studiot

Of course there is, both in maths and reality. Neither you, nor I, nor anyone else can redefine the circle in mathematics, which is what you are trying to do, and imatfaal has already commented on. Mathematically a circle is defined as a particular locus. In the real world we can actually trace out this locus, so the 'real' circle has real existence. It does not matter that there is not perfectly circular real world object. A (circular) cut through an object will do if you want a concrete noun. If you want abstract nouns then consider a range-range navigation system. The position lines of a rover in this system are perfectly circular. And yes the position line exist as much as other abstract nouns such as the colour red or the emotion joy.

Crafting a mundane common or garden example to describe a concept is never as perfect as a carefully worked out formal definition scheme.

duplicate thread http://www.scienceforums.net/topic/80092-properties-of-negative-lengths/

There are two answers to you question on ripples. Mine was posted within 11 minutes of your original post. I do not see a response. I do not see any response to the several members who have taken the trouble to answer your question about mirrors either.

Let us say you are on a circle. By that I mean you are confined to travel a circular path as for instance if you were on the circle line on the London Underground. You only need to know one number to reach any other part of the circle. Now this number may increase steadily and evenly eg 1 km along the line, 2, km, 5 km and so on, or it may change in jumps eg 1 stop, 2 stops, 5 stops along the line. In this second case the distance along the circle represented are uneven. From this point of view the circle is 1 dimensional. But now examine the situation from the point of view of the cleaning contractors for the stations, who, suprisingly, are not allowed to travel on the trains, but must arrive by road. They need two numbers to specify the position of each station, easting and northing coordinates on their map. So they regard the circle as 2 dimensional. Yet again look at it from the point of view of the train authority, wanting to plan an extra line connecting two of the stations across the circle. Yes, they need the easting and northings of both stations, but that is not enough. They need a third number for each station to make sure the new line comes in at the right level. So they think of the situation and their circle as existing in three dimensions. This point of view of the dimension being how many numbers you need to specifiy something works well in ordinary geometry, but recent advances in mathematics has rather upset this. The discovery of fractal geometry to be precise. This is why I urged you to look up Euclidian, Hausdorf and topological dimensions. Mathematically there may be more than one dimension of an object or situation, depending upon the point of view. But a number for instance 1 or pi is the same from whatever point of view you look.

That's a very posh title. I can just about make out some marks on the concrete surface on your photos. What are you proposing that is special (ie more than just random marks) about them? Why do you think that zig zag shapes cannot arise randomly, in a similar fashion to the leylines fallacy? Incidentally a small tip: It is always a good idea (and conventional) to include a scale in technical pictures. I used to use a plastic ruler obtainable from young children's shops when I used to photograph concrete features. This has highly coloured centimetre and decimetre sections, which is adequate for concrete. go well

delta H, delta G and so forth tell you nothing about the rate of reaction, only whether the process will produce or consume energy (heat energy, free energy etc) They do not even describe the activation energy that may be necessary. I just watched the latest Tom Cruise and he made great foofah about Horatius at the Bridge. The 'rate determining step' is a bit like that - only soe many molecules at a time can participate. Yes a reaction will continue to proceed if new material (reactant) is supplied or if products are removed eg as a precipitate in a solution reaction.

Don't ever try something like this with the AC mains.

12% battery is too low to reach before recharge. The 32 minutes is based on moderate use, no cds running, no wifi running, and a last gasp into hibernation. Hibernation is unfortunate because if the battery runs out of steam before the hiberfile is fully created (and they are huge files) the computer will be unbootable next time. I get a regular stream of student's laptops to fix where said student has fallen asleep with the laptop in bed and it has not has enough juice to go to hibernation properly. The first rule of engineering is don't run a machine above 75% of capacity. Use the 25% as spare for special occasions. Turn on a valve fully and turn it back a fraction etc. Incidentally, imatfaal, my DVD writing box, P3 with customised windows 2k, would give your linux a run for money any day.

The crux of this question is "What is the concentration of a solid?"

That is a very good question, which shows some good scientific observation, well done. You are right a mirror is not lke a television or video screen wall. You see what is called a virtual image in a mirror, because your viewing distance from the mirror is large. This image is a reflection of light from all the objects that the mirror receives light from. (Note this is not all the light the mirror reflects since some bounces off (reflects) at such an angle that it misses your eyes.) The width (and height) of this is called the field of view. A small piece of (broken) mirror or just a small mirror has nearly the same field of view as a larger one. Incidentally I replied to an earlier question of yours, but did not hear any more. Do you not review your threads for answers? http://www.scienceforums.net/topic/79701-ripples/

I am sorry we seem to be in widely separated time zones, but you must have a genuine interest as you have returned several times to this. This may catch you quickly but I will return with a more complete post later. The most important thing to understand is that the energy (or work done) in stretching a spring is not force x distance. It is One half of that. Espring = 0.5 x Force x Distance.

Question 1 Part b Rethink. I did suggest the frictional dissipation be taken as zero since I though there was insufficient information given to calculate it. However on reflection I see that you can simply add another term for work done (a la swansont) against the sliding friction. So strain energy of the spring = gain in potential energy of the block plus the work done against sliding friction. Tis second work term is simply force (Fsliding* distance) = d = extension, and Fsliding = coefficient of sliding friction times normal reaction,. When the energy equation is set up it only contains the terms required plus the coeficient of sliding friction. So this can easily put put in terms of the other terms. You should be able to complete the question from here.

