studiot

I have already answered your question. Is English your first language? I will try to repeat in simpler terms. It is impossible for a vortex to develop in motionless air. There is energy of motion in moving air. During this motion the moving particles of air contact obstacles in their environment. These obstacles provide sideways forces on the moving air and sometimes also forces parallel to the flow. These interactions develop the necessary impetus to kick off the curving motion that leads to a vortex.

A body in orbit is not in equilibrium, therefore no 'balancing force' is required. That is the whole point of the laws of dynamics, or the second sentence of swansont's post#2.

It is one of the properties of waves that they can interpenetrate without loss of identity. That is two (or more) waves can occupy the same space at the same time. Massive objects (matter and particles) are not supposed to possess this property. The consequences of viewing massive objects as waves has interesting implications which are worth discussing.

Mike, have you considered they may have a point? It could be suggested that the comparison in your post fails to address the central (pun intended) issue viz that the introduction of fictious forces reduces a problem from one of dynamics, whether lagrangian or other, to one of statics ie equilibrium. It also allows Newton's third law to be presented as "to every force there is an equal and opposite counterforce".

Amaton, You may like to took at my post #73 in this thread, where I show how to construct the ordinals from nothing at all. http://www.scienceforums.net/topic/69384-what-is-nothing/

Yoyu nearly had something there! Here is how to (mathematically) construict something from nothing. 1) Start with literally nothing - the set with no members - [math]\emptyset [/math] 2) by successively adding more sets with no members form the transitive sequence of sets [math]\emptyset [/math] [math]\{ \emptyset \} [/math] [math]\{ \emptyset ,\{ \emptyset \} \} [/math] [math]\{ \emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} \} [/math] [math]\{ \emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} ,\{ \emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} \} \} [/math] and so on Note I have now formed a sequence with, 0, 1, 2, 3, 4......... members So I have now constructed the counting numbers plus zero!

Not sure what you mean by this question. If you mean what causes streamlines to curve perhaps you should study the Magnus effect. If you mean where does the energy come from to drive a vortex then it must come from the interaction of a fluid with its surroundings and Hamiltons principle. It is, however, important to distinguish between force and energy.

The Clausius-Clapeyron equation is far too heavy to understand dew point (DP). To understand DP and relative humidity (RH) you really need to start with the difference between saturated and unsaturated vapours. Are you familiar with these?

Indeed the best answer is yes and yes. a parcel of fluid pursuing vortex, or other curved or rotary motion is no different from any other particle in mechanics. You have two choices for the analysis viz to calculate the actual forces acting in which case you would use centripetal force and Newton's laws of dynamics ( and viscoscity in a fluid). Alternatively you could go for D'Alembert's quasi static solution in which case you would apply the fictious intertial centrifugal force and consider the resultant equilibrium of the fluid parcel. For instance within a centrifugal pump, this force balance is provided by the reaction force of the casing on the pressure developed within the fluid by virtue of its motion.

No it would not be correct to say that the square root of 16 is plus / minus 4. That is simply because you are using slack wording. "the square root" is singular whereas plus / minus 4 is plural. Yes the number 16 has two square roots +4 and -4. What you are thinking about is the square root function y= sqrt(x). Functions are defined to have only one value so we take the positive ones by default and define a second function z=-sqrt(x) to access the engative values. 16, of course is a number, not a function.

You can't rely on Wikipedia to be 100% accurate, it does not have the benefit of the level of scrutiny enjoyed by conventional textbooks. In the case of that article I think the authors have confused the zero (or null) matrix with a matrix that has no entries. You cannot create a matrix on a computer (or anywhere else) that has no entries. So wht happens is the matrix which has every entry as zero is employed. This also has to happen in matrix operations viz you have to select the entries from a field, which has a zero element, so you can do conventional arithmetic. Remember the rules of matrix operations: Let E be a matrix with no entries, a be a matrix conformable to additions with E and B be a matrix conformable to multiplication with E. Then what matrices do you think are the results of (A+E) = C and EB = D ? C cannot equal A and D cannot equal B since E is neither the additive nor the multiplicative identity. However the rules require that every sum and product can be formed. therefore E is not in the set of all matrices, ie it does not exist. Further, remember that computers start counting from zero, we start counting from 1. ie a byte is 0-255, not 1-256.

I really think that the reference I gave in post#12 explaining inertial mass is most elegant and accessible. In pages 1 -6 Kermode provides a wonderfully chatty exposition of introductory mechanics, including simple versions of Newton's laws (N1;N2:N3) along with a good explanation of why it can be misleading to say simply "force is proportional to acceleration, the constant being mass" His description of the broken railway coupling, designed by a designer who believed this, is a prime example. It may be noted that Kermode taught practical people (pilots) who need an intuitive but practical understanding of mechanics, rather than the ability to manipulate mathematical formulae.

