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I A System with Velocity More than Speed of the Light

  1. Jan 1, 2016 #1
    A Case of Mechanical Waves

    Case: Is There Any Mechanical Oscillatory System Where Maximum Velocity of Resonance Will Increase More Than Speed of the Light?”

    Consider a child, who is playing with a swing. During the period of the time, he learns to apply the optimum force to the swing in order to minimize efforts and maximize the amplitude of the swing. How? The answer is that driving force should be applied periodically and should be timed to coincide closely with the natural motion of the swing.In other words, a driven oscillator responds most strongly when driven by a periodically varying force, the frequency of which is closely matched to the frequency with which the system would freely oscillate if left to it. This frequency is called the natural frequencyof the oscillator.

    Assume we are designing an open system which has the damped harmonic motion and are also forced by external oscillatory forces under harmonic motion. The parameters of designing are as follows:

    Fm = External force (N)

    k = Restoring constant of system (N/m)

    m = mass of system (kg)

    b = Damped force constant of system (kg/s)

    ω'' = Angular velocity of external force (rad/s)

    A = Amplitude (m)

    ω = Natural angular velocity (rad/s)

    We are willing to know if there is any mechanical system with FHM in which maximum velocity of this system will go up more than 3E+8 m/s. What is the range for parameters of designing?

    I found 17 types of the parameters where maximum velocity of our system is equal to 1E+9 m/s > c = 3E+8 m/s. It means that we can have 17 types of design for our system to reach maximum velocity more than speed of the light. All parameters for designing have been arranged into attached Table.
    As we can see, the most crucial thing is that our system will reach to maximum velocity more than speed of the light, if external oscillatory force goes up more than 1KN and damped force constant decrease less than 1E-6 kg/s. In fact, the boundary conditions are:

    Fm ≥ 1KN and b ≤ 1E-6 kg/s

    Case questions:

    1. Is there really above system in the nature?

    2. If your answer is positive. Have you any real examples?

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  2. jcsd
  3. Jan 1, 2016 #2


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    You used nonrelativistic formulas - they are good approximations as long as the speed is slow compared to the speed of light, but they do not work in the range where you try to apply them.
    Use the proper relativistic formulas and the speed will be below the speed of light.

    There is no need to invent oscillators for that, you can simply consider a rocket that keeps accelerating in one direction. Will it ever exceed the speed of light? The answer is no, for the same reason - with nonrelativistic formulas it would, but those are not valid for relativistic speeds. With the right formulas you see the speed stays below the speed of light.
  4. May 30, 2016 #3
    Let me tell you another example about velocity of Gases.

    According to below links, Physicists at CERN generated ions with temperatures of more than 1.6 trillion degrees Celsius. At the Brookhaven National Laboratory in Upton, have set a new record for the highest temperature ever measured: 4 trillion degrees Celsius.


    Assume, we have a 0.500 mole sample of hydrogen gas at 1.6 - 4 trillion degrees Celsius.

    By using of Maxwell–Boltzmann speed distribution function, we can calculate number of molecules with velocity between 300000 km/s to 400000 km/s at1.6 trillion degrees Celsius which is equal 1.01326*10^21. It means that about 0.34 % of total molecules of hydrogen have velocity more than 300000 km/s. At 4 trillion degrees Celsius, number of molecules which have velocity between 300000 km/s to 700000 km/s, is equal 4.23143*10^22. It means that about 14.05% of total molecules of hydrogen have velocity more than 300000 km/s.

    In record of CERN, it has been stated: "… ions together at close to the speed of light…"

    The question is: Had all (100%) ions the velocity close to the speed of light?

    Therefore, we have only two alternatives:

    1. If the answer to above question is positive, then all reference books should apply the limited velocity of 300000 km/s for Maxwell–Boltzmann speed distribution function.

    2. If the answer to above question is negative, then we can use from special theory of relativity only as a simulation model in which limited speed of light is an assumption of this model.
  5. May 30, 2016 #4


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    Same problem again, you try to use nonrelativistic formulas for energies where those formulas do not apply.
    Yes. The ions in the collider move at something like 99.9999% the speed of light. Those ions are not thermalized, the temperature applies to the collision zone after the ions collide with each other. Particles there have speeds between 0 and nearly the speed of light.
    The relativistic form of the Maxwell-Boltzmann distribution is called Maxwell–Jüttner distribution. It is not as simple as capping the velocity. Books about nonrelativistic mechanics don't have to care about that because they only discuss speeds where relativistic effects are negligible.
  6. Jun 1, 2016 #5
    Yes, previous experiments showed that Newtonian mechanics is contrary to modern experimental results and is clearly a limited theory in which velocity of the particles in the Universe always remains less than the speed of light.

    But, here there is a strange case and the interesting point. Because this temperature which has been generated by Physicists at CERN (1.6 trillion degrees Celsius), is approximately the boundary between using of the Maxwell-Boltzmann distribution and Maxwell–Jüttner distribution. It means that all particles (100%) in temperature less than 1.6 trillion degrees Celsius have velocity between 0 to less than 300000 km/s and we can still use from Maxwell-Boltzmann distribution instead of Maxwell–Jüttner distribution.
  7. Jun 1, 2016 #6


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    There is no boundary, the Maxwell-Boltzmann distribution is an approximation to the Maxwell-Jüttner distribution for "low" temperatures.
    No we cannot, as your previous post demonstrates already: a notable fraction of particles moves faster than half the speed of light, which means the MB distribution is a bad approximation.
  8. Jun 1, 2016 #7
    Therefore, you believe that we should not apply MB for velocities close (but less than it) to the speed of light. But indeed, what is upper limit of velocity in which MB gives us a good approximation to calculate number of molecules? Because the companies still use MB to design Turbo Molecular Pumps.
  9. Jun 1, 2016 #8


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    Obviously it depends on how "good" you need the approximation to be. There is no hard limit.
  10. Jun 1, 2016 #9


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    Turbomolecular pumps deal with gas particles with typical velocities of 1 km/s. That is 1/300,000 of the speed of light, and MB is fine.

    As jbriggs said, there is no hard limit, but typically you want to take relativistic effects into account if speeds exceed 1-10% of the speed of light.
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