Remember the French dictum which translates into “Long Live France! Long Live Diversity!"?
Well for the new year I Picked up this English-language article from three French physicists (Christophe Clanet, Fabien Hersen, & L. Bocquet).
It’s about skipping.
Skipping is fun.
Skipping stones, that is.
How do we model the sportful bounce of a stone in water?
Our spider sense might tell us that our modeling will lead us into the graphical realm of parabolae (I myself am a parabolaphile, I am not given to superstrings, and “more advanced” models of our universe which are still quite hard to feed into dyslexics such as myself.)
The stone streaks a path comprising of successive, inverted parabolas of decreasing size (or converging foci and diretrices--- that is each single focus-and-directrix tandem for each parabola displacementally achieved approaches the surface of the water over the course of mere fractions of seconds).
How do we model the stone?
The little stone we use (nor even the fake mono-elemental stone the three Frenchmen used, not even with its less-than-half-centimeter thinness), apparently is not little enough to be designated as what Einstein and others call a point mass.
The point mass is the initial model we Physics-learners (from ages six to ninety-plus) have for a body of mass. It serves well for bodies which do not rotate while undergoing rectilinear motion (i.e., a feather falling in a man-contrapted&confined space approaching vacuum quality.)
Our stone is NOT a point mass, but is best be taken as a rigid space-taking body, sure there is some microscopic deformability to a soft stone, but between stone and body of water, it is the stone which is a rigid body, and the lake (for example) a deformable body.
Our physical modeling of the stone’s volumenic shape is facilitated by the logistical desires of stone-throwers for sport.
It is the stone throwers who sought TO SIMPLIFY the shape of the stone… Thus making the task less arduous for physicists and physics-learners, for it is the duty of physicist to press for the simplest; engineers head for the complex. Physicists seek nature. Engineers regard nature as something ever to be contended with, in sundry simultaneous respects, the product of their work leading into machines useful for humanity's perseverance.
Simplest does not mean shunning calculus, for there is always more usefulness in drawing out the SI and cgs integral, divergence and curl forms of Maxwell’s equations than in showing to kids and older recipients of "pedagogical packages" his bearded face (for how are we to tell him for Marx or Freud).
Without calculus, general physics is debilitated, and pedagogy over-pressed.
Studying physics without entertaining even the most fundamental of first derivatives is like exerting politeness to someone whilst dreading the thought of even being their supper friend.
Particular physics phenomena however need not the affair-sweeping eros of calculus. Not even the entire article of the Frenchmen (“Secrets of successful stone-skipping”) ever brings mention of the slightest differential. The math of Chris and company is all high-school-friendly square root and proportions, and of course the ubiquitous pi. The math is so simple, because it’s the physics that counts (which is a virtue of our very own Phys-e1).
The flat stone.
When I was yet a schoolboy reporting to campus in khaki shorts, my close, personal friend then Dave Tinio coached me into picking flat stones for the stone-throwing we were to perform upon the basin of rain accumulated onto what was our grammar school’s Japanese-inspired rock garden. The garden was named after demi-boulder sized rocks of ovoid contour. Dave and I were not Titans capable of heaving the Goliath-size ornamental columnar stones (which would only be seen as small in relation to Stonehenge's or Easter Island's rocks) into the water (which would be too shallow to host the fall of boudlers in any case). It was not the rocks for which the rock garden was named after that Dave and I threw, but rather wimpy ones we picked around the garden's periphery.... little stones, calculi so small that they could not even be used as lever pivots for our ethnic tree-branch swing-and-hit game [i]siyato[/i] (Make light not of this kiddie game siyato, which has cost now and then the otherwise-safeguarded integrity of the pupil of the eyeball of some reckless bourgeois kid). Siyato is a more injurious sport than is stone-throwing, although both are technically not "contact sports" it is nature's and the hand-of-man's contact forces which make such ever-practiced games possible.
A flat stone translates by physical modeling into a thin disk.
And even us Phys-e1a’ers know at least a key property for such a thin disk: the rotational inertia, which is just some pure number times the mass times the square of radius.
What the Frenchmen themselves used was not a stone out of nature (although their work made its way into [i]Nature[/i]--- the magazine, that is) but an aluminum disk engineered to be unnaturally thin given the ample almost two-inch diameter.
Although most stones I know of (well, not here in post-post-industrial anti-erotic deadline-barbed penta-literal cemetery square Cambridge but back in the more rustic and lively Philippines) looked like rounded cubes, little boullions or if not, chips of cubes….
Such natural shapes are not entertained here in our modeling of the SUPERMODEL-thin flat stone.
The stone’s skipping is a collision of two bodies of mass--- well, not just a collision, but a quick series of collisions, empirically less than thirty-eight, all in all per expert throw. Although the order of magnitude of parabolas/collisions was hard put to reach even half-ten for me and Dave.
Before stone and fingers of boy parted ways, they made a solemn vow to algebraically jointly conserve the momentum they had enjoyed together when they held each other so tenderly.
The plus sign "+" lingered in the intervening air space for m-sub-boy times v-sub-boy and m-sub-stone times v-sub-stone.
Momentum is conserved as always. The stone has got angular and translational momentum, the body of water does exhibit any translation or angular momentum, displacement of velocity when gridded within a coordinate system itself invested, too, onto the surface of the earth (The coordinate system thus moves, with rotational acceleration besides curvlinear displacement due to Earth's relation around the sun.... and I am too junior to look beyond our meagre Solar System to consider displacemens extramural to our solar system, but such displacements in the Giga--and-greater-kilometer scale no longer carry any bearing to the mere metric action of our stone to be involved with the surface of a body of water, but the host body of water within the coordinate system neither moves nor accelerates in relation to the coordinate system, which will, like a big brother or father, do all the moving and accelerating for the body of water) (Not even if we used the small angle approximation would we find the slightest theta). The body of water only rotates along with the rest of the very complexion of the earth’s surface around the earth’s center of mass.
