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> We want to find the force at each tire. So by rearranging, we get F = PA. The pressure is gotten by a tire pressure gauge.

Can someone explain why the air pressure inside the tire is the same as the pressure that the tire applies to the ground? Ignoring sidewall stiffness as pointed out to others, why must this be true?

Because the tire would deform, if there is a net force on it. Therefore the force from the inside is the same as the one on the outside. (However, that argument has only any chance to hold at the contact point from the tire to the ground, since obviously everywhere else the tire does not perform because of stiffness of the tire material.)

It actually isn't true at all in practice. The inflation pressure and contact pressures vary dramatically even for a static tire due to curvature, tread depth, and many other factors. See e.g. https://www.sensorprod.com/news/white-papers/tcp/wp_tcp.pdf

The force must be the same (because the net force must be zero, otherwise the tire surface would move up or down), and the area is the same on both sides so the pressure must be the same.

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