studiot

Is this your own theory? 'functional' has a particular well defined meaning in maths and mathematical physics. Neither time (your other post) nor energy meet this definition.

Let us start with a body at rest at the mean position and then apply a disturbing force. For instance let a pendulum hang freely at rest, and be drawn aside a small distance and held at this side position. When at rest the (total) energy of the pendulum is zero. That is its KE and its PE are both zero (not equal and opposite). As it is drawn aside the disturbing force raises the pendulum against gravity imparting potential energy and kinetic energy to the pendulum. The kinetic energy is undefined, but when the pendulum is again at rest at the side position it has zero KE and a defined PE. Up to this point the pendulum is not executing SHM. If the pendulum is now released it will fall back to its original level under gravity. As the pendulum swings its PE is converted to the KE of motion until it reaches its lowest point at which time it again has zero PE, but now has substantial KE. So it swings on through the lowest point and start to climb back up. As it climbs it looses KE and gains PE, until in a frictionless system it has gained the same height on the other side of rest as it was originally released from. At this point it again comes to rest. All the KE has now been converted to PE, which is again positive. The pendulum is now executing SHM and the cycle continues indefinitely. At all times in this cycle the PE (and the KE) are non negative ie zero or positive. The pendulum starts off SHM with a specific PE and zero KE. Initially, the change in PE is negative, the change in KE is positive. Once this negative change in PE has reduced the PE to zero the change becomes positive and the PE increases again. If we can agree these factual observations we can proceed to discuss how this is implemented as a force-displacement-work calculation.

Agreed you would have far better pay and job security if you bacame a plumber and kept the theoretical physics as a hobby. If you must stick with physics then one growth area for the future is the manipulation of microscopic entities. This has applications in biology, chemistry, materials science, micromachinery as well as physics and the potential for you to create a company and reap the benefit if you discover anything. go well with your career.

A further distinction in law is worth mentioning, and again I appeal to imatfaal for amplification. Several posters have mentioned being sued for damages. There is a big difference between UK and US law about this, I do not know about Dutch or other EU laws. The difference is that in the US so called 'punitive damages' can be awarded and are often substantial. Further they are awarded by juries and their amount are often a political act. In the UK you have to be able to prove you have suffered material harm and are only liable to receive compensation for that harm and nothing more. That is why compensation in the UK is some much less than in the US for corresponding actions.

A simple distinction. The study of how fast, how far and for how long bodies travel, without worrying about the forces involved, is often called kinematics. The study of motion including forces, energies etc is called dynamics. Your problem above links the two together. Kinematics includes equations such as distance = speed x time and the more complicated one above that you mention. Have you studied kinematics yet?

A qualitative explanation? First to remember that the applied field and the field due to the material add (vectorially) to form the total field. This diagram may help. For diamagnetic materials there is no material field so the magnetic moments a zero without an applied field as shown in the first diagram. If we apply an external field the induced moments oppose the applied field reducing its intensity as shown in the second diagram. For paramagnetic materials there is a material field that averages zero without an applied field because every molecule has a random orientation. When we apply a field to a paramagnetic material all the random molecular moments align with the field, reinforcing it and increasing its intensity. Paramagnetic materials also exhibit diamagnetism, but paramagnetism and diamagnetism spring from a different mechanisms and the numbers just work out that paramagnetism is quite a bit stronger so with paramagnetic materials the diamagnetic effect is masked.

Thank you , John, for your reply. I fully appreciate that mathematically the direction of rotation is a matter of convention and perhaps choice. But the question I asked was about teaching the subject to youngsters. You may have been lucky with your professors or older or both but this question was prompted because I looked at the teaching material supplied to some 9/10 years olds about rotational symmetry., ie in primary school. They are being taught that the 'natural' direction of rotation is clockwise, having not long been taught about clocks. They will also be taught about bearings in Geography, which are also measured clockwise. I have seen much confusion arise in secondary schools when the maths teacher attempts to teach anticlockwise angles. So my question still stands for discussion.

