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So, the total energy of thegas does not change. However, the average
energy ofeachmoleculedoes, ofcourse, increase, andthat iswhat defines
the temperature(and part of the comfortlevel of the occupants). At the
sametime, thetotal number ofmolecules in theroomdecreases. In essence,
we burnwood toforcesome of themolecules toshiver outside theroom
(this problem was first discussed in Nature 141 , 908 (1938)).
4.2 Space Station Pressure (MIT)
The rotation of the station around itsaxis isequivalent to the appearance
of an energy where is the mass of an air particle and
R is the distance from the center. Therefore, the particle number density
satisfies theBoltzmanndistribution(similar to the Boltzmanndistribution
in a gravitational field):
where is the number density at thecenter and is the tem-
perature inenergy units. The pressure isrelated tothe numberdensity
simply by So, at constanttemperature,
Using the conditionthat theacceleration at the rim is we have
4.3 Baron von Münchausen and Intergalactic Travel
(Moscow Phys-Tech)
The general statementthat aclosed systemcannot accelerate as a whole
in the absence ofexternal forces is not usually persuasive to determined
inventors. In thiscase, he would make the point that theforce on the rope
is real. To get an estimate of thisforce,assumethat theballoon is just
above thesurface of theEarth and that thedensity of air is approximately
constant to 2 km. Archimedestells usthat theforce on the rope willequal
the weight of theair,mass excluded by theempty balloon (given a
massless balloonmaterial). We then may use theideal gas law
SOLUTIONS