# Would a three wheeled vehicle be faster than a four wheeled vehicle of the same weight?

If I have a four wheeled vehicle (small wooden block with metal nail axles and plastic wheels) and apply a force X to it, would it be made faster by keeping one wheel off the ground in order to reduce friction? My thought is that the remaining three wheels would then have more weight on them, and thus more friction -- but is this added force more than offset by the loss of friction in the missing wheel?

Update: After some back and forth with Ruben I think I have gathered the following --

The friction per wheel exists in both contact with ground and to a much greater degree contact of axle to wheel. There is a small and most likely negligible wind resistance component.

                      4 wheeling........3 wheeling

• Friction per wheel = F ...........................4/3 F
• Wind resistance = W ...................................W
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What people usually learn in freshman physics is a model of friction due to Amonton and Coulomb. Let's call it the AC model. The AC model works for solid, rigid surfaces in contact. It fails for fluids and wetted or lubricated surfaces, and it doesn't describe rolling resistance such as what you get with a car or bike tire. At the axle you have a lubricated bearing. At the ground you have rolling resistance. Neither of these is described accurately by AC, so I would be suspicious of any answer that doesn't specifically invoke something beyond AC. –  Ben Crowell Aug 18 '14 at 2:04
Also, you have the additional problem that static friction is actually what makes the car go. You WANT friction. –  Jerry Schirmer Aug 18 '14 at 3:43

This is an interesting question. It looks to me like three-wheeled version is probably faster.

Your vehicle's top speed is set by the relation $P=Fv$, where $P$ is the maximum power available, $F$ is the total frictional force, and $v$ is the top speed. The frictional force has three contributions:

1. rolling resistance

2. friction at the axles

3. wind resistance

Wind resistance probably isn't changed very much by a 3-wheel versus 4-wheel design, and friction at the axles is probably not as big as rolling resistance. Therefore let's focus on rolling resistance.

As described in more detail in my comment, you can't use the standard Amontons-Coulomb (AC) model of friction for rolling resistance. As far as I can tell from wikipedia, an appropriate relation for rolling resistance can be written in the form

$$F_f=C(F_N)F_N,$$

where $F_f$ is the force of friction, $F_N$ is the normal force, and $C(F_N)$, unlike in the AC model, does depend at least somewhat on the normal force.

If $C$ were independent of $F_N$ as in the AC model, then it wouldn't matter how many wheels we had. Four wheels would give a certain amount of friction per wheel, which would be multiplied by four. Switching to three wheels would increase $F_N$ at each wheel, and therefore the friction per wheel, by a factor of 4/3, but this would only be multiplied by 3 wheels, so the effect would cancel out.

But $C$ does depend on $F_N$. The WP article has some information on how $C$ depends on $F_N$ for railroad cars and for pneumatic tires that have been optimally inflated for the load. The result appears to be that $C$ decreases with an increase in $F_N$. Therefore the result appears to be that the three-wheeled vehicle would be faster.

If fewer wheels give better efficiency, the question would be why we don't all ride around on unicycles.

For cars, I think four wheels are chosen for stability (and maybe handling?).

For train locomotives, the large number of wheels is so that the locomotive can be heavy (and get good traction) without damaging the tracks.

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To answer your question on unicycles, I think you'd likely find that as you try to loose wheels, you increase the weight that bearings have to support, and this increases their drag: in the extreme, bearing drag would become the dominant factor, which of course your model can't see. –  WetSavannaAnimal aka Rod Vance Aug 11 at 23:39

Friction is what keeps the wheels from spinning (i.e. traction), the friction that you want to reduce in order to gain speed, is air resistance. Removing a wheel adds more strain (friction) to the axes, probably making the car slower (keep in mind that the 4rth wheel still has some friction even though it is removed).

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This vehicle is small-scale (pinewood derby car) so I'm not worried about air resistance. Your comment about added friction to the remaining three axles makes sense but I think that is part of the question -- is the added friction there more than the friction lost by removing the fourth wheel from play? –  Ethereal Jan 10 '14 at 13:47
I do think the added friction is more, because in this case the car becomes unstable too (which increases the normal force the axes apply on the wood at the 2nd and 3rd wheel (hence increasing friction)). So unless the car is made stable somehow, I suspect that removing a wheel will cause the car to move slower. –  Ruben Jan 10 '14 at 13:57
Also, removing the 1st wheel will probably cause the 4rth wheel to bear a much lower load, meaning that 2 wheels are essentially doing nothing and just adding extra friction. A complete redesign of the car would be needed to compensate for this fact. –  Ruben Jan 10 '14 at 14:01
What do you mean when you say that the car will become unstable? The weight will be concentrated toward the back of the car, and I think that I can assume that the rear wheels will both remain in contact with the surface at all times. As for the wheel(s) not in contact, why would they be adding friction? Are you referring to the wind resistance present in the unmoving wheel(s)? –  Ethereal Jan 10 '14 at 16:08
@ethereal Say the top left wheel is removed (1st wheel), if you were to push down on the car a bit, it would pivot on the 2nd and 3rd wheels. That is why they will have most of the weight on them (because the car is longer than it is wide). As for the friction: I imagine the axes as small metal bars connecting the 1st and 2nd, and 3rd and 4rth wheels. That is why the axle would still hinder the movement of the other wheel. –  Ruben Jan 10 '14 at 22:11

