They’ve got me stumped -- how about you?
Just for fun
The mean energy of a particle in an ideal gas is (3/2)kT, so that the total energy of
all N particles in the gas Etot = (3/2) NkT. Consider now one of those particles, and
watch how its energy changes as it keeps colliding with the others. Sometimes that
particle’s energy is large, sometimes it is small.
What is the maximum energy the particle could ever attain? Clearly, the answer is
Etot: there is no way the one particle could ever reach an energy greater than
the total energy of the whole collection.
But this is in conflict with the Maxwell-Boltzmann distribution, according to which
there is a perfectly finite probability of a particle possessing any energy at all --
even one exceeding Etot.
We all believe that a collection of objects, if left alone and undisturbed for long
enough, will reach a state of thermodynamic equilibrium. In this state the
temperature is uniform throughout. The hot things have cooled off, and the cold
things have warmed up, until a final steady state has been reached in which nothing
changes any more and everything has the same temperature.
But suppose one of those things is a greenhouse. Now the ultimate state is steady
and nothing changes -- but the greenhouse is warmer than its surroundings.
We all understand the mechanism whereby this happens: it has to do with infrared
radiation. But isn’t thermodynamics supposed to be independent of mechanism?
Somewhere in the situation there must be something out of thermal equilibrium
with the collection of objects.
What is that “something?”
RUBBER BAND SNAP AND RELATIVITY
The physicist David Griffiths invented this puzzle some time ago. I love it!
Imagine a plank: two nails have been hammered part-way in, and their heads stick
up out of the plank. About those heads a rubber band is looped. Pull the band tight
and then let go: it will snap inwards, striking the nails.
Suppose the left-hand nail is struck by the contracting band first. Then the plank
suffers a slight recoil to the right. This motion will continue until the right-hand nail
is struck-- but during the time interval the plank will have shifted somewhat to the
If we time things carefully the two nails will be struck simultaneously, and the plank
will not shift. But relativity teaches that two events that are simultaneous in one
frame are not simultaneous in another. This seems to say that in one frame the
plank sifts sideways as a result of the snaps, while in another it does not.
FLOATING IN A SWIMMING POOL
When I lie on my back on the floor, I can feel the floor pressing up against my back.
But the physical sensation of floating on my back in a swimming pool is utterly
different – I don’t feel any force on my back at all.
But why? After all, in the pool it is the water that is supporting me, just as in my
living room it is the floor that does the trick.
Here’s a sharper version of the same puzzle. When I stand upright I am aware of the
pressure upwards on my feet exerted by the floor -- I feel a “squish” on the soles of
my feet. But suppose I get into a swimming pool and orient myself vertically. I no
longer feel any sensation at all on the soles of my feet.
But why not?
(Somehow I have the feeling that the solution to this puzzle has to do with the fact
that my own body is mostly water. So me floating is a pool is like a bag of water
floating in the pool. But I have never been able to make this idea work.)
By the way: are you familiar with the famous “Kapitza Problems?” If not, look them up
– they are wonderful! [Pyotr Kapitza was a Russian physicist. I love the fact that the
date of his birth is an anagram of the date of his death.]
Posted: to General News on Sun, Feb 9, 2020
Updated: Sun, Feb 9, 2020