The hanging cable

A interesting problem was put forward by my physics teacher on Friday, relating to material properties - what is the longest possible constant-diameter cable, from a material with a given yield stress and density, that can be hung vertically in the Earth's gravitational field (with its end just touching the ground) without snapping?

In essence the problem is very simple. The point of maximum stress is right at the top of the cable, where it is hanging from some kind of steadfast loop capable of supporting the material in its entirety; at this point the stress is equal to the weight of the cable divided by the cross-sectional area. However when the materials used are strong enough to last kilometres into the air, the changing value of g starts to become a factor in the weight force experienced by the maximum stress point. I did some scribbling today, and came up with a little proof of a nice formula that takes into account the changes in g, with the use of a definite integral as the infinite sum of a series in its limit:

The most important insight to be gleaned from the final formula for the maximum height of cable is that the cross-sectional area of the cable is irrelevant to the height achieved, as long as it remains constant. It depends only on the radius of the Earth (constant), the universal gravitational constant (constant), the mass of the Earth (constant), the density of the material (constant for a given material) and the breaking stress of the material (constant for a given material) - note that it is assumed yield stress = breaking stress, since it is unnecessarily complicated to consider the plastic properties of the cable between yield and fracture.

An example of the use of the formula is to consider a material - steel is a popular choice for high-stress cabling. Wikipedia tells me the breaking stress of a certain type, 2800 maraging steel, is 2617MPa and the density is 8100g per cubic metre. The formula returns a H_max value of 33072.8304159 ≈ 33.0km. If the uniformity of the Earth's gravitational field was assumed, the H_max would be significantly different: 

Cross-sectional area of cable = 0.25πd²
Volume of cable = 0.25Hπd²
Mass of cable = 0.25Hρπd²
Tension at top of cable = 0.25Hρgπd²
Stress at top of cable = Hρg
H_max = σ_frac/ρg = 32934.3954896 ≈ 32.9km

There is an absolute difference of 138.43493 ≈ 138m here, which converts to a surprisingly sizeable percentage error of 0.419%!

Now it has been established how important the non-uniformity of the Earth's gravitational field is in such a matter, one must now wonder how the shape of the cable could maximise the value of H_max - my hypothesis is that having a slight linear taper on the cable such that it is fatter at the top than at the bottom, with the cross-sectional area scale factor per metre climbed matching the scale-factor for decrease in gravitational field strength per metre climbed, would be optimal. This is because having less mass at the bottom is essential for marginal gains, where the gravitational pull is strongest, and a taper to this specification should account logically for the inverse-square nature of Newtonian gravity.

Steel was a fairly old-school example for testing the value of H_max: a huge focus of the material science field is the applications of carbon nanotubes, tailor-made materials based on graphene and its chemical derivatives. Wikipedia informs me that one particular type, Armchair Single-Walled NanoTubes, has a breaking stress of 126.2GPa, nearly 50 times the strength of 2800 maraging steel. Ignoring the considerable strain produced by such a breaking stress (0.231), an oversight it seems, the H_max values can be calculated with the calculus and simplistic methods, given an approximate density value for Armchair SWNT of 1660kg per cubic metre (from here):

Calculus H_max = -35976055.5613
Simplistic H_max = 7749653.04644

This is pretty stunning as a result. The nanotubes are so strong that the calculus formula breaks down to form a negative result - this implies the cable can stretch infinitely out into space (assuming it is not affected by the gravitational fields of other bodies, which isn't technically true of course), effectively escaping the effects of the Earth's pull, without fracture. I suppose the combination of the cable being nearly 5 times less dense and 50 times stronger means there is an effective relative increase in potential height of 250 times. 

Hyperbolae 2 - introducing the third dimension

This is a summary of today's musings. In essence I have done two fairly basic things that marginally extend the 2-dimensional hyperbola: I have taken the infinitesimal-lamina principle behind solid revolution and used it to model how the function will rotate around the x-axis. Then I have returned to the basic calculus of revolution to derive an expression for the volume of a lobe, given values of the stretch-coefficients a and b, and the limit of integration c. Enjoy!

Investigating hyperbola-type implicit functions 1 - asymptotes

"Which way round was it again?"
During one of my FP1 past-papers, of which I have been trawling through the entire lot, it struck me that there was one part of the course that was very much a matter of "here's the result, learn it". Hyperbolas are only touched-upon in the AS further maths specification, and for me that makes the given theory more difficult to engage with. Hence, I found in one question that I could not remember the generalisation for the asymptotic gradient of a hyperbola. Well, with two options in mind, I thought asking for help somewhat non-proactive so I decided to see if I could work it out with a little bit of limit notation.

