studiot

It is not the tangent, it is the hyperbolic tangent [math]tanh\left ( x \right )= \frac{e^{2x}-1}{e^{2x}+1}[/math] I need to go out now, but post again later when you need more.

I am sorry, I apologise I was too hasty the acceleration is not constant. In order to derive your formulae you need you to set up and solve a differential equation. Do you know anything about differential equaions and their solution by the method of separation of variables? A common assumption is that the resistance is proportional to the square of the velocity. In this case the equation of downward motion is then [math]mg - kv^{2} = m \frac{\mathrm{d} v}{\mathrm{d} t}[/math] Separating the variables the equation becomes [math]kdt = \frac{mdv}{a^{^{2}}-v^{2}}[/math] Where a2 = mg/k Integration yields [math]c + kt = \frac{m}{a}tanh^{-1}\frac{v}{a}[/math] Can you put the values given to obtain the constants?

Is there some reason form making life so difficult? What is the only acceleration of the body and is that constant?

When you study the physics and mathematics of oscillations the first thing you learn is that there is a disturbing force and a restoring force. Without these oscillation cannot and will not take place. The oscillation is controlled by the restoring force, not the disturbing force, which fact is entirely in accordance with Newton's laws. The disturbing force may act once only at the beginning or it may be continuous, in which case it is called a forcing function, but the restoring force must always be available as needed (ie it will be zero at some point in the cycle and vary in strength and direction over the cycle) The disturbing force is considered external and the restoring force considered internal to the system. Once the disturbing force has done some work on the system and thereby input some energy, the system forces take over. The restoring force is normally considered conservative ie it does not loose the energy input. There may or may not be dissipative forces acting, I have already agreed that for the purposes of this thought experiment we can discount friction. The normal way to do this is to say it is negligable so that although we know in the long run it will eventually loose all the energy, this may take a very long time, much longer than the timescale under consideration. There is also a second energy loss mechanism that can act called dispersion, which is not applicable in this case and leads to non linear oscillations. But you do not need to go to all this trouble to present such a system. The simple pendulum will go for a very long time in vacuo before the weak dissipative forces drain the initial impetus. It also demonstrates, quite clearly that Newton's first law requires the presence of the restoring force to reverse the direction of motion at the extremal points of the oscillation. There is also a thermodynamic viewpoint as it is possible to (theoretically) construct thermodynamic oscillators and the thermodynamic considerations are interesting as they link in with Newton's laws. Take an adiabatic pipe containing an ideal gas and a frictionless piston. The oscillator is made by giving the piston a displacement in the pipe. The piston will oscillate back and fore indefinitely in the absence of dissipative force. This is an interesting example in terms of entropy and energy considerations. Your magnetic linear oscillator has similar characteristics. However, Klaynos is right in the real world there will always be dissipative forces acting. For instance inductors have resistance and therefore resistive losses.

Yes thank you for an interesting viewpoint, enthalpy, particularly the non contiguous bit. Even the maths of that situation is still nascent. I further agree with your comments that multiple slit experiments are over emphasised. +1

Yes you can move the dirt, but not (hopefully ) the patio concrete. All that shows is that in some cases momentum but no energy is transferred, in others both are transferred. Is that a problem?

In my water stream example the water looses all its 'sense' of its former direction as it streams off against the wall in all directions. It could have come from lots of different places. Here is another example. Consider a pendulum. It has lots of kinetic energy as it swings through the low point from left to right. It also has lots of momentum towards the right. It also has zero potential energy at this point in its swing. When it swings back the other way it has exactly the same kinetic energy, But its momentum is exactly opposite from the left to right swing.

No that is the opposite of what I said. The stream of water comes from one direction, carrying its energy and momentum with it. The stream hits the wall and transfers all the momentum but none of the energy.