Well I suppose it rather depends upon what you mean by 'outside', as the man said. If you mean that are there any physical quantities that are totally independent of or unaffected by both space and time, that may neverless be observed in physical objects or entities within space and time, then yes we can assign meaning to the idea. For instance the temperature at some point in space or time has this characteristic. Look for other physical quantities that do not contain either T or L in their Buckingham dimensions.

Don't you think that a definition that admits the possibility of division by zero is kinda worrying?

This is exactly correct. But it does not then follow that there is anything at all wrong with the equation. All equations, theorems, etc come with conditions of applicability or validity when we apply maths to physics (or other disciplines). That is to be expected because the maths is only a model of the other discipline, it is not the same thing. Obviously we choose the maths that accurately models our area of interest. It is a common failing to ignore these conditions, often with disasterous results. A simple example would be the calculation of stresses and deflections in beams. The 'normal' equations assume that the span is greater than 20 times the depth. Short or deep beams that do not meet this criterion require the use of a different formula. In such a beam, the standard formula will offer a perfectly possible stress that can indeed be generated by a (slightly) different load, but will not be the actual stress experienced by the deep beam. The standard formula will, however offer the correct stress in a standard beam. In such circumstances we say that the standard equation 'breaks down' in the case of short or deep beams. A more spectacular example of breakdown along the lines of ydoaPs' infinity would be the equation of specific energy in a travelling fluid. This becomes infinite at a discontinuity such as a hydraulic jump. This is typical of any equation that has a 1/(x-a) term which therefore involves division by zero at x=a. Does this help?

Whilst I agree that the OP hasn't offered anything substantial or scientific to support his proposition, I cannot agree that dimensions are per se orthogonal. Buckingham dimensions have no orthogonality and, conventionally, there are six of them. I note Bignose echoing my comment about generalised coordinates.

Did my post#9 get lost in the war?

In general, science and technology tries to have a single consistent definition in use for any specific word so that scientists can readily communicate with each other. Unfortunately 'dimension' is one of those exceptions where there are at least three different uses in science plus one more in mathematics. Normally which one is meant is obvious from the context, but I have witnessed many a silly argument between people who are using different definitions of a particular word. One thing that 'dimension' does not mean in science is something akin to the eastern mysticism of etherial planes, which seems closest to what you are implying with your list. go well

Please explain.

Apologies, I see I made a spelling mistake the integers, including zero should be the ordinal numbers. For a more mathematical example consider the set of possible numbers of row or columns of a matrix, M. If the matrix is square then M is also the dimension of the (vector) space spanned by the matrix. In any event consider whether the elements of M should be drawn form the ordinals or the natural numbers. If we allow zero what is the effect of multiplying a matrix with another with zero rows? What does a matrix with zero rows look like?

In post#1 you asked as short specific question without qualification. This seems a serious question and four members, including myself, took your question in that vein and responded seriously. You responded to each member in such a way as to make them believe your original intent was facetious. So far this process has been pretty counterproductive and we have reached your post#11. I am going to take post#11 as serious and supplying some of the missing context. There are two concepts of mass in classical physics. Gravitational mass as used in Newton's law of gravitation. Inertial mass as generally so far offered here. On of the deep questions of physics is to understand the reason that these two concepts coincide. Engineers don't bother to worry as to why they just attach constants to make bring about this coincidence. If you want an introductory discussion of this issue in modern physics both classical and relativistic/quantum, look at The lightness of Being by Frank Wilczek. The book is largely about the subject of mass in physics. If you want a good pragmatic development of inertial mass look at The Mechanics of Flight by A.C. Kermode, OBE Which developes the use of mass as the constant in Newton's laws. If you would like to post further details of your viewpoint classical or relativistic/quantum we can discuss the matter further. Yes you are correct you need to establish some givens to develop a formal system of mechanics. One of these concerns motion, which necessitates the concepts of space and time. The concept of a body brings in the issue of what is moving and leads to mass amongst other properties. Formal mathematics eg time derivatives are also permissible.

In the UK the Permian and Triassic are usually lumped into one system and the permo triassic New Red Sandstone outcrops from the South west diagonally across the country towards the North East, running from Exeter to Watchet, up the Severn valley, through and under the heart of the Midlands and the Vale of York and into the North Sea at Middlesborough. In part it marks the boundary of the former Zechstein Sea of that era where marine limestones were laid along its edges.

Well in classical physics ie mechanics mass is the constant of proportionality that connects force or momentum and velocity.

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