# Physics Applied to Aristotle's Wheel

A $$20 \text{ kg}$$ wheel consisting of a tire of radius $$R$$ and rim of radius $$r$$ makes contact with two rough surfaces. The coefficient of kinetic friction between the rim and the surface it contacts is $$\mu_{k_1} = 0.2$$ and the coefficient of static friction between these two surfaces is $$\mu_{s_1}= 0.4$$. The coefficient of kinetic friction between the tire and the surface it contacts is $$μ_{k_2} = 0.3$$ and the coefficient of static friction between these two surfaces is $$\mu_{s_2} = 0.6$$. The tire makes exactly one revolution as it moves at a constant speed between the positions shown in the figure. The normal forces, $$N_{1}$$ and $$N_{2}$$, are equal and $$R=2r$$.

What is the magnitude of the force $$F$$ (in Newtons) that must be applied at the center of the wheel to keep it rolling at constant speed?

Use $$g = 9.8 \text{ m/s}^2$$ for the acceleration of gravity.