# Angle of rotation of a wheel over time

I am modelling the motion of a wheel that should rotate at a constant speed over time. Think of a Ferris wheel or a car wheel.

I am trying to determine at what angle the wheel will be at about the center at any given time.

Rotating in a friction free, gravity-less perfect environment, such that r and m etc. are irrelevant.

My first guess would be something like this, but I have a feeling this won't hold true for all quadrants of the rotation.

$$\arctan\left(\frac{\cos(t)}{\sin(t)}\right)$$

Can anybody help? It's driving me wild and I won't be in front of a computer to help me figure it out for a good few days!

Furthermore, I then want to work out an equation to accelerate and decelerate the angle of the wheel rotating. So any help on that would be great too!

• What information do you have? For instance, does the wheel rotate at a constant angular speed? Do you know that angular speed? – Dan Rust Sep 28 '13 at 11:30

Do you have angular speed of your wheel? If yes (let's note it $\omega$), then the angle is $$\phi=\phi_0+\omega t,$$ or, if you want a value in $[0,2\pi)$, $$\phi=2\pi\left\{\frac{\phi_0+\omega t}{2\pi}\right\},$$ where $\{x\}$ denotes the fractional part of $x$.
Let's suppose we have a wheel of radius $R$ rolling on a straight line without sliding (i.e. without wheelslip). Suppose also that the coordinate of center of our wheel is $x(t)$. For simplicity we take $x(0)=0$ and we take the angle at $t=0$ to be zero.
Note that the angle of rotation is a function of coordinate: indeed, if the wheel rolls a length $l$, then its angle changes by $\frac{l}{R}$. Thus the angle is $$\phi = \frac{x}{R}.$$ As previously, we can take it in $[0,2\pi)$:$$\phi=2\pi\left\{\frac{x}{2\pi R}\right\}.$$
The speed of our wheel is $\dot x$, hence local angular speed is $\frac{\dot x}{R}$. If you take zero acceleration (i.e. constant speed), you will reduce the formulas to the case above. By derivation with respect to time you can obtain expressions for angular acceleration as a function of $\ddot x$, etc.