The force of friction is directly proportional to area of contact. But then why do the tyres of trucks and other large vehicles have big gaps in them? It reduces the area of contact so frictional force should be else. But by observation it is seen that it provides more friction. Can you please explain the reason for it?
[Physics] Friction on tyres
everyday-lifefriction
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It's a surprisingly complicated question. Given your mention of friction, probably the main point is that for a car tyre the friction is not linearly dependant on load. Wikipedia has some information about this here.
If you had perfectly smooth surfaces the friction is actually proportional to the area of contact and independant of the load. This is because friction is an adhesive effect between atoms/molecules on the surfaces that are in contact. However in the real world surfaces are not smooth. If you touch two metal surfaces together the contact is between high spots on the two surfaces so the area that is in contact is much less than than the apparent area of contact. If you increase the load you deform these high spots and broaden them, so the effect of load is to increase the real area of contact. The real area of contact is approximately proportional to the load, and the friction is proportional to the area of contact, so the friction ends up being approximately proportional to the load.
However a rubber type is a lot softer than metal, and a road is a lot rougher than a metal plate. Even at low loads the tyre deforms to key into the irregularities in the road, so increasing the load has a lesser effect. That's why you get the sub-linear dependance described in the Wikipedia article.
But this is only the start of the complexity. If you use a wider tyre the contact patch area isn't necessarily bigger. A wider tyre has a wider shorter contact patch while a narrow tyre has a narrower longer contact patch. The contact patch area depends on the tyre pressure, the deformation of the sidewalls and probably lots of other things I can't think of at the moment.
And anyway, if by "grip" you mean grip when cornering, the grip isn't just controlled by the contact patch area. When a car is cornering the contact patch is being twisted. This is known as the slip angle. The wider shorter contact patch on a wide tyre has a smaller slip angle and as a result grips better.
It's not correct that runners lean forward to begin a race in order to increase friction. They lean forward because otherwise, they would experience no propulsion whatsoever because static friction is zero when the runner is completely upright.
When the runner leans forward and flexes his leg muscles, he exerts a horizontal force on the track in the backward direction. The track responds by exerting an equal and opposite frictional force (unless there is slipping) on the runner in the forward direction that propels him forward.
Generally speaking, the more a runner leans forward at the start, the larger the horizontal component of the force exerted by his legs against the ground, and the larger the frictional force he will experience. As a result, his initial acceleration will be greater.
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This is largely counteracted by the reduction of the normal force per unit area with a larger area of contact. Taking this to the extreme yields the physics 101 assumption that area of contact plays no role in the frictional force. Strictly speaking, this is of course incorrect. It's physics 101, after all.
One need only to look at cars built for competitive racing to see that this is indeed incorrect. The designers of those cars go out of their way to shed all excess weight. Race cars would use the skinniest (and lightest) tires possible if surface area didn't matter. Race cars from 100 years ago did indeed use the skinniest of tires. Developers of race cars quickly found that tire size does matter. The relationship however is nowhere near linear.
If you know the road will be perfectly dry, the best tire to use would be a slick. What if the road isn't perfectly dry? Slicks and wet roads are a dangerous combination. With slicks, the coefficient of friction can be over 0.9 on dry pavement but less than 0.1 on a wet road. Tires intended for everyday use have grooves. These grooves do two things. One is that they give the water somewhere to go so that there is some water-free contact between tire and road. The other is that they channel water away from the tire. While those grooves do hinder friction on perfectly dry pavement, they drastically improve performance (compared to slicks or skinny tires) on wet pavement.
Reference: Personal. The tires on my car are 295/35ZR18 99Y (rear), 275/40ZR17 98Y (front). That means if I was foolish enough to desire to do so, I could drive in excess of 300 kph on dry pavement without worrying that the tires might melt (and my car supposedly can do that and more). Wet pavement is a different beast. My car has a desire to go sideways on wet roads -- and I don't have slicks.