studiot

That still has to obey energy conservation.

That would not work well due to the Law of conservation of Energy, assuming the panels were optimally laid out in the first place.

And did you ask Chat GPT why this method would yield fallacious answers ? AI 0 ......... Surveyors 1

Here is some explanation that may help. The Earth is not 'all there is'. In other words there is space and time along with contents and the stuff that makes it all up beyond or outside the Earth. However the Universe is defined as 'all there is'. In these circumstances both space and time may be part of this 'all there is'. We don't know if either space or time are finite or infinite. We don't even know the nature of this infinity if it is there. We do know of other infinities which which have no boundary but yet can't be reached. For instance absolute zero of temperature. As you approach AZ it becomes harder and harder to take the next smaller and smaller temperature step towards it such that it can never be reached. Another example, if you know any maths, is exhibited by numbers. Take any real or rational number you will find there is no such thing as a nearest number to it. You can always find a nearer number than any propest 'nearest' and can go on indefinitely getting closer and closer to the proposed number, without ever actually reaching it. So if our finite is bounded in space, why not also finite in time ? Nature has shown itself more varied and unwilling to fit into our human conceived boxes and is therefore constantly suprising us with new stuff.

Your question - A good one - seems the sort of question that might (should) pop up in class when you have all the background of the course and its notation. So it is a very good idea to give as much information as possible about where you are coming ffrom and where you want to go. We normally like to place replies into the body of a thread, to benefit any member who might be interested. There is nothing confidential abolout this. The honest answer is that we are looking for solutions to Airy's equation. There is no pat answer as with say linear differential equations, each solution must be worked out for the specific situation. Airy's original was an inspired guess as are pretty well all other known solutions. This is the drawback for this method and the reason it has never proceeded very far. Complex analysis and conformal mapping techniques have largely replaced it as an analytical technique. Anyway here are a few pages from a book that is ideal for engineers studying stress analysis. It works from an engineering point of view and explains, when it comes to it, that we 'guess' the form of the solution and back substitute to check it fits the equations of constitution and discusses this for several example cases. Are you familiar with the idea of the equations of constitution and compatibility ? For Constitution either the equations of equilibrium or the equations of motion may be employed, but these alone will not give enough equations to produce a solution. That is where compatibility (or configuration or constraint) come in. Advanced Strength and Applied Stress Analysis R G Budynas

Is the 'Gap' getting smaller ? I invite you to explore this report on a computer closing the gap. Note this is only one of a line of bionic implants. Is the computer-brain one an advance on the well established computer-heart one ? https://www.bbc.co.uk/news/articles/ckgn49r069wo

I caught you previous howl of anguish over my previous post, but it is taking a few days to prepare something simpler. I had hoped that by pinching someone elses' drawing I could offer a quicker answer. Never fear, we will get there.

Not sure if you want a physical or mathematical explanation. The physical one is easier so I will start with that. When you integrate the differential equations you get many or infinitely many solutions that differ by an arbitrary constant. (actually since you should be using partial derivatives for this, you get an arbitrary function). Physically what we want is for that arbtrary function to affect all parts of the region and its boundary equally. Then its effect may be discounted. Finding a suitable function requires differentiating ande substituting the derivatives into the proposed stress function as you have indicated. The only way for the arbitrary part to affect all parts equally is for it to a constant function or at most a linear function, whose derivatives are zero or constant.

Well I am not claiming that there is a single object, but a simple example would be a shadow.

Given that the building blocks of life are molecules, it seems reasonable to assume that life arose from molecules in a "primordial soup". Well I agree with Luc on this. But only partly since he has made the basic mistake of trying to combine two separate statements, each with their own independant veracity, "Arising from molecules" is about source. "In a primordial soup" is about place.

Is anyone claiming that there are no immaterial objects in the known universe ? I claim that there are such objects, some of which actually admit of some Science of their own. Further, at least some of these objects can affect material objects.

Most approach this by extending Timoshenko theory. Here are some links. https://www.sciencedirect.com/science/article/pii/S0020768316300725 https://ntrs.nasa.gov/api/citations/19910000994/downloads/19910000994.pdf https://jresm.org/archive/resm2015.19me0827.pdf file:///C:/Users/somerset_2/Downloads/Mit_2001-06.pdf https://www.ias.ac.in/article/fulltext/reso/007/10/0054-0058 https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/gss/default.htm?startat=ch06s02.html

Pages 700+ are right in the middle of a 1450 page book so it doesn't scan very well, but here is a table from chap14 if it is of any use.

