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Choosing the right tyre pressure is important in order to optimise the tyre setup of a F1 car. The best tyre pressure is one that results in the greatest area of the "contact patch" (the area of the tyre that makes contact with the track surface). An under-inflated tyre will have a large contact patch but will not support the car very well, increasing the risk of the bottom of the car scraping the surface of the tarmac. On the other hand, an over-inflated tyre will have a small contact patch and is therefore not providing the maximum level of grip.
This exaggerated diagram shows the principle well.
Different tyre pressures will also result in different rates of wear of the tyres. This is why in some F1 races two team-mates racing in identical cars can have remarkably different behaviours from the same compound of tyres, simply because they have each chosen a different tyre pressure to race with. The driving style of the driver also plays a part.
A typical tyre pressure in F1 is approximately 1.1 bars, whereas a normal road-going tyre pressure is in the region of 2.2 bars. This is because a F1 car weighs only 600kg and the average family car about 1000kg, therefore less pressure is needed to support the lower weight.
Another interesting point is that most of the F1 teams use dry nitrogen gas to inflate their tyres instead of air. This is done for two reasons. Firstly the moisture content of air is variable depending on the local weather conditions and this differs considerably between some of the exotic locations on the GP calendar. By using dry nitrogen gas the tyres will behave in a predictable way wherever they are being used. The second reason is that air is a mixture of nitrogen (78%) and oxygen (21%). Oxygen gas is far more reactive than nitrogen and at the high operating temperatures of F1 tyres (> 100°C) the oxygen reacts with the tyre, reducing the total pressure inside. Using pure nitrogen removes this problem and tyre pressures remain far more consistent.