studiot

But that velocity is perpendicular to r so are they in relative motion?

Go on.

What were parts A and B? They usually hang together so one leads on to the next. What exactly do you not understand about C and D? You have noted that despite the two axes shown in the sketch this is a one dimensional problem (ie x is not horizontal)? Finally I take it that the spring and dashpot are vertically aligned through the centre of gravity of the mass, so there is no twisting of the mass.

In relation to time and the foregoing discussion about physical and real, consider the following: In space, positions x, (x+10) and (x-10) are all physical and real since I can move my boat from one to the other and back again as many times as I like and place it in any one of them if I am a good enough sailor. But in time I can only observe my boat in the present. I cannot move it to the future or the past or observe it there. So does this make the present physical and real, but the future and past non-physical and non-real?

There is the issue of the motion. Apparent mass gain appears to observers in relative motion. What is the relative motion of two observers at say r/2 and r from the centre of rotation? Does their separation alter?

Of course, John is quite right, the only entity for which A = -A is zero so you are just working on keeping zero to itself. I was thinking about the general trig sum of two angles. CAST is a way of remembering which trig function is positive in which quadrant, here is a list.

Consider alpha = 20 degrees and beta = 50 degrees. Are you really offering that cos (20+50) = - cos (70) ? Some of these statements depend upon which quadrant your sum ends up in, have you heard of CAST?

Does Hooke's law matter? By the looks of things rkt was suggesting calibration. I was going to suggest something like that myself, along with what we used to call a spud gun (that used a coil spring).

[math]I = \int {\frac{{du}}{{1 - {u^2}}}} [/math] [math] = \frac{1}{2}\ln \frac{{1 + u}}{{1 - u}}[/math] [math] = \frac{1}{2}\ln \frac{{1 + \sin x}}{{1 - \sin x}}[/math] [math] = \ln |\tan \left( {\frac{\pi }{4} + \frac{x}{2}} \right)|[/math] [math] = \ln |\sec x + \tan x|[/math]

Well done for doing it yourself. +1

Because you have made d(1/cosx) in your line 1 equal to sinx in your line 2 rather than sinx/cos2x So work your integration by parts out again.

You have the right idea with your 1/1 but with the wrong part of the integral [math]I = \int {\frac{1}{{\cos x}}dx} [/math] [math] = \int {\frac{{\cos x}}{{{{\cos }^2}x}}dx} [/math] [math] = \int {\frac{{\cos x}}{{1 - {{\sin }^2}x}}dx} [/math] If we make the substitution sinx=u [math] I = \int {\frac{{du/dx}}{{1 - {u^2}}}} dx = \int {\frac{1}{{1 - {u^2}}}} du[/math] This has three different (logarithmic and trigonometric) forms. Does this help?

What makes you think there is a single most fundamental branch of mathematics? Why should this be so, after all the Hindus thought that the world was carried on four elephants and even neolithic man placed his lintels on two supports. There are many different ways you could divide maths into branches. One such is maths is the study of number and shape. This was certainly true early maths, although the study of Geometry was indeed codified first, this could not have happened without some numebr theory. Here is an interesting viewpoint from one of the world's most famous mystical poets. Coleridege was also responsible for this poem about the first proposition on Euclid's first book of fomal geometry. http://blogs.ams.org/mathgradblog/2013/06/05/euclid-coleridge-poem-2/ More recently we have categorised the subject into two branches Analysis and Synthesis. Synthesis is interesting because it leads to the idea of cmputability, a modern notion. In modern times a good source for your research would be the Cambridge University text "Computability and Logic" by Boolos and Jeffrey go well

Just out of interest what do you mean by this? To balance the forces (is force balance on the GCSE? syllabus) in an infinite ocean it is a relatively simple calculation as CharonY suggests, although you must know a good deal of underlying physics to do this. But to measure the viscosity using stokes law requires some quite sophisticated corrections for real world apparatus.