Ok so they are related, but they are also different. It is possible to transfer momentum, without energy transfer. Take for instance the impact of a stream of water on a blank wall. The water impact transfers all its forward momentum as a (pressure) force exerted on the wall. Hower since the wall does not move this force does no work and no energy is transferred. Particles bouncing off a surface are another example of this.

The meaning of reality, nothingness and non physical 'objects' have been discussed before on this forum and also the relationships between them. Generally the discussion falls into the province of philosophy, not mathematics. I see you are from Sweden and I know nothing of Swedish. I can tell you that English distinguishes two types of object (noun) Concrete nouns Abstract nouns This is where the difference (in English at any rate) between 'reality' and 'existence' comes in. Objects (ideas) can exist in the abstract but not the concrete. edit clarification abstract objects like ideas can exist in the abstract without also having concrete existence. Reality includes all concrete nouns, but only some abstract ones. So existence has a wider remit than reality since something can exist "in your dreams" to use a phrase. I can verify the concrete reality of an apple, but not of an atom with 300 protons and no neutrons. There are also bordeline grey areas that are worth consideration and we return to mathematics here. Consider a 100mm cubic block of wood. This has a very definite physical existance, and is big enough to contain many 50mm diameter circles, I could easily cut several out of it. But do these circles exist before I cut them out? Since the length of one side is 100, mathematics tell me there must be a point in the wood that is 37mm from one end. So does 37 therefore exist? I bore a 20mm hole through the block. Dole the hole have physical existance? Back to counting. There are tribes in the South Seas that do not possess our counting skills. There number system goes 0ne, two, many. So do numbers exist in Stockholm, but not in the Solomon Islands?

how ? Here is an example of a meaningless near sentence that may mean something if put right, but we don't know what you actually want to say so we don't know how to help.

What do you really mean?

So do the Buckingham analysis it was developed for hydraulics originally.

OK so you mean integers. It is interesting that you have included zero in your list. Mathematically the set of positive integers is usually called the natural or counting numbers. Since zero is neither positive nor negative it is usually excluded. However some do include it in the set. The best origination for these numbers is in the name 'counting' numbers. The idea of a whole of complete unit precedes that of counting and the natural numbers. Once you have a unit the numbers follow naturally from the process of counting ie adding a single unit each time to the last. This will not help much if your objective is to study integer algebra eg Diophantine equations, but it is a start and your original post was very sparse.

Quote from Wikipedia The Colebrook equation is an implicit equation that combines experimental results of studies of turbulent flow in smooth and rough pipes. In other words it is empirical so has no 'derivation' . That does not mean it is without any justification, do a Buckingham dimensional analysis to obtain the 'form' of the equation (but not the values of the constants)

Perhaps you would elaborate on what you mean by the natural numbers?

Well the dog is certainly visible.

I raise my hand in admission of this sin, yer 'onor.

But that has never been in doubt, except perhaps the without bound bit since they will either eventually render the object asunder or crush it. Also it depends what you mean by 'by themselves' The Magdeburg Hemispheres are perhaps to most famous example of this. http://en.wikipedia.org/wiki/Magdeburg_hemispheres

It can be frustrating when one side refuses to engage in discussion and wants to be the only shop open on the block. So unfortunately even if we all modelled ourselves on ajb it would not work. He has many customers on whom his extreme patience is wasted.

If a single force is exerted on the grey material then it must move in response to that force and the OP (who has not responded to my legitimate queries except by changing post#1) has developed a perpetual motion machine of the first kind. It is surely important that the magnet system does not exert a force on the grey material?

Funny I assumed exactly the opposite as a metal surface would greatly complicate issues and the system could never be in equilibrium.

Why not?

No I meant exactly what I said in (1). It is not possible to avoid this moment. Try to understand my points rather than just contradict them. You cannot move a system from its mechanical equilibriium position to a position of disequilibrium without applying a force.

Understanding inertia (physics answers the how but not the why) in respect of linear motion is the easy part. Rotational inertia is rather more tricky.