studiot

I very much doubt the professor said exactly that. The flanges take pretty well all the bending moment, but that is not the only load/stress acting. To keep it simple I will ignore torsion and state that the other stress acting is vertical shear. This is taken pretty well completely by the web. So much so that the beam has to be designed to resist the inevitable tendency to buckling that this vertical shear introduces. You will see 'web stiffeners' welded into beams under particular load for just this reason

I don't quite understand what you mean? Why should ag = bg if a and b are different points of S? What does the condition (group axiom) that every member of G has an inverse imply about a and b and the equality of ag and bg? I find it helps to discuss such statements in relation to a particular example, did you have one in mind?

Do you really believe this figure or is it just empty words? What does it mean? Can I say that because we have only breathed one trillionth of the atmosphere there is any reason to believe the rest is not breathable? Or that if I have only swum in .0001% of the sea I have any reason to believe I would not be able to swim in most of the rest of it? Or that because I have only sampled 5% the rest is not made primarily of water? We know some from direct observation We infer more We test our infererences and refine our knowledge from the test results But we should not give credence to non scientific headline pseudofacts. ophiolite sums it up well

Posted at 10.32, offline at 10.33 ? Commercial computer environments are often air conditioned. Hard drives are not static sensitive, it is other computer components that you would need the paraphanalia for. Just observe sensible safety precautions like don't plug and unplug things live unless specifically designed that way.

Good evening doG

I have tried to offer a simple analysis and explanation. If you want the definitive text it is to be found in the Cambridge University book Contact Mechanics by K L Johnson. He decomposes the motion of contacting surfaces into three parts Sliding, Rolling and Spin This introduces both linear forces and moments acting upon the bodies and contact area. It should also be noted that, although the velocity of the contact point in rolling is zero, the acceleration is non zero so Newton's laws require a force to be acting.

If you think about it there must be a weak dependence upon speed. Deformation is a dissipative action, the energy is not recovered into the motion it is lost as heat. So the greater the rate of deformation the greater the heat loss, and one of the functions of the oil is cooling. Engineers ensure however, that this loss is very very small compared to the energy of motion. I repeat, however, that you need to distinguish between the rolling resistance of bowling balls (as asked by the OP) and vehicles, which is vastly more complicated.

Why should it depend upon speed? Rolling resistance of balls is due to the very slight deformation of the contact surfaces. This, in turn, is due to the force pressing the two together. This force is independent of velocity. This is why very high strength steel is used for ball bearings. It leads to minimal contact deformation. You should further note that ball bearings usually run in an oil environment that offers significantly greater viscous drag than air, without ill effect. Why do you think we use oil not air?

Well I think everyone here is talking about different situations. The Original Post . There is no mention of an axle or other connection, it is implied that the whole object is rolling eg a round or constant rolling profile object, like a bowling ball. If a wheel plus axle and drive is intended than there is a whole raft of aditional sources of retarding friction. Secondly I think the OP was not considering objects such as wheels because the rolling resistance of a tyred wheel is affected by tyre pressure, profile, flexibility and many other complicated factors. Energy is lost flexing any flexible wheel.

I don't really catch where this thread is leading but a couple of comments. The mortar in brickwork does not stick the bricks together. It is there to hold the bricks apart. Masonry in general is a compression based system. Many promising very strong fibres are also very slippery. It is all very well creating a strong fibre but a problem if you can't transfer the load into or out of it. Steel reinforcement in concrete copes with this by the use of hooks and other shapes. This option is not available to ropelike fibres. This problem has beset many existing uses of carbon fibres. Creep is another issue with many materials, particularly glasslike structures, plastic and semiplastic materials.

I am beginning to understand your difficulty. A harmonic oscillator has a constant total energy and partitions this between kinetic and potential energy. The potential energy is a minumum at the mean point and the kinetic energy a maximum. At all points the total energy may be expressed as a function of velocity v; frequency [math]\nu [/math]; displacement from mean position,x ; amplitude a; force constant f in an equation given by [math]E = \frac{1}{2}m{\nu ^2} + \frac{1}{2}f{x^2}[/math] Where the first term is kinetic and the second term potential energy. Note that the potential energy is positive , whether x is away from the centre or towards it since the second potential energy term depends upon x squared. The constant f does not change with direction. So when you come to substitute [math]V = \frac{1}{2}f{x^2}[/math] into the Schrodinger equation There is no question of positive or negative signs, it is always positive. You get to the Hermite solution by introducing a new variable s = x/a and allowing the simple constant of integration to become a function of s. Have you seen this derivation?

What you are proposing is tantamount to proposing that each quantum level is 'fuzzy' ie occupies a spread of energy values equal to the peak to peak variation. What evidence do you have for this?

