DavidAngelMX

Yes, it is a vector Space indeed. A vector space is a set of vectors associated to a group of scalars which fulfills the next conditions: a(V1+V2) = aV1 + aV2 V1*V2 = V2*V1 ; Vi vectors, a scalar Now, R3 is a Vector field, and the quation you present, which is the equation of an arbitrary plane, is a subspace of R3. In fact, it is the space of all R3 planes.

I agree about the slope, but a slight gust of wind could have helped to break the balance of the ball, leaving the rest of the job to gravity. But I think it's improbable that the gust of wind moved it by itself until it was inside the hole, because then the "slight gust of wind" wouldn't have been so slight, don't you think?

Well, not necessarily, because the alpha decay occurs even though the "potential barrier" that bounds the protons in the nuclei is greater that the energy any of the particles inside the nuclei can ever acquire. The alpha decay is a quantum effect, it doesn't mean that the emmiting nuclei has a lower potential or that the particle emmited had a higher energy that usual, it's just a consequence of Scrodinger's quation, only a probabilistic efect. So radiation emmiting atoms don't necessarily have a higher potential.

Well, let's stay on the field of mechanical waves I don't think this is right, look at it this way: When a sound has a low frecuency you identify as a lower tone(like a bass), and when it has a high frequency, as a higher tone(like a violin). So what you're saying implies that if you go to a concert with a bass player and a violinist, you would hear the violin first and then the bass, and even when you hear it from a long distance, this doesn't happen, otherwise there wouldn't be any big concert houses.

Well, I guess this thread is not interesting to anyone anymore, so, Hasta la Vista

Look, we all know that friction doesn't have to do with imperfect surfaces, but with electromagnetic interactions between the surfaces of the materials, so, if these spheres are made of matter, there should be some interaction, and as a consequence, there should be friction. So, if the friction is great enough to force the small sphere to roll and not slide, and assuming that there is no energy loss when a determined point of the surface of the small sphere is separated from the surface of the other one, then the ball should roll forever, since there is no air friction to stop it, just like the imaginary case of a ball rolling on earth without any friction except for that with the floor. Well... I don't get the idea, I don't think this is a thrilling gravitational event.

Yeah, you got it right. Just don't pay too much attention to the gray face, it's just the standard design for a parallellepiped in ms word

Okay, just lemme know.

Yeah, I agree that the events will happen simoultaneously, and what the observers would see (if the were able to) would really look cool. But still both observers would still be together in the same history.

Look, man, some of the people you´re addressing your question are science students (including me) and-or scientists, and the reason we are is because sometime, and maybe at an earlier age that you, we started making this kind of questions to ourselves and-or other persons, so you should try to be a bit more modest, that is a bliss, believe me, and no one wants a cocky genious. As for the question, my opinion is that, like everything else in physics, lines, as they are summoned in scientific work, are just an abstraction of reality. It is pretty easy for us to understand the space we live in as a three dimensional space where the dimensions are represented by three orthogonal straight lines, and it is also easy for us to think of the distance a body travels (in uniform straight movement) in a determined amount of time as a straight line mapped out by the proportionality constant ¨velocity¨. So lines do exist in our interpretation of the world, but you won´t find any dimensionless straight line anywhere.

Of course I did!!! You wanted a clear solution, there it is. I hope you believe me now and give us some opinions. Either way, it seems like the subject´s been missing for a while, but anyway.

I don't think this part is right. You would travel through time at a different rate than someone in a different inertial frame of reference, but it's not like you would travel to the future and see the future of that other person, it more likely means that, for that other person, your mevement, and your "history" would seem different, and your life would pass at a different rate than his.

Look, first I'll explain a problem like the one I posted first, then the Method of Eigenfunction Expansions and Finite Transform. I know my fingers will hurt afterwards, though. Problem.pdf Eigen.pdf

I don't know how to write any math expressions here, but if you help me out or gimme an e-mail address, you'll get a full explanation, so you can trust me.

