What is Length Contraction? (Thought Experiment)

another very interesting prediction of the special theory of relativity is that of length contraction now what is length contraction if there are two observers who are in relative motion and they try to measure the distance between two common points then they will end up measuring two different distances in fact a person who is moving with respect to some reference stream will measure a distance which is shorter compared to another person who is measuring those distances in a rest frame of reference moving objects are shorter compared to the same objects in a rest frame of reference so this kind of a contraction in the length along the direction of motion is known as length protraction or sometimes also Lorentz control contraction and we can prove this with a very simple thought experiment let's imagine that there is some kind of a train which is moving with a certain velocity V in a given direction and let's suppose V is constant and there is a certain set up inside this particular train so there is a pulp let's suppose on one end of the train and there is a mirror on the other end of the train and what what happens is that this bulk emits a short pulse of radiation or a light photon and this light photon can travel along the length of the train they reach the mirror get reflected backwards and travel back and reach this initial point now if I want to calculate the total amount of time period that the light ray takes in its round-trip from its initial point to the mirror and back to its initial point it can be calculated in a very simple manner so let's suppose first I want to make these calculations with respect to some kind of an observer who is inside the tree - let's suppose that for this person inside the train the length or the longitudinal distance between these two points is given by then X now let's suppose that according to him this is the distance between these two points and according to his watch the total amount of time period that is taken by this light ray to make this kind of a round-trip is given by Del T knot okay so I'm going to make use of the fact that the tacking portion rate of SGR tells us that the speed of light in any given inertial frame of reference is a constant so for this person the total amount of mediate that is taken by this light rate in his round trip is simply the total amount of distance divided by the speed of light so total amount of distance is 2 del X naught divided by the speed of light okay fine quite simple but now let's look at the same physical process but from the perspective of another observer so let's let's say that in the different case there's an observer in the platform so he is not inside the Train as opposed to the earlier case with a person was inside the Train and for this person everything inside the Train was a terrace stream of reference for this person this train is now in a moving frame of reference okay now he's also going to look at the same kind of a process but the light is being emitted from one point it goes to the mirror gets reflected and comes back and he also let's suppose wants to calculate the time period it takes for the light to make this kind of a round-trip let's say that for this person the train has a long needle length of del X so let's suppose the length between these two points the mirror and the initial point of the pulp is del X for this person okay now what happens is that when light is emitted by the bulk at this point the train is moving in a certain particular direction so that by the time the light reaches the mirror the train has already moved a certain distance in the forward direction yes light is emitted by the bulb the train is in this particular position by the time the right leash light reaches the mirror the train has moved to this forward direction that means the light has now traversed a particular path which is greater than the length of the train and this extra distance that the light has travelled can simply be given as the velocity of the Train multiplied by let's say the time period in the forward direction which is Del T 1 so these measurements are made by this person so he is the one who is making the measurement of the length and this is the time period according to his watch now from the middle the light ray is going to get reflected backwards but as the light gets reflected backwards and reaches this initial point the train has also moved forward a particular direction so this particular distance which the train has moved forward v let's suppose del T 2 so del T 2 is the time period for the light to go from the mirror to its emission point in its backward churning so to summarize as the light starts at the initial point the Train is here as the light reaches the mirror the Train has moved forward and I still try as the light comes backward and reaches the initial point the Train has reached this position so the light goes from here to the mirror and comes back and reaches its initial point here so there is a difference in distances because of the motion of the train in the forward direction now if I want to calculate the time periods let's calculate the time periods separately in the forward direction as well as in the backward direction so let's suppose the time period del T 1 is the time period taken by the light in the forward direction to reach the mirror this time period is simply equal to the distance the longitudinal distance of the Train plus the extra distance the light had to travel so the longitudinal distance of the thing is Del X plus V del T 1 okay these are all measurements made by this person so the length is measured for this person and the time is also measured by this person and his frame of reference this is a total distance travelled by the light in the forward Direction divided by the speed of light again I'm making use of the fact that the speed of