Time for a new setup!

Rolling wheels across the floor, I've noticed that especially at the end of their runs, they sometimes seem to pick up a bit of speed from nowhere, rolling several meters on the newfound energy. This shows up in the position-time graphs as a bit of a wobble that has made some data difficult to fit well, and makes for big uncertainties on the accelerations. Yesterday, this showed up when I tried a different analysis on the data. Fitting windows of 5 data points to a quadratic, I determined the acceleration for one of the runs, and got the following:

This seems to be common to all of the data. The average is pretty reasonable, but there's a lot more going on here...

The accelerations are terrible to start mostly because the wheels don't take long to cross one tile, so that quantization error is a big deal. This is aggravated by the fact that getting the acceleration from a position vs. time graph involves taking the second derivative, and derivatives generally degrade data. As a quick aside, this is something that I've always rationalized using Fourier transforms. If we have a function written as:

\[f\left(x\right)=a_0+\sum_{n}\left(a_{n}\cos\left(nx\right)+b_{n}\sin\left(nx\right)\right)\]

then its derivative is given by:

\[f'\left(x\right)=\sum_{n}n\times\left(-a_{n}\sin\left(nx\right)+b_{n}\cos\left(nx\right)\right)\]

Note that the higher frequency components (with big $n$) are multiplied by bigger and bigger numbers. This is true of both the data (which is okay) and the noise (not so okay), so that the data gets noisier and noisier looking the more derivatives we take. At any rate, the problem in the early data might be subdued, if not really solved, by taking more accurate data for position and time.

The oscillations later on are particularly concerning, though. They occur over a distance scale of a few metres, which rules out an unbalanced wheel as a culprit. With a radius of about 5 cm, a poorly balanced wheel should give oscillations about once every 30 cm. It seems more likely that these are due to irregularities in the floor. To confirm this, I've laid out a common coordinate system along the floor (choosing a certain tile edge to be zero, going from there, and using that coordinate system for all videos) and looked for commonalities in the oscillations between experiments. It's hard to tell if they have a lot in common -- maybe yes, maybe no.

Clearly I'm going to need to measure this differently if I want to pick out a difference of 0.005 m/s² in acceleration. I'm thinking homebrew photogates over a one metre test section, and I've ordered a handful of laser pointers to see if I can rig something that'll work well in a hallway (otherwise, stray light is always a pain). More on that soon, I hope!

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!

Greetings!

A mousetrap car is a toy car that is powered by a mousetrap, using the energy stored in a wound-up trap to drive the wheels. Cars can be designed to any of several goals, but my favorite is the distance competition, in which the goal is to build a mousetrap car that travels the greatest distance between a standing start and an eventual rolling stop. There are two reasons that I prefer this one, I suppose.

First, it's open-ended -- there is no theoretical upper limit on how far a car can travel. While the average "first attempt" might travel 8-10 metres, it's easy to find video of a variety of designs travelling 34 metres, 42 metres, 49 metres, even approaching 60 metres and steering to fit that run into a gym. The Doc Fizzix website keeps a leaderboard that includes cars that apparently travelled more than 100 metres -- alas, without video.

Second, the distance competition is loaded with physics. A thorough analysis of any mousetrap car requires an understanding of of static friction, rolling resistance, potential and kinetic energy, efficiency, work by conservative and nonconservative forces, inertia, rotational inertia, simple machines and other physics concepts. Doc Fizzix publishes a beautifully illustrated and very comprehensive guide to building and testing mousetrap cars that discusses many of these concepts in that context (see this sample for some idea of what they're about).

Design at the longer distances can be fickle, with very similar designs differing by a factor of two in distance and seemingly small tweaks resulting in large increases or decreases in distance. This sort of uncertainty is fertile grounds for superstition. So what are the most important parameters for long distances? Should wheels be large or small? Heavy or light? Rubbery or hard? How can energy best be transferred from the mousetrap to the wheels? Where are the "bottlenecks" -- the bits that most urgently need to be improved in a basic design?

You'll find a lot of answers to these questions on the web, but remarkably few good scientific approaches to the problem, even from science teachers. Good science must include two things: experimental work that establishes what is true in the world, and theoretical work that makes an attempt to explain why it's true (or, from another viewpoint, to extrapolate to what else might be true). It's probably not surprising that many student designs are thin on theoretical consideration, and take a trial and error approach in any improvements. Also common, though, and no more scientific, is an approach that is strictly theoretical, discussing only what "should work".

I'd like to take a more balanced approach to exploring these problems here, beginning with some experiments to lay the groundwork and hopefully leading to a better mousetrap car. If you have your own work to share, or comments on what I post here, I'd love to hear from you!