Galileo’s square-cube law explains that a cylinder over a given dimension will collapse under its own weight:
From what has already been demonstrated, you can plainly see the impossibility of increasing the size of structures to vast dimensions either in art or in nature; likewise the impossibility of building ships, palaces, or temples of enormous size in such a way that their oars, yards, beams, iron-bolts, and, in short, all their other parts will hold together; nor can nature produce trees of extraordinary size because the branches would break down under their own weight; so also it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animals suggest a monstrosity.
Suppose we scale some physical system up by some factor f, or down if f < 1. Not only will some of its static properties change, as Galileo points out above, but also some of its other behaviors will speed up or slow down. But by how much?
The mass and thus weight of a body will tend to scale as f³.
The tensile strength of a member will tend to scale as f², so by smallifying an object we make it stronger (1/f times) relative to its own weight. If we consider stress from accelerations, well, this means we can accelerate it faster (1/f times) before it fails in the tensile mode; relative to its own dimension, this is an advantage of a factor of 1/f² in acceleration, but this only works out to a factor of 1/f in frequency.
That is, suppose we have some sort of machine in which a weight is being jerked back and forth along a course inside the machine by a rope, and if we run the machine too fast, the rope will break. We smallify the machine, say by a factor of 1/f = 3. Now the weight is being moved a 3 times shorter distance, it has 27 times less mass, and the rope can withstand 9 times less tension. So the rope can withstand 27/9 = 3 times greater acceleration, but at a given jerking frequency, 3 times less acceleration would be needed to jerk the weight the same distance. But the length of the weight’s course within the machine has also diminished by a factor of 3, so at the same operational frequency, the stress on the rope has actually diminished by a factor of 9.
However, if we run the machine 3 times as fast, so that it jerks the weight 3 times as often, the velocity increases by a factor of 3, but the acceleration increases by a factor of 9. So, in the mode of tensile failure, smallifying does increase the maximum frequency, but only proportionally, not quadratically as one might hope.
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The usual expression for rectangular beam bending stiffness is k = Ebh³/4L³, where E is the material’s modulus, h is the thickness of the beam along the direction of bending, L is its length, and b is its width transverse to the direction of bending. If we scale the beam uniformly up or down by some factor, then the changes in bh³ and L³ leave the stiffness varying directly with the scale.
The general expression for the resonant angular frequency of a sprung-mass system is ω = (k/m)½. So, if a mass is at the end of an elastically bending beam, the mass m changes by a factor of f³ while the stiffness k changes by a factor of f, so its resonant angular frequency changes by a factor of 1/f.
tensile-mass oscillation
shear-mass oscillation
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A resistor’s resistance is ρL/bh, where ρ is the resistivity. If it becomes f times longer and f times wider and deeper, it will thus diminish in resistance by that same factor f. Smallifying resistors thus makes them proportionally higher in resistance; we might say that conductance, rather than resistance, is proportional to scale.
A thin resistive film whose thickness does not change is well known to have a characteristic “resistance per square”: its resistance over 1 mm × 1 mm is the same as its resistance over 1 cm × 1 cm or 1 μm × 1 μm.
At ordinary (macroscopic) sizes it is easy to achieve an enormous range of resistances at any scale, because in a given volume you can either make a long, thin conductor or a short, thick one, and moreover available resistive materials themselves cover several orders of magnitude of resistivity; resistors from 0.001Ω to 1GΩ are all staple articles of commerce, and resistors from 1Ω to 1MΩ are common, and all of those common ones are available at any size from 0402 up to many watts. At the lower end these resistances are limited by the resistances of the commonly employed wires (i.e., not superconductors, but materials like copper) and at the upper end by the leakage currents of materials such as glass, polypropylene, and Teflon, used as insulators.
On chip, the story is different, because although low resistances are easily accessible (just use a metal layer) high resistances such as 100kΩ are not; they occupy an inordinate amount of space and are a nuisance.
A (planar) capacitor’s capacitance is εA/d, where ε is the permittivity of the dielectric. This is proportional to f, so smallifying capacitors thus makes them proportionally smaller in capacitance.
Common macroscopic capacitors cover an even wider range than resistors, from 10 pF to 50 F, or 10000 μF if we exclude supercapacitors.
Perhaps the most common way to mark time in an electrical circuit is with an RC circuit with an exponential decay time constant τ = RC; this is how RC filters and 555 timer circuits work, for example, and how the internal oscillators in microcontrollers work. Since R is inversely proportional to f while C is proportional to it, smallifying an RC circuit will not change its frequency response.
This is an astonishingly different result from Dennard scaling.
A cylindrical air-core coil has an inductance of roughly μN²A/L. Scaling it by f increases A by f² and L by f, so inductance scales with f. So inductance, like capacitance and conductance, is proportional to scale.
The resonant angular frequency of an LC circuit is ω = (LC)-½. So multiplying both L and C by a factor f will multiply ω by 1/f; larger LC circuits oscillate proportionally slower.
electromagnetic relays
electrostatic relays
crystalline structure
heat transfer characteristic time
turbulent vs. laminar flow
matter diffusion
composite materials
aerostats
heat exchanger design
mass transfer design
liquid friction