light is constant it all your frames of reference now let's suppose in the reverse direction the time period taken by the light ray is Del T 2 okay if that time period taken is Del T 2 then in the reverse direction the distance is Del X minus V then T 2 by C because since the train is moving in the forward direction the light ray has to travel a path which is shorter by this comparator this factor now I finding the simple calculations if I if I take the second term in the right hand side and bring it to the left hand side I'll simply get del T 1 minus V del T 1 by C is equal to Del X by C or del T 1 and multiplied by 1 minus V by C is equal to Del X by C again del T 1 C minus V by C is equal to Del X by I can cancel CC here and this will become then t1 is equal to del X by C minus V let's suppose that this is point number a similarly if I perform the same calculations here del t 2 plus V del T 2 by C is equal to Del X or del T 2 1 plus V by C is equal to this is del X by C so del X by C so this is a time period it takes for the light to go from the initial point of the bulb to the mirror in the forward direction and this is a time period it takes for the light to come back in this reverse journey to its initial point so what is the total time period measured by this observer in the entire trip of the light radiation and the total time period would be let's suppose del P is equal to Del T 1 plus del T 2 right so if I simply add these two terms that I just now obtained I'll end up getting u del X by C minus V Plus del X by C + V right I have just substituted the expressions from E okay let's suppose this is be okay and B into here and then I can just add both the shoot'em C - V C + V this gives us del X C plus del X V Plus del X C minus del X V del X VN minus del X V gets cancelled you're left with 2 del X C by C square minus v square okay this is the total amount of time period taken from this person's perspective now I have done a video earlier also where I calculated that I'm violation expression between two different observers in a similar kind of a setup I calculated the relationship between the time period measured by an observer in a restroom of reference inside the setup an observer in relative motion frame of reference if you are interested in finding out how I obtained that expression you can look at the video I will give a link in the description but the time dilation between the time period measured by this observer inside the train and the time period measured by this observer in on the platform is something that I have already obtained before and the expression was given by del T is equal to Del T not by root over 1 minus v square by C square so if I use this expression here I will then obtain del t if I replace with del T not van Gogh or minus v square so del t knot by root over 1 minus v square by C square okay and delta naught is equal to 2 del X C by C square minus v square ok now again if you note I have already obtained here in terms of this particular observer so I'm going to substitute this expression here so this becomes 2 del X naught by C root over 1 minus v square by C square and this is root oh all right v square by C square this is 1 minus v square by C square I can simply cancel it out and 2 2 gets cancelled all so you I end up getting del X naught is equal to Del X by root over 1 minus v square by C square or del X is equal to root over 1 minus v square by C square del X 1 so this is the expression of length contraction where LX here is the distance measured by this observer between these two points the bulk and the mirror and then X naught is the distance measured by this observer within these two points now these two points are the same points right because they are the endpoints of this string but the distance is measured bit by both these two are not exactly the same in fact the distance measured by this person between these two same points is shorter in length compared to the distance measured by this person here so this because this is because of this particular factor root over 1 minus v square by C square this is known as length contraction or Lawrence construction so basically a moving object has a length which is shorter compared to that same object which is at rest and this can be simply proven by this kind of a simple thought experiment where I have only made one assumption and that is that of the second portion at Avast here which is that the speed of light in both these two frames of reference is a constant and using that I have obtained the relationship between the length of these two points now the link contraction is something that happens along the direction of motion it is not something that happens in a direction perpendicular to the direction of relative motion itself and if you look at this particular factor into one minus v square C square it is a factor which only becomes quite significant when we are talking of very very high velocities where very four velocities which is very very low compared to the speed of light this factor is not very significant and this is the reason we do not see like contraction in a normal day to day lives length contraction becomes significant only when they talk about velocities almost equal to the speed of light itself so this is known as link contraction

Channel: For the Love of Physics

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Keywords: length contraction derivation, length contraction equation derivation, what is length contraction, length contraction, relativity of length, length contraction in hindi, length contraction in special theory of relativity, lorentz contraction, gedanken experiment, lorentz transformation, thought experiment, special relativity, special theory of relativity, albert einstein, time dilation and length contraction, time dilation

Id: Ajjf_rIufb0

Length: 11min 46sec (706 seconds)

Published: Wed Apr 18 2018