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Lorentz transformation

  1. Nov 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Hi all I am having trouble working part b of a question that I am currently doing. I have attached the question below in a pdf file. I am really not sure where to start, I have looked in several book and can only think of relating to the light clock thought experiment. But the problem is I cant see where the $$(ct)^2$$ comes from.

    2. Relevant equations
    $$t'=ϒ(t-(ux/c^2))$$

    $$x'=ϒ(x-ut)$$



    3. The attempt at a solution
    I did attempt to set $$t'=0$$ but this got me no where.
     

    Attached Files:

  2. jcsd
  3. Nov 29, 2016 #2

    stevendaryl

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    Well, using your two equations for [itex]x'[/itex] and [itex]t'[/itex], see if you can figure out what the combination

    [itex](x')^2 - c^2 (t')^2[/itex]

    is, in terms of [itex]x[/itex] and [itex]t[/itex]. Then look at the special case where [itex]t'=0[/itex].
     
  4. Nov 29, 2016 #3
    Before I attempt what you have told me. can I ask how u got that equation
     
  5. Nov 29, 2016 #4

    PeroK

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    You need to give some thought to question (a) (i). This does not require any equations or calculations.
     
  6. Nov 29, 2016 #5
    It is because it is the rest frame, so it will be the proper length of the rod.
     
  7. Nov 29, 2016 #6

    PeroK

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    Pretend you've never heard the words "rest frame" or "proper length".
     
  8. Nov 29, 2016 #7

    PeroK

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    @Taylor_1989 In fact, let me ask you a practical question. Forget relativity for a moment.

    1) There's a train standing at a station platform and you are asked to measure its length. How do you do it? Hint: this one is easy.

    2) A fast train comes through the station without stopping. How do you measure its length?

    Remember: this is a practical, non-relativistic question.
     
  9. Nov 29, 2016 #8
    If at rest I would use a tape measure, whilest moving I would use a stop watch
     
  10. Nov 29, 2016 #9

    PeroK

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    That's not an answer! You have to be prepared to think if you want to learn physics.
     
  11. Nov 29, 2016 #10
    I am really not sure. How else would I measure a train standing still? I really don't have a clue. Also I am trying to think but I am struggling quite bad area of physics , so no need for that comment.
     
  12. Nov 29, 2016 #11

    PeroK

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    Imagine I was going to carry out your measurements for you. You said: "use a tape measure". So, I get a tape measure. Now what? You've said nothing about what to do with the tape measure. You've given me no procedure or even any hint. You haven't even distinguished between measuring the height of the train and the length of the train.

    That said, here is a better idea. I would mark the platform at regular intervals (let's say every metre is accurate enough). Then, I'd measure where the two ends of the train are and subtract one reading from the other. For example, if the front of the train is at position ##106m## and the back of the train is at position ##3m##, then I'd have an approximate measurement of ##103m## for the length of the train.

    Now, what about a moving train? Let's assume the plaform is still marked every metre. And you have a stopwatch. Give me a procedure for measuring (approx) the length of the train as it speeds through the station. Hint: this is not so easy to do on your own. You may need a friend with another stopwatch.. Or, as many friends as you can muster!
     
  13. Nov 29, 2016 #12
    I keep thinking that if I start my stop watch at the same time as my freind then when the train passes him then we both stop our stop watch. But that make no sense, it would not give me the length of the train. Erm, I feel like I should have each friend standing at each marker then they start and stop there stop watch when the front and back of the train passes them. But even then it make nonsense to me I still cant figure how to measure the length of the train.
     
  14. Nov 29, 2016 #13

    PeroK

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    You're on the right track with one person at each metre interval. Each stops their watch when the front of the train passes them. If you stand at the end of the platform (where the train comes in), when do you stop your watch?

    It goes without saying that you all synchronise your watches in advance!
     
  15. Nov 29, 2016 #14
    There is something fundamental I am missing here. My lecture note don't really explain it, the book I have describes a light clock and and video on you tube I have been watching uses a graph with ct being on the y axis.
     
  16. Nov 29, 2016 #15
    I would stop my stop watch as soon as it come in right, then I would compare the difference in time intervals with my freinds, correct?
     
  17. Nov 29, 2016 #16
    Am I allso on the right lines look at the
    Minkowski diagram in special relativity
     
  18. Nov 29, 2016 #17

    PeroK

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    The light clock is something different. The question you posted was set for a good reason: to get you to analyse the notion of length for a moving object. Let me help you a bit more with the train measurement.

    You stop your watch when the rear of the train passes. Let's say you are standing at point ##0m## at the end of the platform. This establishes the time that the rear of the train was at this point. Let's say your stopwatch reads ##57s##. Then you go to each of your friends and find which one also stopped at ##57s##. This establishes where the front of the train was at ##57s##.

    So, you have simultaneous measurements of the position of both ends of the train. And, for a moving object,that is as good a definition of its length as you are likely to get!

    For the stationary train you did not need simultaneous measurements. You could measure the position of the rear of the train, then stroll up the platform to the front of the train, and measure its position at your leisure.

    If you read Einstein's 1905 paper, that is the sort of analysis he undertook because, as he put it, "Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.".
     
  19. Nov 29, 2016 #18

    PeroK

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    Our discussion so far was about the meaning of the length of a moving body. We have not yet progressed to SR and spacetime diagrams. This discussion was fundamental, because if you haven't thought carefully about what "length" actually means, then it is going to be impossible to understand length contraction.

    You should be in a position to answer (a) (i) now. Then, you might try (a) (ii).
     
  20. Nov 29, 2016 #19
    Okay I am with you there. So why is t'=0 when really t'1=t'2 or any time for that fact. is t'=0 the difference in time if so is that not delta t. I am trying to get use to the notation as well. t is the none moving frame and t' is the moving frame, correct? So in ur example t' would be me and my friends standing on the platform?
     
  21. Nov 29, 2016 #20
    Yes I understand the question u posted me and thank you for your patients.

    I would like to add that my lecture never went through any of this neither any graph stuff.
     
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