I have had problems with the comments of some here, but not with any of yours that I've seen. No need for apology there mate.

First question Yes swansont is correct. The fact that the extension = zero means that the tension in the spring has reduced from the limiting tension (T) at the initial position to zero. This limiting tension has already been worked out in part A. Since the movement was slow we assume no dissipation due to friction (!?) so all the strain energy from the spring is used in raising the block from its initial rest position up a height equal to dsin(theta) adding to its potential energy. Thus PE gained by block = mg d sin(theta) Strain Energy lost by spring = 0.5 T d Second question Something is wrong here. You cannot have 3 as both greater than/= and less than/=. In any case the formula for U give U = -9 at x=3. U corsses the x axis and become negative at x=2 so are you sure the second condition is not x>2?

No the idea was to give you some (useful) terms you could investigate further (look up). I said nothing about the word 'space' which is another term that needs qualifiers and has particular defined meanings in physics and in mathematics. Numbers, dimensions and space are different technical terms that are independent of each other, just like apples, oranges and bananas are independent different fruit. And equally number, dimension and space can be used in combination to build up more complicated concepts, justs as the fruits can be used combined to build up a more complicated food preparation.

You have described 'flow entering the channel' from the uncharted area (Here be dragons ) to the left. Why would flow enter the channel? I assume that the ocean to the left has a common water surface level at the interface with the channel, even if there is an abrupt and enormous change of bed level. In a real world water approaching such a sharp profile would create vortices in the lee of both right angles and laminar flow euqations would no longer be applicable. Water can only enter the channel if water moves along the channel and something cause that movement. Bignose has already alluded to that. What ever causes the movement imparts changes of pressure etc to the water. Yes, Poiselle's equation is the simplest profile. Real world hydraulic profiles in open channel flow are associated with the names of Chezy, Manning and others. The water surface profile is complicated and full of features and heavily influenced by changes of bed slope (and obstructions). In addition to uniform flow with a surface parallel to the bed (note not horizontal) the flow depth can be gradually varied (also called draw down) or gradually increased to pass over an obstacle (called a backwater curve|) or it can show a step change at an abrupt reduction of gradient. This is called a hydraulic jump and can be quite spectacular. In addition to the NS equations you should study the momentum balance equations. These are the ones that describe the forces that act between the layers of water and the boundaries. I suspect these will supply the information you are lacking. In order to study viscoscity/ friction you need to study momentum transport between the layers.

Let's start with this one since it introduces a complication that is unhelpful until you are happy with the basic flow equations. I am assuming you we are discussing flow of water in an open channel with sides and a bed, and that your diagram is a plan view with water depth measured in the z direction ? As such there will be a particular pressure at any given depth due to the sum of the imposed atmospheric pressure and the pressure due to the water depth. If the atmosphere is constant in the x direction we may subtract the constant atmospheric pressure and deal in what is known a gauge pressure. If the bed is horizontal the pressure is the same everywhere and there water is still. There is no reason for it to flow. Now incline the bed so the left hand end is higher than the right hand end. This difference in elevation will provide the energy to cause flow along the lines you have indicated, except that it will develop a curved profile due to the friction with the sides and bottom. You should look up Poiselle's equation.

+1

Maybe I'm nitpicking but surely both these definitions are flawed? e is the sum of an infinite series, which we can prove by Tannery's theorem converges to the first limit. With the second limit what is the first n in your series ?

I found the idea to be intriguing and interesting. I still do. The only thing important to me about the thought is fixing the problems that arise because of the thought. By straight line; I meant as opposed to a geodesic. Though I may have been willing to play loose with the definition of geodesic by flirting with different possible causes. I asked because we gloss over the question 'what is a straight line'. It is interesting to note that Newton, for instance, did not actually mention straight lines, although N1 and N2 are often couched in terms of a 'straight line'. N1 Every body continues in its state of rest or uniform motion in a right line, unless it is compelled to change its s tate by forces impressed upon it. N2 The change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which the force is impressed.

This difficulty occurs because what is meant by numbers and dimensions is a profound question. For many purposes the physicist's view suffices and Imatfaal's comment sums it up nicely. Mathematicians go much further so neither the plain word number nor the equally plain word dimension are sufficient by themselves. So mathematicians use qualifiers for particular well defined situations. Cardinal number Ordinal number Complex number and so on, there are many more. Equally there are types of mathematical dimension Hausdorf dimension Topological dimension Euclidian dimension again there are more. Each of these have different properties, applicable to the situations in which they are used or found.

For programming, you need Clenshaw's recurrence method, on the trigonometric recurrence formulae: cos(nt) = 2cos(t)cos({n-1}t) - cos({n-2}t) sin(nt) = 2cos(t)sin({n-1}t) - sin({n-2}t) I have already supplied the refernce, but here is another one Fike CT 1968 Computer Evaluation of mathematical functions. incidentally my first reference explains why the series for sin(t) [math]\sin (t) = \sum {\frac{{{{\left( { - 1} \right)}^k}{t^{\left( {2k + 1} \right)}}}}{{\left( {2k + 1} \right)!}}} [/math] is unsuitable for computer programming. This series does not begin to converge until k>> |t| So for large t ( which a computer can be expected to respond to ) this can mean many, many terms.

Good luck!

Yes you are right.