You are posting in the relativity forum but attempting to impose Newtonian mechanics. Apply the correct formula and you will see for yourself why the question is meaningless as posed.

Then work is done on or by some agent. You have asked a very reasonable question that often puzzles folks and received a relatively short answer from swansont, as he has many calls on his time here. I have tried to fill in some of the gaps but you don't seem to like responding to more than a small part of posts. I do not wish to indulge in verbal fencing about this subject. I gave you the answer in post#13. The gravitational energy is inherent in the whole system by virtue of the masses and separations of the particles involved, according to the equations being bandied about. It manifests itself as the (gravitational) potential energy of the system. If either the separation or mass or both change then work is done, perhaps on one or more of the masses or perhaps on an outside body. The work done exactly equals the change in gravitational potential energy. In other words that is the available energy from the change. In order for the system to have arrived at its present state energy must have been input in the past in the form of work by some agent. We do not necessarily know the details of this, only the quantities. So that is where the energy came from. We observe the effects as 'the force of gravity' in Newtonian mechanics, or alternatively the acceleration due to gravity, but as swansont has already noted, we do not know how this force is generated. That is one of the big question curently being attempted by modern physics.

The negative sign is so that as objects approach closer to each other positve work is obtained from the potential energy change. So a falling stone does work, but it takes work to raise a stone upwards. You didn't reply to my last post noting that it does not take energy to maintain the presence of a force. You should also note that the potential is inherent in the system ie both the earth and the stone together, along with their positions and motions. It is not a property one part alone.

The classical view (originated by Ampere) is that magnetism is generated by (very) small current loops. This works for Gauss and Amperes law (without relativity). You should look up the 'Rowland Ring' theory, which is the most modern version. The more modern quantum view is that the loops are replaced by the spin of the electrons. By the way I think you are doing very well for your age.

Forgive the obvious, but for a particle is the 'vibration' not transverse, so it is neither moving away from nor towards the centre? Hermite polynomials have radial symmetry and apply to shells which do indeed vibrate in and out, but then shells are the other face of quantum mechanics and not particles.

This assumes you require energy to generate a force. That is not the case. Forces and energy are different, independent physical quantities. You need to get this very clear before proceeding to study types of energy. A simple example is a brick sitting on a table. The table exerts a force on the brick. No energy is involved. Energy is only involved, as uncool said, when something changes.

I should have added that not every section appears in every paper, and some papers may have additional ones I didn't mention. The list is a guide, not prescriptive. Also as mentioned some publishers do prescribe a particular format.

But fractions too are more than just numbers. What number, for instance, is represented by this fraction? [math]\frac{{{x^2} - 2x}}{{x + 6}}[/math] And what about the many types of fraction Algebraic fractions , continued fractions, partial fractions to name but a few more. Continued fractions are good fun becasue they represent an alternative way of thinking about recurring decimals or irrationals that have been discussed in this thread. [math]1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + ..}}}}}}}} = ?[/math] Is a definite (irrational) number, but is it a division?

Personally I would say that division is a process and a fraction is one presentation of that process and that Myuncle is correct in observing that they are different, even though his arithmetic is a bit rambling. You can apply division (ie operate the verb to divide) to more objects than just numbers. Indeed engineers perfected 'dividing engines' for just that process many years ago. Dividing a circle is exactly what is involved in creating pie diagrams. So yes I agree with Myuncle that even at early stages in teaching we should distinguish between fractions and division as the latter can and will be applied more widely in later teaching. It is an old saw that, however simplified our treatment, we should never teach something that we later have to say "that was actually false".

quote ajb "Right okay" Hopefully this makes now sense of my other posts for you. I'm sure you recognise the integrals fraction/division I posted that confused john cuthber as the x coordinate for the planar centre of gravity.

Are you sure? http://en.wikipedia.org/wiki/Gravitational_potential

Well as you observed the contact force of magnetic attraction is a direct force. That is to say it is a normal force between the magnet and the object. Newton's 3 rd Law says that this attractive contact force is balanced by a normal reactive force which is equal in magnitude but opposite in direction ie still normal. A lateral force applied to the object is at right angles to this and only faces the force of friction. The frictional force against sliding is the normal reaction force times the coeficient of friction. Since the coefficient of friction is less than 1 (typically 0.3 to 0.5 for a magnet and a piece of iron strip) the sliding force is less than the normall pull force.

Quote ajb All you have done is multiply two numbers together. You can think of any number as a map from the real line to the real line by multiplication. I'm sorry you are not understanding what I wrote. All the examples are written using operator notation. Yes indeed the first two examples boil down to the multiplication of two numbers, but the third example is not multiplication in any sense. Are multiplication or division not operations? In the third example I am operating on a circle with the ratio or fraction a/b. This would give me a sector and is purely geometric.