Elastic Bodies. Force.

A certain person named Francis used to ask me repeatedly,

“When my arm pushes against a wall, or when anything pushes against my arm, what causes the force?”

I always answered his question exactly the same way.

When we think of the bone inside the arm, or of components used to build houses and walls, we think of those as ‘rigid bodies’. But in reality, rigid bodies are just special cases of ‘elastic bodies’, and one needs to understand why elastic bodies exist, in order to understand rigid bodies, in the way He asked.

The molecules that make up an elastic body, generally consist of atomic nuclei that are separated by electron-pairs, or chemical bonds. These bonds act as ‘microscopic springs’.

When the distance between two nuclei that are part of a molecule is at its neutral distance, the electrostatic repulsion between the nuclei equals, or cancels with, the electrostatic attraction caused by the electron-pair itself.

If the distance between the nuclei decreases slightly, then the electrostatic repulsion increases, but the attraction caused by the electron-pair – i.e. caused by the chemical bond – stays about the same. And so a net repulsive force results, that tries to restore the distance between them to their neutral distance.

If the distance between the nuclei increases slightly, then the electrostatic repulsion decreases, but the attraction caused by the electron-pair – i.e. caused by the chemical bond – stays about the same again. And so a net attractive force results, that tries to restore the distance between them to their neutral distance.

One reason why the folly is still undertaken today, to teach Newtonian Bodies to Students, prior to teaching more-advanced concepts, is the fact that what happens on the macroscopic scale in Newtonian Mechanics, also tends to approximate what happens on the microscopic scale, and vice-versa. With Quantum-Mechanics, Relativity, etc., this can no longer be guaranteed. And it helps explain why Newton was able to ‘understand his world’ so well, even though the subatomic world wasn’t known yet, in his era.

Elastic bodies are made out of a huge number of atoms, but their macroscopic behavior derives from their microscopic behavior, in that If the distance between their end-points decreases slightly, from its neutral distance, a net repulsive force results, while if the distance between their end-points increases, a net attractive force results. This latter, net attractive force is also due to their ‘tensile strength’.

When such elastic bodies are only modeled as having two end-points, then they are also simplified as ‘springs’. And springs have a so-called ‘force-modulus’, which is a linear factor, by which a small change in distance, results in some change in force. If a spring has a very low force-modulus, then it is very ‘soft’ or elastic. If it has a very high force-modulus then it is very ‘stiff’ or inelastic.

Beyond some amount of compression the spring will fracture, and this behavior is known as ‘brittleness’. If the spring is very brittle, then it won’t compress much, before it breaks.

The fact to understand about rigid bodies, is that they are just elastic bodies, whose force-modulus is so high, that we don’t humanly observe them deform. And the brittleness is also so weak, that to try to apply enough force to them to make them (appear to) deform, is impractical and just results in their breaking.

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Understanding NMR

Under ‘the term NMR’, people may correctly understand two different subjects:

  1. Why do subatomic particles, in this case nuclei, precess?
  2. How do Engineers exploit this precession, in order to form 2D and 3D images, in ‘NMRI’?

In this posting, I am only going to address subject (1).

Precession and spin are easier to understand, when we can simply apply the Newtonian concepts. Quantum Mechanics today tends to obscure the subject of precession. And so for most of this post, I am going to make the somewhat daft assumption that the precession of subatomic particles, is Newtonian.

If a gyroscope is spinning along an arbitrary axis, and if we apply torque to its axis, this torque integrates into the spin vector – at an angle to the existing spin vector. Unless we are accelerating or slowing down its spin. This results in the spin vector rotating – and thus in precession.

But, if we have seen the demonstration in which an off-axis gyroscope is precessing on a passive pedestal, we also observe that eventually the phenomenon weakens, and that the practical axis seems to shift further and further in the direction gravity is pulling on it.

The reason this weakening takes place, is the fact that some additional torque is being applied to the gyroscope, against the direction in which it is precessing. Otherwise, it would just precess forever. This additional torque could be due to friction with the pedestal, due to air resistance, due to magnetism, or whatever.

An artillery shell is aerodynamically designed, so that as long as it has excess spin, interaction with the air will always push it in the direction of any existing precession, and so this type of object will tend to straighten its axis of spin, into the direction with which it is flying. This would be the equivalent to the gyro from before, straightening up and standing up against gravity again.

Atomic nuclei that have an odd mass number, also have a non-zero spin quantum number, thus having spin, and also have a magnetic dipole moment. The wanton assumption could be made that its magnetic dipole moment is always parallel to its axis of spin. But then if we visualize matter as consisting of nuclei that are separated by vast, less-dense clouds of electrons, it would seem to follow that each nucleus is always precessing in response to local magnetic fields.

And even if we were to apply an external magnetic field to such a system, it would follow that precession could not yet be detected externally, because the nuclei are all out-of-phase. Ostensibly, they would also continue to precess, and to stay out of phase, simply due to an applied magnetic field. The only big difference with the practical gyro should then be, that the magnitude of their spin-vector should never change, since this should be intrinsic.

But if we were to insist on this very Newtonian description, then something else should also happen that is not as obvious. Those thin wisps of electrons should not only react to the applied field, but also locally, to the field of each nucleus precessing. So if we assume conservation of energy, there would also be reactive torque acting on each nucleus, in response to its own precession, because the density of the electron clouds is not zero.

After a certain settling period which is measurable, the nuclei end up aligning themselves with the applied field, resulting in the state that has its lowest-possible potential energy. This takes milliseconds instead of the nanoseconds that some of these behaviors should take on the subatomic scale. Precession has still not been detected.

Likewise, the fact that subatomic decay can take years instead of nanoseconds, refutes certain mundane explanations, of what might be causing that.

Well, one thing that Scientists can do is compute what the dipole moment of such a nucleus is, as well as the magnitude of its angular momentum – spin – and to compute as a function of the applied field-intensity, with what frequency all the nuclei should be precessing… This frequency is called the “Larmor Frequency”.

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