Hi ,
has anyone got the possibility to make a graph of γ ?
If we consider (the hypothenuse) h = 1 = c , one leg c1 = β, and λ the angle h-c2
isn't c2 = and the curve of the Lorenz transformation the curve of sec λ (h/c2)
Thanks for your help
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Hi ,
has anyone got the possibility to make a graph of γ ?
If we consider (the hypothenuse) h = 1 = c , one leg c1 = β, and λ the angle h-c2
isn't c2 = and the curve of the Lorenz transformation the curve of sec λ (h/c2)
Thanks for your help
Your question isn't very clear, graph of γ as a function of what?
A function of β (c1 from 0 to 1)
But if you say that it is a secant, there is no need to make a graph, if the lorenz transformation describes another curve, I read "hyperbolic" somewhere, then it would be of great help to have the 2 curves plotted together.
Thanks, mathman
You can use Wolfram Alpha to plot graphs, e.g.: http://www.wolframalpha.com/input/?i=plot+1%2Fsqrt%281-x^2%29
Here is gamma plotted against Beta for Lorentz compared to the secant of beta. (Now to be fair I had to calculate the secant of Beta times pi/2 in order for them to have the same vertical asymptote on the graph.
http://home.earthlink.net/~jparvey/s...antlorentz.jpg
Note that even when you fudge to get the same asymptote, they are not the same curves.
I am a little confused as to definition. Is γ² =1-β² or is γ² =1/(1-β²)?
I was assuming the first.
[QUOTE=whizkid;477254]You might want to look up the history of the Cyclotron, a type of particle accelerator.
a brief synopsis:
A cyclotron works by making a charged particle pass through a magnetic field which causes them to follow a circular path. A varying electric field is used to accelerate them. As the particles accelerate, they move in ever larger circles. Now, classically the increase in size of the circle and the increase in speed combine to give the particle the same period for each circuit around. This would allow the varying electric field to have a fixed frequency. Basically, the electric field gives the particles a boost at the same point of its path around the circle. Since the particles take the same time to make the trip regardless of how fast they are moving, this kick can be given at regular intervals.
However, this only works at lower speeds. Once the particles get closer to the speed of light, their relativistic mass begins to increase, and the above reasoning only works if the mass of the particle is constant.
The result is that as the particles in the cyclotron get closer to the speed of light, they take don't take the same time to complete the circuit. The particles and varying electric field fall out of sync, and you lose the ability to accelerate the particles further. Thus such a design has a limit as to how fast it can accelerate particles.
One answer to this is the synchrocyclotron. The idea being that as the particles are accelerated, the frequency of the electric field is adjusted to keep it in sync with the particles.
The adjustment is
Synchrocyclotrons, by accounting for relativistic mass increase are able to accelerate particles up to speeds unattainable by a simple cyclotron.
If the relativistic mass increase didn't occur, then you wouldn't need to modify the cyclotron to get higher speeds, and if this mass increase didn't follow the Lorentz factor, then synchrocyclotrons, which are specifically designed with the Lorentz factor in mind, would not work as expected either (and they do).