Solutions (^223)
where p is the density of the atmosphere. While
determining the pressure of the atmospheric col-
umn, we assume that the free-fall acceleration is
independent o f altitude. This assumption is justi-
fied since by hypothesis the height of the atmosphere
is much smaller than the radius r of the planet
< r).
Using the equation of state for an ideal gas of
mass M occupying a volume V in the form pV
RT and considering that p = M/ V we
Lind that
=--
RT •
Substituting this expression for p into Eq. (1)
and cancelling out p, we determine the tempera-
ture T of the atmosphere on the surface of the
planet:
T PIPth !IGMh
R (^) Rr 2 •
2.16. We must take into account here that the heat
transferred per unit time is proportional to the
temperature difference. Let us introduce the follow-
ing notation: Toth, Touta and Tri, T.r , are the
temperatures outdoors and in the room in the first
and second cases respectively. The thermal power
dissipated by the radiator in the room is ki (T—Tr ),
where kl is a certain coefficient. The thermal
power dissipated from the room is k 2 (Tr —Trt),
where k 2 is another coefficient. In thermal equi-
librium, the power dissipated by the radiator is
equal to the power dissipated from the room. There-
fore, we can write
k i (T — Try) = k2 (Tn. Touti)•
Similarly, in the second case,
kl (T — Tr2) = k2 (Tr2 — T out2) •
Dividing the first equation by the second, we obtain
T —Tr' Tri — Total
T —Tr2 Tr2—Tout2 •
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