How to Calculate gear ratio for potentiometer I am not a mathematician so please bare this in mind. I am designing a computer steering wheel, using real car parts (fitting the steering wheel from a hatchback via gears to a potentiometer which is wired to a game controller). The steering column needs to turn 900 degrees. However the degrees of turn that my potentiometer has is 270 degrees (135 degrees +/- from centre).
If the gear attached to the steering column has a diameter of (x) and the gear has a diameter of (y). What is the value of X and what is the value of Y.
Ideally although I do want the answer, more importantly I would like the formula about how to calculate two gear sizes given the degrees that both can/should turn.
Also does the physical distance between two gears effect the turn ratio? To my knowledge it shouldn't, but I thought I would check.
Thanks in advance.
 A: So, a full turn in the steering column walks a distance of $\pi x$, then 900 degrees = 2.5 turns, so the steering column walks $2.5 \pi x$ while rotating 900 degrees.
A full turn of the gear walks $\pi y$, so as 270 degrees = 0.75 turns, the gear will walk $0.75 \pi y $. Now as they are attached the walking of both gears are the same, so you have the equation
$$2.5 \pi x=0.75\pi y \Leftrightarrow x = \frac{3}{10} y$$
Of course the diameter and the distance are related, as de distance between the centers is z, you should have $$ 2z = x + y$$
So, solving this equation on $z$ you should get
$$2 z = y \frac{13}{10} \Leftrightarrow y = \frac{20}{13} z$$
$$x = \frac{3}{10} y = \frac{6}{13} z$$
In general, if two gears A, B with diameter $x, y$ and centre distance $z$ have to turn $\alpha$ degrees for A, and $\beta$ degrees for B. The fundamental equations are
$$\alpha x = \beta y$$
$$x + y = 2z$$
And the solution is
$$y= z \frac{2}{1 + \frac{\beta}{\alpha}}$$
$$x = z \frac{2}{1 + \frac{\alpha}{\beta}}$$
