A first look at rolling resistance

Friction of all kinds is the enemy in any distance competition. For a mousetrap car, this takes at least four forms:

• Friction on any parts between the mousetrap and the wheels;
• Friction or rolling resistance where the axles meet the frame;
• Rolling resistance of the wheels on the ground; and
• Air drag against the car

Today, I want to discuss the third of these.

Rolling resistance is caused primarily by the deformation of the wheels where they meet the ground. It's a problem of particular interest to automotive engineers, who have developed some very thorough models for it, and physics students, who would prefer a less-thorough model. For the latter we have this:

\[\begin{align}F_{r}&=\mu_{r}F_{n}\end{align}\]

which should give a constant acceleration that doesn't depend on the mass of the wheel (mass appears through $F_{n}$, but is cancelled in the acceleration since $F=ma$) or the distribution of mass on the wheels. Still, it seems a common view that lightweight wheels are a must on any successful mousetrap car. So are they? First, an experiment...

That's me rolling a simple axle and wheels down a linoleum hallway. I've modified the mass and moment of inertia of the wheels in this experiment using pairs of rare earth magnets fixed to the wheel. With the magnets shown in the video, the mass of the wheels and axle is increased by almost 50%.

The linoleum makes a convenient surface because: (1) it's the surface we most often test on at my school, anyway; and (2) it's got regular markings that make it a convenient and fairly accurate measuring tape. From the video, I've recorded the frame number at which the wheels cross each division between linoleum tiles (at least where I could make it out). This is easily converted to a distance-time graph like the following (which corresponds to the video above):

Here the blue '+' markers are the measured positions at each time (the gaps are where I couldn't make out a division from the video), while the red circles are the positions for motion with an acceleration of -0.0366 m/s². The fit, as you can see, is excellent. Doing the same for wheels with 2 pairs of magnets on each wheel and with none at all, I got the following:

Weights	$a$ (m/s2)
0	-0.0450
2	-0.0411
4	-0.0366

The trend is clear -- placing more weights on the outside of the wheel reduces rolling resistance. What I need to know now is whether this is an effect of the increased mass, or the increased moment of inertia. I suspect the latter.

A simple experiment will discern between the two. Where I've placed the magnets very near the edge of the wheel on the runs above (as shown in the video), I can place them instead about halfway out. This won't change the mass, but it will change the moment of inertia. Results, I hope, on Monday, then time for some theory!