# How to model a driven wheel with no slip

I'm working on a design problem, and I'm having trouble figuring out how to model the torque needed to drive a bike tire at a set acceleration.

Let's say I have a tire with normal force between tire and ground Fn, coefficient of static friction between tire and ground u, and diameter d.

What is the necessary axle torque to drive the tire at desired linear acceleration a from standstill without slipping? How great can a be if you want to avoid slip?

Also, the problem doesn't give me a tire mass or moment of inertia. Is this problem possible without those?

## 1 Answer

Assuming the road level, because no information on slope is given, the maximum force before slipping is $$F_{max}= F_n* \mu =mg\mu$$

Then the torque will be $$\tau = mg\mu D/2=I \alpha$$

assuming the mass of the wheel is concentrated at the perimeter its

$$I= mD^2/4$$

$$\alpha = \frac{mD^2/4}{mg \mu D/2}$$ $$\alpha=<\frac{D}{2g\mu}$$

• I think $F_n$ should be $m*g$, not just $m$. – am304 Feb 25 '19 at 13:01
• @am304, yes you're right. I have modified my answer. – kamran Feb 25 '19 at 16:02