Circumferential Evidence

Shortly before starting our move to Florida last year, we took our car to have all 4 tires replaced for the 1000 mile trip. On our way back from the dealer, the car's tire pressure monitoring system went off, warning that one or more tires were not properly filled. We checked each one after we got home and all were fine – Consulting the manual told us the system often gives false positives after tire replacements, and needed to be reset. Clearly the system was not directly measuring the pressure, so I was curious how these work. I found a nice explanation from Bridgestone Tire: The system measures the revolution speed of the tire and compares to the speed over the road, in effect measuring the tires' circumference. What I wanted to know is, what sort of conversion is needed to get from circumference to pressure?

The effect of lower pressure will be for the tires to ride lower to the ground, flattening against the road. Here's how I imagined it:

Here the tire has inner radius r, outer radius R, and the axel is a height h above the road. Initially, I tried to figure out the pressure directly: Each tire is applying a force to hold the car up, and force is equal to pressure times the area of the contact patch. That was a bit of a mess, but I realized to detect an under inflated tire, we can instead look at the volume of air inside, which is related to the pressure through the ideal gas law. That turns this into a geometry problem: We want to find the perimeter and area of a circle with a wedge replaced by a triangle. Skipping some intermediate steps, the perimeter is

and the area is

To get the volume, we can just multiply by the width, w, of the tire. To put some numbers to this though, we have to enter the arcane realm of tire sizing.

Our car uses tires of size P235/55R19. The first number tells us the width in millimeters. The number after the slash is the "aspect ratio", which is a percentage given by (R-r)/w. Things get even weirder from there, since the third number is the wheel diameter, equal to 2r, in inches! We can combine these measurements (after converting to a single system of units) to get r, R, and w, then plug those in along with a couple values for h. We can then look at how the tire circumference changes with volume of air lost from the tire:

This plot covers the range from fully inflated to riding on the rims, so I was surprised it stayed so linear. While thinking about this, I remembered an episode of Mythbusters where they looked at improved gas mileage from over-inflated tires – Here's a similar analysis from Popular Mechanics. That article attributes the effect to lower rolling resistance, but my results make me wonder if part of it is getting increased circumference, resulting in the same number of revolutions taking you a greater distance: For the two ends of the plot above, each mile could require 693 to 718 revolutions.