studiot

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.

I did ask for a sketch to confirm your flow. Bignose was also unsure. I think you mean what is called plane parallel shear flow given by the flow field. u = u(y,x,z,t) = (u(y,t), 0, 0) Note line 6 of post#2 should read The first equations tell us that p is a function of x and t only. Further since u is independent of x, [math]\nabla .u = 0[/math]

You should also take note that as waves progress to shallow(er) water the wave height increases and the profile departs markedly from sinusoidal, in particular the crest amplitude and trough depth are no longer equal or equal to half the sinusoidal amplitude, a. See the Reid and Bretschneider 'Breaking Index Curve'.

What do you mean by 'a straight line' and why is it important to you?

Have you tried trigonometric substitution. with x = tan2(t) and then use sec2(p)= (1+tan2(p))

Perhaps the correction to my last post will make more sense as an answer.

A picture is worth 1000 words. I think you mean [math]\frac{{\partial p}}{{\partial y}} = \frac{{\partial p}}{{\partial z}} = 0[/math] as conditions for [math]\frac{{\partial u}}{{\partial t}} = \frac{{ - 1}}{\rho }\frac{{\partial p}}{{\partial x}} + \nu \frac{{{\partial ^2}u}}{{\partial {y^2}}}[/math] The first equations tell us that p is a function of x and t only. The second (NS) equation tells us that [math]\frac{{\partial p}}{{\partial x}}[/math] is equal to the difference between two terms that are both independent of x. Thus [math]\frac{{\partial p}}{{\partial x}}[/math] must be a function of t alone. since we have steady state [math]\frac{{\partial p}}{{\partial x}}[/math] must therefore be zero. Am I reading you right? #Edit My apologies I see a serious clanger crept in during the copy/pasting. I said [math]\frac{{\partial p}}{{\partial t}}[/math] must be zero when I really meant [math]\frac{{\partial p}}{{\partial x}}[/math] I have corrected the above.

michel's original post mentioned 'a picture' and asked rather vaguely 'where and when?' What did michel have in mind? A picture would distinguish between northern and southern hemispheres and modern times and Roman times. Many pictures could provide more information, but cameras are not accurate surveying instruments so ascension and declination would have to be estimated in some way from the picture(s). There have been a couple of responses that suggest it is impossible to determine position and time on the planet from astro obs alone. Difficult yes, impossible, no. Accurate knowledge of times is very very helpful, but not essential. Years ago, surveyors used astro time obs to correct their chronometers.

The simple rate equation relies on the reactants being intimately mixed and free to move. This is not the case with solids. Reactions involving solids are multistep. An important step brings the reactants into contact and is often the rate determining step. Here is a guide to solid kinetics. http://www.fhi-berlin.mpg.de/acnew/groups/nanostructures/pages/teaching/pages/teaching__malte_behrens__solid_state_kinetics.pdf

Why not? A given star only transits its zenith over one point at any one time.

What do you mean convert forces? So long as there is no actual movement my lever example provides a 100% 'efficient' force 'convertor'.