Better than Archimedes
Archimedes is said to have declared: "Give me a place to stand
and I will move the earth". The figure explains the way he meant to
do it.
Does it really? Let's invite the Mathematician
and the Physicist to discuss the question
in more detail. Here's how the discussion could have looked like.
Mathematician: Although ingeniously
simple, Archimedes' idea has some serious drawbacks. Here's one of them.
Let A be the point, at which the lever supports the earth as shown in the
figure, and let B be the fulcrum, i.e. the point at which the lever rests
upon a stable support. We may agree for that for practical reasons the
segment AB should have some reasonable length, say, not less than 1 mm.
But then the length of the entire lever, if it were to accomplish its task,
would have to be monstruous. If we use F to denote the force to be applied
at A to move the earth, and G to denote the force applied by Archimedes
at the point C at the other end of the lever, then by the well known laws
of physics the balance of these two forces implies the following relation:
Let's assume the following values for the magnitudes involved: AB =
1mm, F= 6 . 10^{24} kG, G=10 kG, quite reasonable, I believe, for
the case considered. Now you can compute the distance BC and the length
of the lever (which the interested reader will certainly do).
Physicist:Why do you think the value
of 6 . 10^{24} kG is reasonable for F? This is exactly the value
of the entire earth mass and we have no reason whatsoever to claim that
the force would be numerically equal to the mass. If the earth were replaced
in the figure by an object with mass equal to 6 kG, then  knowing that
we are on earth  it would justified to say that F is equal to 6 kG, i.e.
58.8 newtons (it seems that the use of newtons is safer, since it prevents
confusion between the lower case "g" and the upper case "G"
in kg and kG; 1 kG = 9.80665 newtons). The problem would be simple. But
in our case neither Archimedes nor you have said anything about the position
of the fulcrum.
M: Come on, don't hang on details.
What interests me is the problem of using a lever to lift a very huge mass.
After all, I can assume that I am on Jupiter and I want to lever up a ball
of the size of the earth.
P: But Jupiter...
M: I know, I know. Jupiter consists
of helium and nitrogen and I cannot "lay" the earth on it. But
it's the problem that attracts mu attention and not the relation between
earth and Jupiter.
P: OK, though I would prefer to
bring the problem somewhat closer to reality. The earth moves along a free
orbit and has no weight, just like cosmonauts flying on an orbit around
the earth. But, anyway, I can accept Jupiter. Do you know what the weight
of the earth would be on Jupiter?
M: I don't, but you should.
P: I can compute it. The general
gravitation law says that the force of attraction between two masses is
proportional to the product of the two masses and inversely proportional
to the square of the distance between them. If the earth and Jupiter are
tangent, this force is approximately
M: So, we have finally reached an
agreement. Now I can compute that to lever up the earth on Jupiter using
a pression corresponding to the force of 10 kG, I must use a lever of the
length of 1.29 . 10^{18} km.
P: I don't agree that we agree.
I may accept, however, that the length of 1.29 . 10^{18} km is
sufficient to counterbalance the force of attraction on Jupiter's surface
with a force of 10 kG.
M: Aren't you being fussy? If you
can counterbalance the force of attraction, then with a slightly greater
force you can also lift the earth.
P: In what time and by how much
would you want to lift the earth to consider the job done?
M: The time is of no significance
to me and I would be satisfied with moving the earth just 1 mm off the
ground.
P: Well then, I consent to 1 mm,
but the time is very important for me.
M: Why would you care about time?
P: Because it may happen that an
entire lifetime would be too short to carry the task out, as I will prove
to you now. Let's assume that we can afford to dedicate 100 years to lift
the earth 1 mm off the surface. By applying a constant force we shall make
the earth move with uniform acceleration of 2 . 10^{22} m/s^2.
To attain this effect we need a rather small resulting force of 123 kG.
Unfortunately, we must first counterbalance Jupiter's force of attraction,
which is equal to 1.29 . 10^{25} kG.
Therefore the lever must provide a force of 1.29
. 10^{25} + 123 kG, which is approximately equal again to
1.29 . 10^{25} kG. In the conditions
you proposed the length of the lever must have one arm of length 1.29 .
10^{18 }km. Let's forget about the fact that there is no technical
possibility of constructing such a lever. Anyway, if the point A is to
cover the distance of 1 mm in 100 years, the point C must traverse the
distance of 1.29 . 10^{15 }km in the same time, which yields a
mean velocity of 400 000 km/s  much greater than that of light.
M: Well, but the idea was correct,
wasn't it?.
P: I am still in doubt. Can an idea
correct for the mass of 1 kg be considered correct for arbitrarily large
masses?
M: But then, can the earth be moved
at all?
P: Why, yes, but no lever nor place
to stand is needed for that. It would be sufficient to launch rockets into
space while the earth is at some specific positions, so as to act on it
with a mean force of 123 kG in a fixed direction for the time of 100 years.
This would move it by 1 mm and endow it with an additional speed of 6.3
. 10^{13 }m/s.
M: I must conclude that I like Archimedes'
idea better. At least it's unfeasible! The earth shouldn't be played with...