Each wheel needs to be as lightweight as possible compared to the weight of the car, so all the kinetic energy goes into forward motion of the car, not into rotary motion of the wheels. That is an argument for fewer wheels. At the same time, larger wheels will have less rolling friction than smaller ones.

The wheels need to have a hard outer surface, not rubber, so you don't lose energy flexing the rubber.

The wheels need to spin easily with as little friction as possible, and not wobble. Like if you spin them with your finger, they should continue spinning for a while, and not get into a wobbling action that will really create friction.

The weight of the car should be more toward the front, for directional stability. If the weight is toward the rear, the slightest sideways force will tend to turn the car, making the sideways force worse and causing the car to slide against the side of the track. (In airplanes this is called a ground loop.)

ADDED: As I think about it, it might even make sense to go down to just two wheels, to reduce rotational energy even more. Then have the weight slightly forward of the axle, and have a teflon or low-friction skid in front. (Think of it like a dragster car - most of the weight on the wheels in back, very little in the front.)

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The top speed of the vehicle has nothing to do with how much energy you have locked up in the kinetic energy of the wheels. Most of this answer doesn't address the question or connect logically to the issue of 3 versus 4 wheels. –  Ben Crowell Aug 18 '14 at 2:06
@Ben: It certainly does. The car starts with a certain amount of potential energy. At the end its kinetic energy consists of kinetic energy due to forward motion plus rotational kinetic energy of the wheels. The more rotational energy there is in the wheels, the less there is in forward motion. You could look at it as - the necessity to spin up the wheels acts as a drag on the car. The only possible effect of 3 instead of 4 wheels is fewer wheels to have rotational kinetic energy. –  Mike Dunlavey Aug 18 '14 at 12:19
This answer deals with what is likely the underlying issue facing the OP: that of the Pinewood Derby car, in which a small block of wood is made to roll down a ramp. First car to the finish wins. In such a scenario the kinetic energy of the wheels does play a role - but it is not ~friction~ that is responsible for the loss of speed, but an apparent difference between inertia (translation plus rotation) and gravitational force. –  Floris Aug 18 '14 at 12:45

In my experience with Pinewood Derby cars, the issue is that there are several sources of friction: the friction of the wheel against the "axle" (nail), but also the friction of the wheel against the body of the car, and the wheel against the guide strip on the center of the track. The retarding torque that the latter two can create can be quite significant. Making sure that the wheels are well aligned and that the point of contact with the car is well lubricated is essential. Similarly, anything you can do to maintain the alignment of the car with the track (lateral stability) can have a large positive impact.

As for the basic question 3 vs 4 wheels: I suppose that it would be easier to align 3 wheels than 4 - and in that sense the three wheeled solution has a chance of being faster. However, I am pretty sure that any three wheeled solution will have less lateral stability - and if the car has a tendency to shift left and right, the friction from the side of the wheels (against the track and against the car) all quickly become the dominant force.

As for rolling friction: there is a non-linear factor in "real" friction problems, which argues in favor of spreading the normal force over more wheels. This is because when a wheel deforms, the contact area goes up with force (assuming a constant "pressure" inside the tire), but the size of the deformed region (volume of tire that is deformed during contact) goes up faster than the contact area. This means that there will be relatively more power dissipation in a heavily loaded wheel vs two more lightly loaded wheels.

This is an effect that is negligible for a pinewood derby car - but it is very real on large vehicles with inflated tires and heavy loads. In those situations, more wheels is better.

See for example this picture of the space shuttle in LA (photo from http://blog.apt2b.com/wp-content/uploads/2012/10/10-12-12-Space-Shuttle-Endeavour_full_600.jpg):

)

Lots of wheels to spread the load. This limits the force per wheel, and thus the distortion of not only the wheels, but of the road surface. Fewer wheels means each wheel "sinks deeper" into the road below, and tho would mean that the wheel is always "rolling uphill" - a significant factor in rolling friction for heavier objects. This is nicely visualized in this diagram (from http://www.phy.davidson.edu/fachome/dmb/PY430/Friction/actroll.gif)

But as I mentioned - this is unlikely to be a factor in your little car.

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Friction at the axles would work in favor of the 4 wheeled vehicle (more friction per axle) while wind resistance would work in favor of the 3 wheeled vehicle (more surface area). I'm not sure that friction with the ground would play a factor.

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