 Having achieved this, feeling extremely pleased with my elevated understanding of the function despite having not blasted through the 3 or 4 AS level questions I would have otherwise spent the time with, I began to consider what would happen if the same general form of hyperbola was used with higher-order indices. My findings are on the A4 scans below.

The second page deals with more trivial cases, where x and y are raised to different powers - there produce fairly uninteresting polynomial-type curves, which could effortlessly be rearranged to express the first-quadrant branches explicitly. Nevertheless an implicit-differentiation method for proving the limits of the gradients is still fairly stimulating, and the result is pretty nice, if quite obvious.

A beginner's attempt at modelling air resistance and restitution - a concerted attack on the dogma of modelling assumptions

As the title of this article may suggest, I thought it might be interesting to see what mathematical knots I could tie myself in once I begin to remove modelling assumptions from a seemingly-simple physical system. The answer, somewhat predictably, is very very many indeed.

This all stems from some work on differential equations I did over the Christmas period, where I first came into contact with the idea of modelling resistive forces as a function of a differentiated variable (acceleration usually). After doing a physics PAG (assessed practical) on determining the terminal velocity of a cupcake case falling under acceleration due to gravity in October, I realised how interesting an investigation into exactly how the object accelerates between t = 0 and terminal velocity could potentially be and with a new mathematical skillset, I began to do some scribbling...

Several months later, with some on-and-off periods of work on the paper, I have come up with this 48-page pdf file documenting how I have somewhat tangentially approached ideas such as drag and restitution with a certain level of naivety. It has not been extensively proof-read, so is most definitely an unpolished version which serves to demonstrate how I have leapt from one problem to the next, and no doubt errors will be inherent in some of my working as a result. Nevertheless, I am immensely proud of some of the derivations, as well as my real-life experiments to test them.

The following list contains all the external links to Desmos graphs, which feature throughout the document:

p13 - http://www.tinyurl.com/hmdd5jc - CTDM
p20 - http://www.tinyurl.com/jezmcxo - light-gate data
p30 - http://www.tinyurl.com/za2cq2f - multiple cycles of the CTDM in a bouncing-ball situation
p31 - http://www.tinyurl.com/hncm9su - fragility of the CTDM
p32 - http://www.tinyurl.com/grd873b - stability of the SQTM
p36 - http://www.tinyurl.com/hl32bt2 - collision impulse and force
p47 - http://www.tinyurl.com/jrmk6ye - strobe data and restitution

Introduction 

I begin the investigation with some theoretical modelling - first I take drag to be proportional to velocity, then velocity squared and finally velocity cubed. After this, I investigate how a model could be developed containing both a linear and a quadratic term in v, called the combined term drag model (CTDM). Next, I conduct some experiments with falling objects: one such experiment involves a custom-made tube of 10 phototransistor-LED light gates to observe in detail how a falling muffin-case's displacement varies with time; another determines the spring constant of the average tennis ball; another uses a strobe-light and a long-exposure camera setting to collect displacement-time data for a bouncing tennis ball. I use all this data to test the efficacy of the various models, with a good deal of running commentary and analysis.

The paper can be found here: https://www.dropbox.com/s/bgdq0vwqa0wumxp/PAPER.pdf?dl=0

An intermediate attempt at modelling interference graphically

This modelling work, which I have spent most of my weekend developing, stems from a very simple question which came up in my AS physics textbook, on the topic of wave interference. It went something along the lines of "Draw 6 diagrams, 1 second apart, to show how these two approaching waveforms interfere". I was surprised at how difficult it was to faithfully do this, even with two simple waveforms in wide discrete time jumps and it got me thinking - how could I use a computer to do this better? The result is in an 8-page investigation, where I have endeavoured to develop some theory about how to model passing and reflecting waves and how they interfere - it begins easy, with some simple linear interference by summation of instantaneous displacements, but soon I found that the mathematics became more complicated (particularly in stage 3, where I began to model how waves slowly escape from a trap, like a solid-state laser but without the quantum stimulated emissions).

I have tried to explain myself fully throughout, although not enough proof-reading has been done to keep it from being a little haphazard, and the last page of the document requires some following-up when I have a little more time. However, I am pretty proud of some of the derivations.