Actually my graphs are merely a plot of the correct formulae, which are also stated on my diagram. A body executing simple harmonic motion is subject to two forces. A disturbing force and a restoring force. The equations that lead to SHM and the energies involved are as a result of the restoring force only. It is a common mistake to try to calculate with the disturbing force. The restoring force always acts towards a fixed point. That is the restoring force changes direction halfway through the cycle. We normally place the origin of coordinates at this point. In your analysis the restoring force is kx for half the cycle and -kx for the other half. You do not need to worry about the angle between the restoring force and the displacement since they are along the same line (the x axis). Because the restoring force direction reverses you need to separate the work integral into two integrals, each with its correct sign. See if you can work through this now and come up with the correct formulae. This is a bit like asking the question What is the integral of sinxdx from x=0 to x=2pi? The answer is zero, but the area between the curve and the x axis is obviously not zero. Again you have to split the integral to obtain the area. So far this is all mathematical, but thinking about the Physics may also help. What exactly is x? ie what does it represent? Well not only is SHM motion under the action of a restoring force always directed towards a fixed point, but further the value of that restoring force is proportional to the distance from thatpoint ie the displacement. x is the displacement from the mean point. Now if you wish to consider work and energy we can say that When the body is moving towards the fixed point the restoring force and displacement are in the same direction. So the restoring force is doing work on the body, accelerating it, ie increasing its KE. When the body is moving away from the fixed point the body is doing work against the restoring force, storing the work energy in the source of the restoring force as PE. This energy comes from diminishing the KE of the body.

1) I am no expert in life sciences, even I have heard of Darwin's Finches. I will leave you to do some research about this. 2) I understand Mendel was a contemporary of Darwin and published his (query more mathematical) rules not long after Origin. Again I will leave you to look up the comparison.

I don't understand the comment. The moving demonstrations clearly rotate clockwise.

Since light is just a form of EM wave, in wave terms, one of the simplest to measure might be to use radio waves rather than light and use lecher lines. Wiki offers them to measure wavelength but they can also be used to measure amplitude. http://en.wikipedia.org/wiki/Lecher_lines Of course a light intensity meter provides a measure of amplitude.

Because shining two light sources on the target produces a single combined light beam not two separate ones side by side. And of course a single light source produces only one light beam.

When we discuss angles in mathematics we consider anticlockwise angles as the positive direction of rotation. In particular this is carried through to phasor theory in electrical engineering and many other fields. But when mathematicians and physicists discuss rotational symmetry the clockwise rotational direction is chosen as positive. eg http://www.mathsisfun.com/geometry/symmetry-rotational.html Is this anomaly not potentially confusing for young minds?

Remember first that most materials are made up of molecules, not atoms. Many, if not most, materials exhibit little or no magnetism because the atomic effects tend to cancel out when the atoms form molecules. All materials exhibit an inherent diamagnetic effect. When a material is put into an external magnetic field, the orbital electrons are acted on by a force which changes the orbitals in accordance with Lenz's Law. Thus the effect is to oppose the external fied, thereby reducing it. Those molecules where the atomic magnetism does not cancel possess a permanent magnetic dipole moment. Without an external field these dipole moments are oriented randomly so again cancel. However when there is an external field the dipoles tend to align with the field, strengthening it. This effect is called paramagnetism. Paramagnetism does not, of itself, change the orbitals. Ferromagnetism is a particularly strong form of paramagnetism.

This whole discussion appears to be a rerun of this thread http://www.scienceforums.net/topic/77993-selective-nature-of-schrodinger-equation/#entry761150 The diagram I mentioned is called harmonic1 and appears in my post#13 there.

I would still like to be able to post the diagram if someone could help please.

Come back and ask questions about it when you like, I agree it's a lot to digest. If you can begin to make sense of it we can talk about the lagrangian formulation necessary to cope with electric forces.

With respect. Imatfaal, sidharath specified simple harmonic oscillation, not other forms of PE. I have posted the quadratic diagram here before and I see from my settings page that it is still there. If I knew how to do it I would refer to it here, can you help?

No you need to think again. The sign of potential energy (PE) itself is always positive, unless it is zero in which case it is neither positive nor negative. But you are talking about change of PE, which can indeed be positive or negative, depending upon whether the objects are gaining or loosing PE.

