In article <vlejo4$15i8k$
[email protected]>, Luigi Fortunati asked
But then, why do two extraordinarily different systems like the Earth's
mass (6*10^24kg) generate the force of 90kg-weight on my body (mass
90kg) and my body generates the *same* opposing force of -90kg-weight
on the Earth?
and later, in article <vljh6k$28t5l$
[email protected]>, Luigi asked
I repeat: the claim that my miserable gravitational force can attract
the Earth with the same force (90kg-weight) with which the Earth
attracts me (as the equality between action and reaction claims) is unacceptable!
To generalize Luigi's questions, if we have a pair of masses M1 and M2,
fixed in position (with respect to a Newtonian inertial reference frame
(IRF), to keep things simple) some distance apart, with M1 not equal to
M2, is there any good reason to think that the gravitational force of M1
acting on M2 is equal in magnitude and opposite in direction to the gravitational force of M2 acting on M1?
There are actually a couple of useful lines of reasoning, each of which suggests that the answer is "yes":
To start with, notice that Newton's law of gravitation specifically
states that the answer is "yes". So we're basically asking whether
Newton's law of gravitation is in fact an accurate description of
reality (in the domain where we expect it to work, i.e.,
non-relativistic non-quantum systems).
(1) We can directly measure the gravitational force between masses in
a laboratory, and do indeed find them to agree with Newton's laws.
See
https://en.wikipedia.org/wiki/Cavendish_experiment
for an introduction to the "classic" experiments on this (dating
back to the late 1700s), and
https://en.wikipedia.org/wiki/E%C3%B6tv%C3%B6s_experiment
for an introduction to experiments verifying one somewhat subtle
aspect of the Newtonian gravitational interaction.
(2) Think about what would happen if the force of M1 on M2 were NOT
equal-in-magnitude-and-opposite-in-direction to the force of M2
on M1: the only way the sum of two vectors can be zero is if the
two vectors are equal-in-magnitude-and-opposite-in-direction, so
if the force of M1 on M2 were NOT
equal-in-magnitude-and-opposite-in-direction to the force of M2 on
M1, then the (vector) *sum* of these two forces, i.e., the total
gravitational force on M1+M2, would be nonzero.
Let's now imagine that M1 and M2 are held apart by a light stick
so as be at a fixed distance from each other, forming a "dumbbell"
(still at reset in a Newtonian IRF, and let's say floating out in
space far from any other masses). Then (if the force of M1 on M2
were NOT equal-in-magnitude-and-opposite-in-direction to the force
of M2 on M1), that nonzero "total gravitational force on M1+M2"
would accelerate the dumbbell with respect to the Newtonian IRF,
violating the law of conservation of momentum.
Moreover, that acceleration would result in the dumbbell having
kinetic energy, so we've also violated the law of conservation of
energy. For example, if we put our dumbbell sideways on a turntable
(e.g., if we're looking down on the turntable, put the dumbell
oriented horizontally, attached to the turntable's 12-o-clock
position), this acceleration would start the turntable rotating.
So, if we put an electric generator on the turntable's axis, we
would have a "free" source of energy, i.e., a perpetual motion
machine.
So, to summarize, we've shown that if the force of M1 on M2 were
NOT equal-in-magnitude-and-opposite-in-direction to the force of
M2 on M1, then you could violate the laws of conservation of momentum
and conservation of momentum, and build a perpetual motion machne.
For a variety of good reasons that I won't go into here, we think
that the laws of physics forbid violations of the laws of conservation
of momentum or energy (and hence forbid the existence of perpetual
motion machines),
so this argument (strongly) suggests that in fact the force of M1 on
M2 *IS* equal-in-magnitude-and-opposite-in-direction to the force of
M2 on M1.
[In the context of general relativity "conservation of
momentum" and "conservation of energy" are rather tricky
concepts, because there's no good way to add up energy/momentum
"here" and energy/momentum "there" to get a total energy/momentum.
And in general relativistic cosmology things get trickier still.
I'm going to ignore all of these subtleties here, and stick to
Newtonian gravity/mechanics.]
(3) In my gedanken system of M1 and M2 being joined by a light stick,
the stick isn't actually necessary. You could actually have M1 and
M2 in orbit about each other, and if the force of M1 on M2 were not
equal-in-magnitude-and-opposite-in-direction to the force of M2 on
M1, then the center of mass of M1 and M2 would oscillate around at
the orbital period. And, if the M1-M2 orbit were eccentric, then
if the force of M1 on M2 were not
equal-in-magnitude-and-opposite-in-direction to the force of M2 on
M1, the center of mass would have a net acceleration with respect
to a Newtonian inertial reference frame.
We can measure the motion of the center of mass (by timing the radio
signals of binary or millisecond pulsars arriving on Earth) for the
case where M1 is the Earth and M2 is the moon, and for the case where
M1 is the Sun and M2 is Jupiter/Saturn/other planets.
[In fact, my very first published scientific paper was
about this measurement. It was only 2 pages long, and
had exactly one novel idea in it. My Ph.D advisor looked
at it and said "that's one more idea than in some papers
I've seen". :) ]
Experimentally, we find that the M1-M2 center of mass does NOT
oscillate or accelerate in this way, suggesting (strongly) that
the force of M1 on M2 *is* in fact
equal-in-magnitude-and-opposite-in-direction to the force of M2
on M1.
So, in conclusion, the basic answer to Luigi's question "*Why* is the
force of M1 on M2 equal-in-magnitude-and-opposite-in-direction to the
force of M2 on M1?" is "conservation of momentum/energy".
--
-- "Jonathan Thornburg [remove -color to reply]" <
[email protected]>
currently on the west coast of Canada
"[I'm] Sick of people calling everything in crypto a Ponzi scheme.
Some crypto projects are pump and dump schemes, while others are pyramid
schemes. Others are just standard issue fraud. Others are just middlemen
skimming off the top. Stop glossing over the diversity in the industry."
-- Pat Dennis, 2022-04-25
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