
The Attempt at a Solution
Firstly, I can draw and attach a picture of the setup of the two wires if anyone feels I need to clarify it. My professor said I am thinking of it correctly.
For 1, I have a feeling that if I can use the above I can relate R and theta. The problem is that I'm not fully understanding how the potentiometer circuit works, or what x represents and if x is analogous to theta. Surprisingly, I'm having a hard time finding some resources online which further explain the concepts I need.
2 is the scary one. I don't even know where to begin with this one. I am not sure what concept my professor is looking for here but I have no idea how I can design such a circuit. The thing is, I know that 3 shouldn't be bad with 2, I think I OK with deriving circuit equations.
I would really appreciate any help or hints or even resources someone could give me for this problem. Its very weird for me to so completely lost on a problem.
Edit: I am not sure if this should go in engineering or advanced physics. Could a mod help? Sorry I am still new to this.