Blogs, Calculus, Modeling

All that (air) pressure

Over the past few days, I have had the opportunity to spend time with my brother and his girlfriend.  They were driving through on their way from Colorado to Florida, and they were gracious enough to stop for an extended weekend and spend time with me.  In order to take advantage of the long weekend, Paula, Artie and I drove over to Charlottesville, Virginia and shared a hotel room with Joe and Stella.  From there, we were able to drive up to Monticello and see Thomas Jefferson’s home.  We also got out to Shenandoah and  did a little hiking.

While there were many exciting things that happened throughout the weekend that I could and may talk about later, the one thing I found myself thinking about were my tires.  In particular, while I was driving, I was thinking about all the time they were spinning around and getting us where we wanted to go.  This wouldn’t have stuck in my mind so much, except, while I was driving one day, I managed to run right into a curb.   This then caused the following,


Well, this in fact ruined our Saturday afternoon plans because my brother and I spent the next few hours calling places and driving around in his truck with the flat tire in the back in order to find a place that was open on a Saturday and actually had a tire that would fit.  We did finally find such a place, Charlottesville Tire, and they were able to very quickly get the tire changed with the exact tire we had on the car already.  They even had a used tire with about the same amount of tread, so I don’t have to worry about the new tire grabbing better than the others.  All together, I was extremely happy with the ways things turned out, I just wish I hadn’t blown up the tire to begin with.

While I wasn’t going that fast, I couldn’t tell you exactly how fast I was going.  I also couldn’t tell you what angle I hit the curb or other such details to try to determine the force the tire experience that led to this blow out.  I had wanted to be able to determine this, but lacking this information, I instead settled for determining the amount of a force the tire feels under normal circumstances.  At least this would give me a baseline of what a tire experiences, and I could then say the force had to be at least that much.

In order to determine this force, we will need the tire pressure and the surface area of the tire.  That is, the pressure is applied to the entire surface area of the tire it comes in contact with, so if we multiply these two we will get the total force felt by the tire.  For the tire pressure, I will assume that I was at the recommended value of 36 psi (from the sticker on the door).  While there are variations in tire pressure, I do try to keep it at this level.  Here I would note that this is the relative pressure of the tire to the air.  That is, at sea level, the absolute air pressure is 14.7 psi, therefore the absolute air pressure of the tire is 50.7 psi.  However, since the force due to pressure will cancel between inside and outside pressure, the total force can be found using the difference, that is, the relative pressure.

In order to calculate the surface area of the tires, I had to find exactly how big the tires were.  At Tire Size, I was able to find the dimension of my tires.  They have an inner radius of 9.5 inches, an outer radius of 11.6 inches and a width of 9.3 inches.  We would now have that the surface area representative of the tread portion of the tires is then going to be 2π(r)(w)=2π(11.6)(9.3)≈677.83 square inches.  In order to find the surface area of the side wall, we would then have πR2 -πr2=π(11.6)2 -π(9.5)2≈139.20 square inches. Since you have an inner and outer side wall, this will give a total surface area of 677.83+2*139.20=956.23 square inches.  Note that I’m not including the inner portion of the tire, since the air is pressing against the wheel and not the rubber of the tire.

Combing these together, we see that the tire feels a total force of 34,424.28 pounds of force.  Since the tire also states that the maximum air pressure should be 50 psi, we would get that this would result in a force of 47,811.5 pounds of force.  This would then suggest that the accident supplied at least the difference of 13,387.22 pounds of force to the tire which then caused the tire to rupture.  While this is not necessarily true, as the force may not have been applied to the whole tire, I feel like it can provide me insight into how hard I hit the curb.

I hope you stay safe on the road and don’t have to experience such a thing.  However, since I did go through it, I hope you enjoyed looking at the relation between forces, pressure and surface area.  I also plan on working this experience into some upcoming lecture, because students seem to love to hear these types of stories as they do impose a human element into the lecture.

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