The inertial component of the stone we need to consider are BOTH the stone's density AND its mass. All the participants in the Newton-introduced relation m = rho times V (Well the [i]Principles of Mathematics[/i] which opens up has got in Latin sentences, BUT THE CABOT LIBRARY COPY OF THE COHEN/). That the quantity of matter for a compositionally homogenized body (whether pure substance or homogenized solution or mixture comprise it) is equal to its density and bulk taken conjointly (i.e., multiplied). m, rho and V for the stone all physically matter in the collision between.
Although two bodies (stone and water collide), what we have here is not a situation so homologous with a football player tackling another football player------ because in the collision between stone and water…. We DON’T GIVE EQUAL REGARD to the physical properties of the two bodies involved.
In a Giancoli or Urone football tackle problem, we give equal regard to the bodies of mass that is each a football player (Ironically, we may choose to ignore everything about the football held by the tackled player altogether).
As for Stone [i]vs.[/i] Water…….
The body of water's ([i]i.e.,[/i] lake)’s mass if OF NO CONSEQUENCE TO US WHATSEOVER, nor is the same fluid body’s volume or shape of any concern. The flat disk that is a stone need not be introduced into any particular chori of water.
Whether the body of water be half the size of Harvard Yard or just the size of our lecture hall HAS NO BEARING as to the concerned inertial property of water, just the density does, for it is such density of water which the stone encounters. The stone, with all the minsiculity of its size, has no physical disposition to receive any contact knowledge of the body of water’s girth or depth, unless we take the limiting case of a host pool not even one foot deep through which may propagate to and fro the pool floor a matter wave. A matter wave returning towards the water surface for a shallow pool of water (well, deeper than a puddle) might, I guess catch up with the intruder stone and mutter some flick of reaction, I do suppose.
I don’t think that Chris, Fabian and Derrick’s mathematical model will hold for a body of host liquid much different from water. We may use different qualities of water, [i]i.e.,[i] sea water/salt water/mineral water/distilled water/lemon water and adjust the rho subscript H2O accordingly----- but to throw the Chris&co.’s alumnim chip into a gourmetically-homogenized pool of thickened liquid chocolate (for bar-molding) would necessarily draw in considerations of viscosity, and not just some simple arithmetic substituting in of a higher rho sub host-liquid value and subsequent click-submission to WebAssign (which I do love, too, albeit less so than I do love calc A & B, flesh-and-soul TF’s, and the Divine One).
Giancoli informs us in that latter chaper of our e1a coverage that inertia (whether be mass or density---- in Galileo’s word, gravita) is the denominator inversely proportional to an elastic modulus whenst we may observe a confined oscillation of matter,
Although all us e1a’ers would necessarily figure in the weight force (F sub wght, … or simply mg) on ALL stages of thrown stone’s being (within and without the state of motion) in doing a free body diagram…. The Force most pronounced in Chris&co.’s brief discourse is the lift force.
What is a lift force?
I might not have an air tight definition of what a lift force is, but I'm quite sure that a lift force, like the buoyant force is a contact force, rather than a field force.
In fact the physics phenomenon of a stone skipping in water does at least flirt with the buoyant force, for several dates in fact (indicated in milliseconds, however, not calendar days)
When Dave or Ramuel holds a stone, the stone experiences the a normal force of chiral aspect, besides the “mg” which gives the Phys-e1a’er coveted points for the exam. Only two forces need be modeled.
Well metaphysically, the stone does experience a prayer force (Dave and Ramuel having both been schooled by Jesuits), but this need not be quantified.
When either young gentleman Dave or Ramuel chucks the stone, a third force external to the stone plays right in.
This force is expressible as the product of the impulse contributed J and the brief fraction of second that the boy gives the hurling motion.
The upper-level air whcih is venue as well
OK, so all througjout the
The buoyant force of air is not pronounced
The French physicists Cristophe Clanet and Faien hersen
Although Cristopher, Fabian and Derrick in their single-page article furnish us with time-step photos, for our own schematic modeling purposes we may choose to only depict three sutuation frames
Due to copyright, and I will translate some of into symboli Giancolici
i.e., Clanet’s tau into the Roman t for time we 1’ers are
I am not autjorized due to copyright to encode the mathematical relations
See, in the real world, even in the English-speaking world, you can designate a time value by the greek symbol little tau rather little Roman t, and alpha as angular length, even when depicting a physical phenomenon for which Angule ACCeleration I s well involved to (as in whar we’ve got above,
Try using tau for time and alpha for angular displacement in a test-room environment for an high school or college (I won’t. I’d rather wait long for grad school to afford amnesty.)---- and you will be diminished….. and at least deemed dyslexic.
If you used capital Roman or Greek T for time value, a grader might even accuse the pauvre eleve of misconstruing a not per oscillation per se time value as the Period.
We do need an orbe crystale.
What is a gyroscope? What is the gyroscopic effect? I know its got to do with a space-taking body’s well-pronounced rotation. And I gather that the gyroscopic effect serves as yet another grand instance of Nature preserving
Like a spinning top, our spinning stone stays upright in relation to an axis piercing its center of mass/gravity
What is physics?
Physics has got to do with approaches, connectionsm, relations
Church Street lab's closing sorry!
To be continued...
In the meanwhile, I ardently pray for all us students to get 83's or higher for our final.