Any form of engineering is about compromise and balancing one competing (and often conflicting) requirement against another. If I listed some characteristics to think about for a general engineering product, in no particular order, what would be your reaction? Reliability, Length of service life, Serviceability, Efficiency, Recycleability/Disposal requirements, Safety,

OK I will have a go at a Hitchhiker’s Guide to Hamiltonian-Lagrangian mechanics. If I spread too much jam on it I’m sure Bignose will apply the necessary corrections. First a mathematical observation. I assume you know some elementary calculus. So I will make the statement that every process that can be modelled mathematically by a differential equation can also be formulated as an integral equation. Now Newton’s second law is a differential equation that models the motion of a body as it travels along some trajectory or path. At any point on that path Newton’s Law provides a connection between external forces and the local conditions (at the point) by way of the time rate of change (differential coefficient), to enable us to determine the actual path traversed. I stress this equation is local ie applies individually at each point, but each individual application may be different. Now an alternative view is to ask Is there a function that if applied to the whole path will define that path for us? The answer is yes. There is such a function that we call ‘the action’. If we integrate this action function along any entire proposed path it turns out that the action integral is minimised by using the actual path traversed. This introduces our first Big Name is known as Hamilton’s principle, also called the Principle of Least Action. Now two things. Firstly proving this is the province of some advanced mathematics. Secondly deducing the actual path uses a process ( some more advanced calculus) known as the calculus of variations. I will not attempt either here. The function itself introduces our second Big Name, the Lagrangian. [math]Action = \int_{{t_1}}^{{t_2}} {Ldt} [/math] and for simple conservative fields like gravity the Lagrangian is simply the difference between the kinetic and potential energies L = K – P A parting thought in support of Bignose’ desire for agreement with real world observation. So let us start back with Newton and his law of gravitation. [math]{F_{grav}} = s\frac{{{M_1}{M_2}}}{{{r^2}}}[/math] This states that for two masses there is a force between them proportional to each mass and inversely proportional to the square of the distance between them. s is the constant of proportionality to make this an equation. After it became possible to measure this force, it was noticed that for certain objects there was an additional force in action over and above that given by the gravity equation. This was found to follow the same form (Coulombs law) [math]{F_{elec}} = t\frac{{{Q_1}{Q_2}}}{{{r^2}}}[/math] Well physicists are lazy customers and like to recycle equations rather than introduce new ones. So when magnetism was being investigated another additional force was proposed and the relationship experimentally verified by Michell in 1750. [math]{F_{mag}} = u\frac{{{P_1}{P_2}}}{{{r^2}}}[/math] A quantity known as the pole strength was introduced. Do you notice any similarity? Note I have only used some of the standard symbols for clarity of comparison. Note also that all these three are experimental results. Observations on the physical world.

An excellent suggestion for sevral reasons. Since our last posting here Externet has posted some video references in his thread on this subject, and I replied selecting an axial flow turbine to achieve exactly this. http://www.scienceforums.net/topic/65768-paddle-wheel-calculations-please/ Incidentally one of the advantages of many small installations using horizontal flow is financial. In order to use water head you have to impound water and expend the capital sums to do this (ie build expensive dams or whatever) before there is a penny piece of £return. A small, low cost, installation, placed into an existing flow tidal or river, will immediately start generating and thus providing a £return to help fund a second installation and so on.

ajb Zero can be rather abstractly understood in terms of ring theory. Loosley a ring is a set for which we have addition and multiplication with some natural conditions. In particular we have zero as the additive identity, that is a +0 = 0+a = 0 for all a in the ring. That is zero is the identity element of an abelian group where we write the multiplication as a +. From the properties of the ring you can show that 0.a = a.0 =0. Here . is the binary operation of a monoid. These are the two algebraic interpretations of a zero that I am aware of. The zero ajb describes above is a valid member of the set and is not, I think , the meaning that O'Nero ascribes by the use of the word 'nothingness' (post#5). I think he is referring, not to a member of any set, but to the set which has no members ie the empty set.

If that is so why does Microsoft provide the arrowed buttons? I don't see how the product of two numbers by logarithms is helpful to this homework help.

That is not an equation. You did write an equation out before and I quoted it from your post #3. Your equation tells you what x is equal to in terms of y. That is it gives you a way of calculating x if you know y. What you need is a way of calculating y if you know x. So you rearrange your equation.