I commend CapriRacer for his/her experimental method and for not being put off by the pontifications of so many people. His result has the “ring of truth” to me. I hope anybody inquiring about his method will keep it civil and do so in the spirit of learning.
I got in on the tail end of this discussion and only heard the radio program because we taped it last weekend and I finally got around to listening to it.
Some comments:
There’s nothing wrong with “using math” to calculate an estimated solution–what you need to do, though, is make sure you are modeling “the entire system” and not making any erroneous assumptions Math, by its definition, is a way of describing reality. And if you get an answer that doesn’t match reality, then you must have something wrong in your model or in the equations you’re applying to the model. One approach in the “scientific method” is to do the math based on a model, and then do an experiment to see if you get the same answer. If the two results are close, you have SOME reason to believe that your model is close to representing reality–though it could still be wrong! Anyway, this is why I offer kudos to CapriRacer for doing the calculations well AND doing the experiment and comparing results.
Other comments:
It would seem to me like the Ideal Gas Law (PV=nRT) ought to apply and be accurate enough in this instance. Yes, a real gas does turn non-linear at high pressures, but unless I’m not remembering rightly, in the neighborhood of 30 psi (plus 15 psi atmosphere) it still compresses pretty linearly.
The initial physics professor appears to have made an assumption that the tire will not change in volume when you set the car down on it. It may not LOOK like it expands all over, but keep in mind that any bulging of the tire will be distributed over the whole thing and thus might be pretty slight and hard to notice by eye. Anyway, if the tire did not change in volume, then his analysis using PV=nRT would be correct. If variables V,n,R,and T didn’t change, then Pressure couldn’t change either. He might have had a clue though, about the assumption that the tire doesn’t expand: anybody who has inadvertently put too much air in a tire and then suddenly found that the tire started rubbing on the side of the wheelwell…Also, didn’t you ever change a flat tire, and find that you have to jack the car up higher to get the pressurized tire on. Yes, a tire does change volume when you put more air in it–when you increase the pressure.
The ideal gas law still is valid, though. If you recognize that the volume of the tire does change with pressure, if you measured (or somehow knew) that change in volume, then that number should lead you the change in pressure (by calculation).
One thing to note about the above method is the definition of “the system.” In that instance the system is defined as just the air inside the tire. If you were to draw a picture of the system (a FREE BODY DIAGRAM!) you would draw a dotted line around the air in the tire and define it as the system. No other components involved. And if your equation pertains only to that system, then that’s just fine. So PV=nRT should be a valid mathematical approach. The difficulty is in knowing what the change in volume is.
Other calculation methods were suggested in all the above discussion. Some of them were valid, too. Using a “balance of forces” equation is certainly valid–when/where it applies. So putting the car down on the pavement should produce an equal force in the opposite direction–somewhere. Note that in this first bit of the chain of logic, you define the system as the car (draw a dotted line around it)–as a rigid body-- being let down onto the pavement. If you model this in terms of force vectors, then there has to be some vector pointing upward, which increases in magnitude by, say, 800 pounds at each tire… This force comes from the pavement. Fine. Then you have to recognize that in your next logical step you are then selecting a different system (you have to be a little systematic): Now draw a dotted line (your system definition) about the tire, instead. The pavement is pushing up on it with 800 lbs. Correspondngly, by balance of forces, there has to be something about the tire which is pushing down on the pavement with 800 lbs. The difficulty in this approach comes from the mental model of what happens inside the tire during all this… that the pavement pushes on the tire, and the tire in turn pushes on the air. If it were that simple it would be great. But this model neglects the stiffness of the rubber. I can sit on an unpressurized tire and hardly deflect it–the rubber is pretty rigid. So the 800 lbs pushing on down on the pavement comes partly from the rubber and partly from the air pushing back from the inside. How much comes from each? Dunno. So it seems pretty hard to guess how much the air pressure will have had to increase–and how much downward force comes from stiffness in the rubber. This is another instance in which the model was OK and the equations were applied properly–but there is some unknown (force from rubber stiffness) that keeps you from deducing the answer.
If you have a Finite Element Analysis package in your CAD system, and you input the material properties of the rubber (and the air), you can model the whole tire and get a very accurate estimate of how much force comes from the rubber and how much the air volume changes, etc. You just can’t do it with a simple balance of forces equation.
Not to put too fine a point on it… a Balance of Vertical Pressures method doesn’t hold up in this situation. Pressure upward from the pavement and pressure downward from the air do not balance. I won’t bore everybody further by talking about when balance of vertical pressures would work–but you can see that the difference is in effects from the tire.
Another mathematical method is also valid: Balance of Strain Energies. This a “before and after” balance. Energy is conserved (or, close enough for our purpose). You have the energy of setting the car down on the pavement. That is calculate-able. It’s just force times distance. That number has to match the energy that goes into straining (distorting) the rubber and the air. You could get a good estimate of the energy required to strain the rubber–again if you used an FEA package on your computer model of the tire. It would also give you the energy of compressing the air. (you could also calculate it from the gas law) The two strain energies together have to add up to equal the energy of dropping the car to the pavement.
Any other engineering types out there have other valid methods in mind?
Anyway, do I know the right answer? No. My GUT FEEL (even engineers have gut feel) is that when you drop the car to the pavement there is a little expansion of the tire as a whole (like a stiff balloon), and that the bottom of the tire flattens, and that there is a net small decrease in volume, and a net small increase in air pressure. But I’d have to do a test to verify that. But CapriRacer already did the test! And thanks!