Thursday, January 22, 2009

Physics of the Derby *Bonus*

Here is a bonus post on my thoughts on the physics of a pinewood derby car rolling down the track.

There is really only one thing that makes the car move-gravity.
I drew a simplified diagram of the situation. I simplified the car to just a point mass. There are 3 forces on the car: mg- the force from the gravity of the earth, N- the normal force from contact with the track, and f- the frictional forces acting opposite the direction of travel.

The car only accelerates in the downhill direction of the track, I have designated this direction as the +x direction, which makes the normal force in the +y direction. The only force with a component in the direction of acceleration is mg, we can break it into x and y components to get a better look at what's going on.

Since the car doesn't move in the y direction we know that the magnitude of mg in the -y direction is equal to the magnitude of the normal force in the +y direction and those forces cancel.

We are now left with only x forces:

We want to sum these forces and set it equal to ma, or mass times acceleration by Newton's famous F=ma. So we watch our angles carefully and discover that the magnitude of mg in the x direction is mgsin theta.

The rolling frictional force from the wheels on the axle and wheels on the track can be represented with a constant k multiplied by the magnitude of the normal force which is the same as the y component of mg. If we assume that the race will take place in a vacuum then this is the only friction that we need to consider. This gives us an interesting result:

The acceleration of the car and thus how fast it will go down the track does not depend on the mass of the car! Reducing the friction of the wheels is the only thing you need to do to build a fast car. So why does everyone say that if you add weights to your car it will help you go fast?
Well, it might be that I have neglected important factors in my simplifications. I should put the race back in an air filled environment so the spectators don't have to bring their own oxygen. This will introduce another frictional force from the air but from what I've seen so far in my classes, this force has little effect in many cases and often times is neglected. Either we have in vain been putting weights on our cars for years or conventional wisdom is correct and air drag makes enough difference for us to bother with attaching weights.
It all depends on what the coefficient of friction is, which I believe involves the density of air and the shape and density of the car. This can be found experimentally. With that term added our expression now looks like this:

The force of the drag is proportional to velocity. This makes me very curious now and I want to get that track and do a bunch of tests with varying weights and other things to find out how much effect this really has.
The placement of the weight may have something to do with the stability of the car also, to figure all these things out I'd have to do a more complicated analysis and look at the car as something more than just a point mass. I'm not going to do this, and now that I think of it, the urge to do a bunch of tests is leaving me too. I'm feeling content enough to just get those wheels turning smoothly, put weights on the cars till they are 5 ounces and enjoy the race.
I would like your input on the matter though. If you happen to have any insight on these things please let me know, I'm sure I've missed something.


Brandon or Michelle said...

This is Brandon (I belong to Michelle, Jon's younger sister).

As a fellow engineer, I find this a very curious result. Acceleration being independent of weight is consistent with the drop experiment, in which a heavy ball and a light ball are simultaneously dropped from the same height, only to arrive at the ground (or any reference datum) at the same time.

My discussions with "hard core" derby dads (you know the type), generally focus on limiting frictional forces. Things like limiting the car to three contact points with the ground or shaping the wheels to reduce contact area. These strategies imply that friction is a function of contact area, not just mass is often assumed.

I haven't pondered to deeply on these claims, but as the "hard core" guys tend to have the fastest cars, I think they may on to something.

This would be cheating, but I want to build a car that has some initial strain energy. Or maybe an operable mass that could change position during the race? It could start out in a high energy state (relative to the car's Center of Mass), then end in a state that is lower (closer to car's CM). Of course any apparatus that did this would certainly introduce additional drag which would offset the marginal energy gain from the moving mass.

You should revisit this after you take fluid dynamics. How about a reverse air foil to provide uplift to counter the normal force and thus reduce friction? Are the velocities fast enough to make this effective?

Jonathan G.Thwaits said...

lucky me, I don't really have to study fluids. I'm getting deep into Maxwell's equations instead, plus this semester I'm doing a bit with thermo, optics, and relativity just for good measure. If you ever come up with all the answers do let me know! and thanks for the comments!