In the paper, I have referenced some equations with numbers in square brackets. These match-up with equations on the corresponding Desmos file (here: http://tinyurl.com/zlmzg9d), to illustrate exactly what each equation looks like in practice.

The paper can be found here: http://tinyurl.com/zwegna4

Refraction in the third spatial dimension

In my AS Physics classes, we have reached the wave mechanics point of the course - one of the most fundamental observable properties of waves travelling through media of differing densities is refraction; however, we only ever seem to consider refraction in a coplanar sense (where there is only an incident angle against the plane of the boundary in two dimensions). My investigation of the morning is to develop a little theory, which will hope to extend my understanding of refraction to apply to the third dimension, which up until this point in my education has been ignored.

Firstly, let's review Snell's law. This states that the product of the refractive index of one medium and the sine of the angle in that medium against the normal is constant throughout the refraction process. In algebraic terms:

This provides the basic relationship between the two angles on the diagram below, which demonstrates the coplanar refraction I am talking about:

Similarly, I could represent the effect of refraction on the third-dimension in the same way, by reducing the problem into a coplanar one and ignoring the second dimension in the above example. However, considering all three dimensions at once will involve some basic vector geometry. The direction of the light ray, where the perspex-air boundary is an arbitrary straight line with the equation x = C, can be represented by a 3-D vector U:

This vector can then be divided into two components, dealing with different planes - the x-y plane and the x-z plane. The angle in the y-z plane is irrelevant because it is parallel to the boundary plane.  I have worked through this problem, and here is my result:

From here, I used Desmos graphing calculator to demonstrate the result of this. I spent much time working with the z-axis, trying to produce the most convincing one possible by adjusting the positions of the x and y axis, all parametrically under the control of the rotation parameters A, B and C. I had some success with this, using the sinusoidal nature of rotational perspective to produce a set of coordinate axes which can be rotated about the origin (i.e. the real z-axis) in the x-y plane, and the image of the x-axis, allowing enough movement to achieve most angles on any 3-D subject. A current demonstration of this somewhat rudimentary and flawed model can be found here: http://tinyurl.com/hk4d3l7. In the future, I hope to be able to understand how not only full rotational freedom can be achieved through the parameters A, B and C, but also how I can faithfully project real points (x,y,z) onto these simulated axes while they can stand up to rotation.

However, since I have yet to truly crack the matter, I have for now settled with a simple pseudo-z-axis. However, it still demonstrates my mathematics fairly well. Furthermore I have paid attention to detail diagrammatically by programming Desmos to draw proportional arrows for the magnitudes of each component ray, as well as the resultant incident and refracted rays. The online version can be found here: http://tinyurl.com/jthre74.

Overall, I feel I have opened a metaphorical can of worms for myself when it comes to the true complexities of this physical phenomenon. Further research has yielded information regarding Fresnell's equations for transparent materials, which determine the relative amplitudes (and hence intensities) of reflected and refracted light at such a boundary between two media, which could be programmed into the graphing calculator. Also, I could isolate the mathematics from the assumption that the boundary is parallel to the y-z plane, such that refraction and reflection on a 3-D surface of differential-determined gradient could be modelled. Perhaps a follow-up to this will be required in the future.

The Simplex Algorithm

This rather complex algorithm was the latest subject in my Decision Mathematics 2 course at school. It involves taking a linear programming problem, and solving it without the need for graphical methods - the beauty of this is that it allows the logic to be extended to higher-dimensional problems which cannot easily be represented on multidimensional Cartesian coordinates. In fact, there is theoretically no limit to the number of variables that can be dealt with!

Anyway, the most important reason for the existence of the Simplex algorithm, as I see it, it that it can be more easily programmed into a computer. So this is what I did!

My C++ skills would seem rudimentary to a seasoned programmer, since I became a novice programmer several years ago and haven't touched an IDE since, but I'm quite pleased with the result of these 692 lines of code. It may spring bugs, but I have tested my version of the algorithm on the majority of the problems in the D1 and D2 textbooks and the results have been successful. This has allowed me to build on the pretty shocking SIMPLEX 1.0, developing a more robust SIMPLEX 2.0.

The various files can be found on my Dropbox:
- DevCPP source/project file (http://tinyurl.com/h7ck5ow)
- Readme in a text-file (http://tinyurl.com/gr4r56x)
- Executable SIMPLEX 2.0 file (http://tinyurl.com/zo8hbat)

Since the program is freshly programmed, it is likely that there will be faults in the code. Any error reports would be welcomed.