IMHO , yes we have the 3 factors as described by Neels.
But in the scenario of a tyre on asphalt we do not have the mass of a smooth block - sliding down a smooth plane on an angle , generating a prescribed Coefficient of Friction, as we all saw in Physics 101.
This is a specific context ( and the only one ) where contact area is classically deemed irrelevant.
But with one element having the highly deflective " sticky " characteristics of " rubber " and the other being an entire surface of solid peaks and valleys , any downward force on the rubber increases the mechanical adhesion
between these two very dissimilar surfaces.
Increase the area of interlocking adhesion ( contact patch area ) and the number of adhering points of mechanical contact increases.
Thus increasing the mechanical grip level.
This gives rise to the concept of the traction circle , wherein a downward force acts on an area of rubber in contact with the asphalt , and this generates an available level of grip that is a finite metric.
Increase the traction used in the direction of travel ( braking or accelerating ) then the level of traction available perpendicular to this ( sideways cornering ability ) is reduced , and visa versa.
Proof of this lies in the fact that a softer rubber in the contact patch area , will be torn off the underlying carcass , and be left behind on the asphalt faster than a harder version.
This torn off rubber adheres to the asphalt to such an extent that in Moto GP , and even more so in karting , there is , in certain parts of the track no asphalt to be seen.
Then we have sticky rubber adhering to even more sticky rubber , the result being high sides in MotoGP , and karts tipping up on the two outside tyres, due to so much mechanical grip between the surfaces.
Thus a wider tyre , or a softer tyre , does increase the Coefficient of Friction ( grip ) , between the two surfaces - as neither are smooth by the classic definition used in Physics 101.
Ive got a thing thats unique and new.To prove it I'll have the last laugh on you.Cause instead of one head I got two.And you know two heads are better than one.
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