I am having real trouble reconciling GCSE with first year university, which is where a student would normally meet the Stokes falling ball viscometer. Google provides many hits on stokes visometer here is a university lab from one. http://www.engr.uky.edu/~egr101/ml/ML3.pdf This gives the fully corrected formula. A simpler one is viscoscity, f =[math]\frac{{2{r^2}g}}{{9\nu }}({\rho _s} - {\rho _L})[/math] ps and pL are the sphere and liquid densities respectively r is the sphere radius v is the terminal velocity. I can't see that you could be expected to produce a derivation of even this simplified equation for GCSE

Actually that was the question. The kilograms were an example, as shown by the opening words "for example". The infinitesimal in the example was dm, which dimensionally correctly refers to mass. Since you have revised your question (1) from how many numbers make 1 kg? to How many infinitesimals (I will take that as dm) make 1 kg? and I said I would answer a properly posed question here is my answer. (1) An uncountable number. Perhaps I should point out here that is because the domain is R3. For the series I posted on the other hand the answer is a countable number. I already did answer your second question, quite specifically. You still have not answered any of mine.

Actually it is on topic since it addresses the OP. You have already been told by a moderator that your approach is off topic. Although you have not answered my question, I will answer yours (1) The question is flawed since the units are different on either side of the equation. Couch your question correctly and I will try again. (2) For the purpose of many physical models (already described) yes that is true. It is not a belief system, but a matter of definition.

Are you saying that the following are not true and should not therefore be taught to final high school / tech college / first year undergrads as they have been for more than the last hundred years? [math]\bar x = \frac{{\iiint {\rho xdxdydz}}}{{\iiint {\rho dxdydz}}}[/math] [math]{I_{xx}} = \int {\rho ({y^2} + {z^2})dv} [/math]

No, physics was not implied, maths was explicity called for since the OP not only posted in the maths section as opposed to the physics one, but further chose an area of pure maths over applied maths. In any case your argument that an infinite number of physical points cannot make up 10 kg is suspect. A vast area of physics, including most of classical physics is predicated upon the premise that you can indeed either infinitely divide a finite piece of matter or alternatively assemble a finite piece from an infinite number of parts. This underlies classical statics and dynamics, continuum mechanics, and even the angular momentum of quantum particles is derived from continuum mathematical analysis. Can you tell me any reason why, if each of the numbers in my series posted above was a mass coefficient, I could not assemble 1.6 kg from the first series and 1.2 kg from the second and any other value by suitable scaling?

I don't have a full solution, but try the substitution [math]n = \sqrt 2 p[/math] Which leads to the condition [math]p = \sqrt {\frac{{1 + {m^2}}}{4}} [/math] Which should be easier to handle.

This site may help with your general study. Look around beyond the particular page linked. http://sciencepark.etacude.com/chemistry/law.php Yes you are on the right track with your thoughts, that atom arrangements are changed during a chemical reaction, forming new compounds (molecules) from the old ones. Nothing is gained or lost.

Electrolysis? Split Elements? Perhaps you are talking about bond energies of compounds? Would you like to rephrase your question?

Here is areference that may assist http://en.wikipedia.org/wiki/Exergonic_reaction

Remember that the terms exothermic and endothermic refer to heat energy only, and does not include entropy effects directly. There are more general energy balances, Chemists tend prefer the Gibbs Free Energy, which does include entropy in the TdS term.

Your question brings up an interesting bit of history because Archimedes was the first to successfully study this question and he wrote down his mathematical treatment in the middle of the third century BC. Unfortunately this document (called The Methodf) was lost in antiquity and only rediscovered in 1910, when and old parchment was cleaned. Anyway suppose we consider this thought experiment: Take a ruler and pencil and draw a thin line 25mm long. Draw another line right along side the first line so that you cannot see any gaps between them. Continue drawing lines for about three hours. You will then have a rectangular area on your paper. No imagine sharpening you pencil and repeating the experiment. It will now take you many more lines to draw the same rectangle, say six hours work. Sharpen again and repeat. Perhaps you can see where this is going. The thinner the line the more you need to create the area until. Until the line is so thin the number is ver very large indeed. This is what is meant by tending to infinity. This was exactly the process by which Archimedes derived his famous mensuration formulae, and the process by which we add up a very large number of very small contributions to create a whole. It is well known that we can add up an infinite number points to obtain a finite total. Mathematically that is what taking limits is about. However you ask how can an infinite number of points add up to different values? Well the simplest way to see this is to look at and compare a couple of infinite series. [math]\sum {_1^\infty \frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \frac{1}{{{5^2}}}.............} [/math][math] = 1.645[/math] [math]\sum {_1^\infty \frac{1}{{{1^3}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + \frac{1}{{{4^3}}} + \frac{1}{{{5^3}}}.............} [/math][math] = 1.202[/math] You can see by direct term by term comparison that these two series have the same (infinite) number of terms, but their sums to infinity are different.