Look again, it exists. Method f eigenfunction expansion

Look, the model I planted is not a wave equation, it is called Heat Equation, and it was proposed by Fourier, and it just differs from the wave equation in one derivative respecto to time. So, in the stationery state, U doesn't depend of time at all, so dU/dt = 0, leading to an equation which is analogous to a Potential Field. I think I wrote the equation wrong, the div is not zero, the Laplacian is, div(grad U) = 0, sorry .[div(grad U) = d2U/dx2 + d2U/dy2 +d2U/dz2] Well, for a start, we propose a solution of the form U=X(x)Y(y)Z(z) [we forget about time, since U does no longer depend on it]. Thes, we substitute on the equation and simplify, yielding: X''/X + Z''/Z = -Y''/Y ; It's easy to see that, for this equation to be valid, since X depends only on x and so on, every term must be a constant [X''/X + Z''/Z = -Y''/Y = k], allowing us to separate in three ordinary DE, say: i)Y'' + Yk = 0 ii)Z'' - (v-k)Z = 0 iii)X'' - vX = 0 ; k,v are real constants Solving all of these equations applying some boudary conditions, and discarding all trivial solutions, we can deduce that k is quadratic and negative (k = -r^2), and v is quadratic and positive (v = s^2); this is to avoid all trivial solutions, and it could be slightly different depending on the way the separation is done, leading to equivalent eigenfunction expansions as solutions for U, but let's do it this way so the solution comes in the form I proposed before. Now, the obtained functions are: i)X(x) = ASin(n*pi*x/a) ; A is A sub n ii)Y(y) = BSin(m*pi*y/a); B is B sub m iii)Z(z) = C(e^rz - e^-rz); C is C sub m,n So, since U=XYZ we have: U = Sum{Sum[C(e^rz - e^-rz)*BSin(m*pi*y/a)*ASin(n*pi*x/a)]}; where the sums are over n and m. Look, I probably made some mistakes so long since I have been solving this directly with my keyboard, and the resulting form will not be exactly the same I proposed before, but it will be equivalent, I can assure you. But now you have the steps to do it yourself more thoroughly. In the end, you can join together the constants A,B,C, into a constant D which depends on m and n. The next step is to apply the initial condition (that one of the faces in mantained at the temp. u; you choose the face), and substitute to the solution U, and so now you have only two functions inside, and a big constant with exponentials in it; this two functions are called Eigenfunctions, and the constant, as the form of the solution suggests, is the Fourier Sine Series Coefficient, in two dimensions, whis is obtained using a double integral of the eigenfunctions multyplied by the condition u and a constant which I think to remember is 4/ab. Solving this, and giving the exponential functions and constants in the resulting "thing" the shape of a Sinh, since it is a linear combination of exponentials, and so it generates the same "space", just substitute in the solution for U (the one with the sums), and Voila, it's done. Look, this might not have been very instructive, so I recommend you to check out Mathematics for Phycisists, Arfken; or Physical Mathematics,Eugene Butkov. Or you can also search the web for "the method of separation of variables", obviously for partial differential equations. You might be interested in investigating a little bit about Fourier Series and Transforms, and "The method of eigenfunctions expansion", which can help you interpret the result in terms of the Fourier discrete spectrum of the coefficients in this solution, and the continous spectrum in the case of the solutions using Eigenfunction Expansions. See ya

I guess many people was scared away with those equations in my first post, but everyone, it's actually pretty simple (at least the physical purpose of it, the mathematical demonstration of the principles behind the solution method are a bit more tricky though). Either way, if Math is the most universal Language, then Physics is the most beautiful kind of Poetry.

I agree with you, it was not my intention to suggest that QM was derivable from Classical Mechanics, I just imply that the idea of quantizised energy shoulden't be such a head-braker when this kind of situations are often seen in classical physics.

Let's analyze the case of a rigid body, say a paralelepiped (a x b x c), which presents a certain temperature distribution U(x,y,z) in the stationery state (the distribution doesn't change over time). We can keep the lower face to a certain temperature u, and the other 5 faces at zero degrees, in whatever temp units you like. Then, Fourier's law of heat transfer says: dU/dt (1/c^2) = div(grad U) ; which can be simplified as div(grad U) = 0, since ?U/?t = 0 By solving (using separation of variables) this equation, we obtain: U(x,y,z) = (16u/pi^2)Sum[ Sum[ (1/mn)(sinh(q*(c - z))/sinh(q*c)) sin(m?x/a) sin(n?y/b) ] ] ; You can see "Mathematical Physics, by Eugene Butkov if u don't believe me. -The sums are over m and n respectively. -The coefficient q is actually q sub mn, since it reads: q = sqrt( (m/a)^2 + (n/b)^2) As you can see, this is a series of eigenfunctions, summing over a discrete spectrum depending on m and n. Don't you think this is a very strong evidence of the quantization of energy, since heat here behaves as a discrete spectrum of permitted values (wave numbers)? Actually the eigenvalues m and n come out of the solutions of the ordinary DE in the process of solution, so it has to do with the physical meaning of this model. Opinions, please.