This large more modern book deals with both analytical and FE methods for warping, and has tables along with some discussion and some useful references. Pilkey chapter 14 page 705. Please note the special notation used in this chapter as there are cross references to the rest of the book, including more twisting and torsion. This chapter includes a tables of equivalent symbols. This large more modern book deals with both analytical and FE methods for warping, and has tables along with some discussion and some useful references. Pilkey chapter 14 page 705. Please note the special notation used in this chapter as there are cross references to the rest of the book, including more twisting and torsion. This chapter includes a tables of equivalent symbols.

Glad you came back, because I am not sure what you mean by warping functions. I think I must call them something different. I'm sure there must be an old name for what you want. So Are you referring to torsion ? Are you doing this for analystical solutions of finite element analysis ? Are you aware that any analysis is only valid for a range of width to length or depth to length ratios. There is things called wide beam theory, slenderness ratio and so on. or are you referring to products of inertia as opposed to moments of inertia ? Please elaborate a bit. Can you give an example? My original references came off the top of my head, near my midnight as I was about to shut down. Just some I could grab quickly.

Look at your formula estimate for maximum speed. I make it less than 2m/s or walking pace ie less than 4 miles per hour. How much air resistance do you anticipate at this speed ? Look at this google page in response to a question "how far should a pendulum swing ? Gravity only acts vertically. The string tension has to have both vertical and horizontal components. What hppens if you increase the angular displacement to 90o ? Basically I am encouraging you to think about the mechanics of the setup.

OK

@exchemist Di you mean to quote me ?

Useful, but doesn't answer my questions. A google search of the use of hydrogen in diving gases is interesing. Here is an authoritative publication. https://pubmed.ncbi.nlm.nih.gov/38507913/

Perhaps you misunderstood my comments. I was questioning the importance of the drag effect. Further I am having trouble understanding how a displacement of 30o could keep the swing arm properly constant. Wouldn't the string be slack ? And you should make clear whether the 30o was measured from the horizontal or vertical.

Since you have quoted less than 5% of a mixture you must have had a source of information to start from. So for instance what is in the rest of the 96% ? And how is it applied ? Please post it here for comment.

I don't know where in the world you are, except that you don't seem to be in my time zone. Consequently I don't know what you mean by 'ice cream' ? I remember from a few years back a major row between the EU and the UK about this question. Whatever, if I wanted any, why on earth would I ask a Chemist ?

Perhaps a trawl through this book might help

In many ways this is true and indeed there is even a great deal of theory in science and engineering that take great advantage of this. However we do not live an a 2D world. In fact such a world is incomplete as we can deduce the exsistence of at least a 3rd dimension mathematically. So there are times when it is necessary for 'solid' thinking. Ok we will do some trigonometry when we have done the next bit about sigmas. And perhaps reinforce stuff about vectors as we go along. I take it you are happy with labelling the axis with X, Y and Z in some order or other ? We will deal with the order in a minute. The diagram shows that we can create or imagine vectors i , j and k respectively parallel to or along the 3 coordinate axes, X Y and Z. (bold letters are often used to denote vectors) Further we make each of these vectors of magnitude exactly one. So i , j and k each have magnitude exactly 1. We then call these unit vectors in the directions of the axes. Having done this we can create any vector whatsoever that is parallel to one of these axes by invoking the first law of vectors. That is a larger (or smaller) vector is a scalar times a vector. So if I have a scalar a and a unit vector i then the larger (or smaller) vector is ai That is the meaning of the second figure I posted where the vector A is the sume of the three components A = axi + ayj + azk - where ax, ay and az are usually different say a1, a2 and a3 So our vector, A = a1i + a2j +a3k Note they have used capital As where I have use small ones. OK to move on one stage further Instead of naming the axes X, Y and Z as we learn in school we can call all the axes X-something. If we call them X1, X2, and X3 something rather clever happens. We can bring in the sigma notation A = [math]\sum\limits_{n = 1}^3 {{a_n}} {X_n}[/math] Where the index, n runs from 1 to 3 for both the coefficient scalars and the axes. My reference to Einstein was that he proposed to simplify even further and for get the sigma alltogether So A = [math]{a_n}{X_n}[/math] Where the summation is 'understood'. And there you have your first @Mordred type tensor.

I believe I said don't woryy about the rest of the maths, there is more there than we need for now. The main message is about splitting a vector into components one parallel to each axis. (hence the i, j, k) Or if you prefer components combining components to make a single vector ( they call the single vector A on that webpage)