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My question is whether, for a pre-determined linear speed, 4-wheeled cars have greater mechanical efficiency than cars with $2 n$ wheels where $n \geq 3$. I take it for granted that the car rolls on a flat surface.

I'm not a physicist but here are my arguments for this hypothesis:

  1. A four wheel car is more stable than a two wheel or three wheel car.
  2. Given that $ F= \mu N$ where $\mu$ is the coefficient of friction and $N$ is the normal force, the area available for contact is unimportant for traction.
  3. Whatever advantage additional pairs of wheels might offer would need to off-set the heat lost in additional pairs of wheels and the extra angular momentum required to move those wheels.

Now, let's denote the masses of the first two pairs of wheels as $m_1$ and $m_2$ located at the front and back respectively. Additional pairs of wheels shall be denoted by $m_i$ where $i \geq 3$. For a car with $ 2 n$ wheels and pre-determined mass(700 kg), I argue that:

  1. $\forall i \space m_i \geq min(m_1,m_2)$
  2. Assuming constant linear speed $v$, the extra angular momentum due to $m_{i \geq 3}$ is approximately:

$ \Delta L=\sum_{i=1}^{n-2} 2 L_i = 2 \sum_{i=1}^{n-2} I_i w_i= 2 v\sum_{i=1}^{n-2} r_i m_i $

where $r_i$ is the radius of each wheel.

  1. In a bend/arc that's approximately circular, the extra work done would be approximately:

$\Delta W =\sum_{i=1}^{n-2} 2 W_i = 2 \sum_{i=1}^{n-2} \frac{m_i v^2}{r} 2 \alpha r= 2 v^2 \sum_{i=1}^{n-2} m_i 2 \alpha $

where $\alpha$ is the angle of the bend, $v$ is the linear speed and $r$ is the radius of the arc.

Extra pairs of wheels might help with improved weight distribution on the axles which might seem useful but this is not an issue for F1 cars which aren't massive. It follows that we would now want to minimise $ \Delta L$ and $ \Delta W$ which is possible when $m_{i \geq 3} = 0$. Note that I've so far said nothing about heat loss as it's not clear to me how this would be modelled.

I suspect that heat loss is a non-linear function of the angular speed of a wheel and varies linearly with the a wheel's width. In fact, I'm particularly interested in hearing how a physicist would model this problem.

Note 1: Once upon a time Formula 1 had six-wheeled cars, the P34. However, as the following article recounts, they didn't do too well.

Note 2: I think my arguments might be incomplete. If so, I'd like to know how they might be improved.

Note 3: I think physics is the right degree of abstraction for this problem as I'm not interested in actually engineering a particular car. Like Carnot, who modelled heat engines, I'm interested in modelling a general class of machines using physics.

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closed as unclear what you're asking by David Z Sep 16 '16 at 11:34

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    $\begingroup$ Have a look at this en.wikipedia.org/wiki/Tyrrell_P34 $\endgroup$ – Farcher Sep 16 '16 at 9:57
  • $\begingroup$ And vaguely related, for serious petrol heads only though en.wikipedia.org/wiki/Brabham_BT46 $\endgroup$ – user108787 Sep 16 '16 at 11:21
  • $\begingroup$ Stability, as with the latest military jets, is a two sides issue. Too stable and vehicles are slow to respond, too unstable means increased response at the price of twitchiness $\endgroup$ – user108787 Sep 16 '16 at 11:26
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    $\begingroup$ I don't actually see a question in here. $\endgroup$ – David Z Sep 16 '16 at 11:34
  • $\begingroup$ @DavidZ I'm sorry for leaving any ambiguity. My question is whether, for a pre-determined linear speed, 4-wheeled cars have greater mechanical efficiency than cars with $ 2 n$ wheels where $ n \geq 3$. $\endgroup$ – user29305 Sep 16 '16 at 12:12