Don't you think this situation is pretty much like having a coin sipinning on its side? While we keep the book spinning, it has a rotational momentum L due to ratation horizontally, which we can say points up, and there is a gravitational torque T, tending to produce a rotational momentum L' pointing horizontally. By summing L and L'(t) we obtain a changing ?L that points diagonally, adjusting the object's rotation in a way that fits this new changing torque, just as an always orthogonal force deviates a particle instead of changing its speed. This new rotation implies something like a lifting force, but I can't realy explain because I can't write any equation here, so I guess you'll have to solve this or look at it in a book. I recommend University Physics, Sears-Zemansky-Young, or its "Scientific-Tecnical Collecton" Mechanics, Heat and Sound, Sears. This is where I've seen it best explained. You can also try Physics, Vol.1, Tipler. The thing is that it is (I suppose) the net rotational momentum ?L that makes the book easier to hold, it has nothing to do with air, though it might add a little lift too, but that depends on the shape of the book and the way you hold it.

First, about the question of the two bodies 1mt above the ground, you gotta specify what kind of Potential you mean; the fact that you pointed out that they are 1mt above the ground, makes me think you are talking about Gravitational Potential, but that just doesn't have anything to do with radiation, which is more likely related to Weak Nuclear interaction, or in the case of electromagnetic radiation, to electromagnetic potential. For a veri specific answer to the first question posted, I can say that alpha and beta radiation, which are related to particles being emmited from unstable nuclei, can be described with Kinetic energy. As for light, x rays, Gamma, etc... they can be described as an oscilating electromagnetic potential field. Perhaps this can only describe these particular cases, but I guess radiation cannot be fully described as only one of these kinds of energy.

I think you slightly misunderstood my words, what I mean when I say "nature always tends to spend the least energy possible in all of it's prodesses" I mean that nature doesn't waste any energy, every natural process uses only the necessary energy, like when a toy car moves in the grass: If you add an initial speed, the car will follow a specific trajectory, depending on the sections of the grass that offer more or less opposition. You can note that the car will follow the less tough path, instead of wasting energy struggling with the grass. About the transverse waves being less destructive... I don't really know, but common sense tells me so. Perhaps we should ask a sailor

I think your answer is right, the energy is lost due to several interactions with the surroundings, including friction, and if there were no dissipative forces, then the oscilatory movement would last forever. Now, to answer the question "why does the system try to get back to equilibrium?", the answer is in perhaps the only law in Physics that is truly general*: The Principle of Minimal Action, which states that nature always tends to spend the least possible amount of energy in it's processes, and all natural systems try to achieve the least energetic configuration possible, which in this case is the so called "equilibrium". *I say this because even the mass and enery conservation principles, the always-increasing enthropy (2nd law of thermodynamics) and many other principles are nearly falling apart as they meet the newest consequences of Chaos Theory and Non-Linear Physics, but that's another story.

I think your explanation is right, but just to give you another point of view, I'll give you mine. Water is an elastic system, do you agree??? It's particles are bond to each other by a conservative force, which is the electromagnetic interaction, so, if the system is deformed by an external force, eg: a rock falling into the water and facing it's surface tension, the system gains potential energy. Because the system is elastic, this potenetial energy transforms into kinetic energy as the system attempts to get back to equilibrium in a less energetic state, as it was before. This is what we call Simple Harmonic Movement: a restaurating conservative force attempting to bring the deformed system back to the state of less energy possible. Now, the water waves stop oscillating because they are inside a dissipative media, and energy is lost as the surface of water collides with air particles, and suffer many other losses of energy. I hope my explanation was satisfying, but if not, just let me know. David Angel R. Tijerina Universidad Autónoma de Nuevo León, México Faculty of Physical and Mathematical Sciences Undergrad

When a substance reaches the state of plasma, most of it's atoms become ionized, and the freed electrons move freely inside the substance. this is achieved when the substnace reaches pretty high temperatures (except for some other phenomena where plasma state is reached by applying a high voltage), much higher that those of a material burning. I think fire is only light emissions from the atoms of the overheated that come as a result of combustion. That wiggly shape is only the consequence of the motion of these gases as they raise away from the material burning. PS: I'm not sure if "combustion" is actually a correct word in english, but if it's not, just translate it from spanish.