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Physics Quiz Question
Dallas wrote:
The origin of the question comes from the current edition of the Jeppesen Private Pilot manual. [...] http://img.photobucket.com/albums/v1...allas/Jep2.jpg Context helps. The kicker, and misleading, aspect is this: "...assuming that all other variables remain constant...." First, what the Jeppesen manual states is self-evident from the ideal gas law: P*V = n*R*T So their text is technically correct, but really incorrect as far as describing what *really* happens when some part of the atmosphere heats up more than the surrounding atmosphere. In the real atmosphere the volume V0 that is occupied by some n moles of gas does not "remain constant". But neither does a volume V0 of gas undergo "free expansion" as the temperature rises from T0 to T1, as I've seen some people suggest. To be clear: "free expansion" means the total energy of the n molecules in V0 remains the same as the volume goes to V1 - that is, as the gas expands it does no work, and no work is done on it. (Or put mathematically, "free expansion" says that P0*V0 = P1*V1; that is because the product of pressure and volume yields units of energy!) Of course it is not the case that the energy remains constant because as the gas expands it has to do work against gravity (i.e. it has to push against the surrounding atmosphere). So in general what actually happens is something _in between_ what happens when V0 = V1 (volume held constant) and P0*V0 = P1*V1 (energy held constant). (And to determine P1 and V1 is why thermodynamics texts are filled with imposing looking differential equations! :-)) BUT THE BEST FORMULA (and most relevant to aviators) DOESN'T DEAL WITH VOLUME CHANGES! IT DEALS WITH DENSITY CHANGES! Here is how it is derived: First divide both sides of the ideal gas law by V and rearrange variables: P = R*T*(n/V) But n/V is just a density! So density = n/V and we get: P = R*T*density But R is "just" a conversion constant to make sure all the units work out. If we only want to understand proportionalities we can discard the R. Then dividing both sides by T yields: density = P/T *** SO IMHO THE BEST FORMULATION OF THE IDEAL GAS LAW FOR PILOTS IS: *** **** **** ****** DENSITY = PRESSURE/TEMPERATURE ****** So when Jeppesen said "assuming all other variables remain constant" it was basically saying "assuming the density remains constant." But air density is the ultimate variable of interest to pilots! So Jeppesen was effectively posing an example that said "assuming pressure and temperature vary such that it doesn't affect the lift produced by your wings!" Now *that* is a misleading (and useless) example IMHO! |
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