Galilean velocity transformation

Transcription

1 Galilean velocity transformation u If an object has velocity u in frame S (note: velocities have a direction!), and if frame S is moving with velocity v along the positive x-axes of frame S, then the position of the object in S is: v x' ( t) = x( t) The velocity u of the object in frame S is therefore: A) u + v B) v - u C) u - v D) u E) -v x x vt

2 Comparing inertial frames x v x Here are two inertial reference frames, moving relative to one another. According to S, S is moving to the right, with v = 1 m/s.

3 Comparing inertial frames v x x Here are two inertial reference frames, moving with respect to one another. According to S, S is moving to the left, with v = -1 m/s. But again: Both observers are right.

4 Important conclusion Observer in S measures velocity of S to be +1 m/s Observer in S measures velocity of S to be -1 m/s Q: Who is right? a) Observer in S b) Observer in S c) Both are right d) Neither is right e) It depends! Two observers in different reference frames can give a different description of the same physical fact (in this case, the relative velocity of the other reference frame.) And they re both right! The two frames are moving relative to each other.

5 Comparing inertial frames v At time t = 0, the two frames coincide. A ball is at rest in frame S. Its position is x = 2 m in S x = 2 m in S

6 Comparing inertial frames v Frame S is moving to the right (relative to S) at v=1m/s. At time t = 3 sec, the position of the ball is x = 2 m in S x = -1 m in S

7 Important conclusion Where something is depends on when you check on it and on the movement of your own reference frame. Time and space are not independent quantities; they are related by rel. velocity. Definition: An event is a measurement of where something is and when it is there. ( x, y, z, t)

8 Comparing inertial frames v At time t=0, the ball was at x = x = 2. At time t>0, the ball is still at x=2 in S but where is it in S at the time t>0? a) x = x b) x = x + vt c) x = x-vt

9 Galilean position transformation If S is moving with speed v in the positive x direction relative to S, then its coordinates in S are x y = z = t = x = t y z vt Note: In Galilean relativity, time t=t is the same in both reference frames; why wouldn t it be?!

10 Galilean velocity transformation u v Same thing as before, but now the ball is moving in S, too, with velocity u = 1 m/s. Is the ball faster or slower, as measured in Frame S? A faster B slower C same speed

11 Galilean velocity transformation u If an object has velocity u in frame S, and if frame S is moving with velocity v along the x-axes of frame S, then the position of object in S is: x v x x' ( t) = x( t) vt The velocity u of the object in frame S is therefore: A) u + v B) u - v C) v - u D) u E) -v

12 Galilean velocity transformation u If an object has velocity u in frame S, and if frame S is moving with velocity v along the x-axes of frame S, then the position of object in S is: x v x x' ( t) = x( t) vt The velocity of the object in frame S is therefore: u' = dx ( t) dt = d dt dx( t) ( x( t) vt) = v = u v dt

13 Important conclusion Two observers in different reference frames can give a different description of the same physical fact (in this case, the velocity of the ball.) And they re both right!

14 So far we have talked about how observers from different reference frames describe the location (x) and the velocity (v) of objects. What other quantities did we talk about in fundamental physics? F = m a

15 Dynamics F S F F v S

16 Dynamics In inertial frame S, we have (in x-direction, say) F = ma How about in inertial frame S? Well, F' = F since you re still applying the same forces, and du' d du a ' = = ( u v) = = dt dt dt a à no additional acceleration in an inertial frame.

17 Inertial reference frames V Now, you re playing pool on the train. The balls roll in straight lines on the table (assuming you put no English on them). In other words, the usual Newtonian law of inertia still holds. The frame as a whole is not accelerating.

18 Galilean relativity The laws of mechanics (good old F = ma) are the same in any inertial frame of reference.

19 Einstein s First Postulate of Relativity The laws of physics (including electromagnetism) are the same in all inertial frames of reference.

20 Huston, we have a problem!! From fundamental Physics you might recall that Mr. Maxwell told us the speed of light c would be: 1 8 c = = m / ε µ 0 0 s Now, Mr. Einstein tells us that ε 0 and µ 0 are the same in any inertial frame of reference. But Mr. Galileo just told us that c = c - v What gives??

21 Galilean velocity transformation u If an object has velocity u in frame S (note: velocities have a direction!), and if frame S is moving with velocity v along the positive x-axes of frame S, then the position of the object in S is: v x' ( t) = x( t) The velocity u of the object in frame S is therefore: A) u + v B) v - u C) u - v D) u E) -v x x vt

22 Today Maxwell vs. Galileo Strange things about the speed of light Is there a luminiferos ether? Let s find out! Interferometers Light: the ultimate yardstick.

23 we found a problem!! Mr. Maxwell told us, the speed of light c is: 1 8 c = = m / ε µ 0 0 s Mr. Galileo told us that c = c v If the laws of physics are the same in all inertial frames then ε 0 and µ 0 (and c) have to be the same in all inertial frames. So let s make up the luminiferous ether to fix Galileo s velocity transformation law (u = u - v)! (We will have to check if there is a luminiferous ether!..)

24 Peculiar light-waves A sound wave propagates through air, with a velocity relative to the air (~330 m / sec) A water wave propagates through water, with a velocity relative to the water ( m / sec) The wave propagates through a crowd in a stadium, with a velocity relative to the audience. An electromagnetic wave propagates through... Answer (19 th century physics): The luminiferous ether.

25 Ideas behind Einstein s relativity Is there an ether? (There where various other motivations for special relativity, but for simplicity we will focus here on the quest for detecting the ether.)

26 The ether v c Suppose the earth moves through the ether fixed in space with speed v. A light wave traveling at speed c with respect to the ether is heading in the opposite direction. According to Galilean relativity, what is the magnitude of the speed of the light wave as viewed from the earth? (Assume the earth is not accelerating). a) c b) c + v c) c - v d) v - c e) something else

27 Quiz on reading (closed book, no talking) The Michelson-Morley Experiment tests if the speed of light in all inertial frames A is not the same in air and in vacuum B is not the same in accelerating frames C is not the same in all directions D does not depend on the wavelength or color E does not change when reflected by mirror.

28 Michelson and Morley How can we do that, Morley? You tell me, Michelson performed a famous *) experiment that effectively measured the speed of light in different directions with respect to the ether wind. *) some say, the most successful failure

29 Frame of reference Observer on the sun: Ether v Ether viewed in the laboratory on the earth: -v -v

30 Ether in the laboratory frame v u'-v L v How can we measure the speed v of the ether? If the ether would be a river, we could measure the speed of the water using a boat that travels at a known speed u. (u is the relative velocity between the boat and the water.) If the boat travels the distance L within the time t, then we know v: L=(u -v)t, therefore v = u L / t But: Very difficult with light! u = c à t ~ 10ns and v ~ *c. à We would have to measure t with an absolute precision of ~ s and we have to know c very precisely!

31 Measuring only differences in c -v -v L u +v B A u -v L Compare the round-trip times t A and t B for paths A and B. This has the great benefit, that we do not have to measure the absolute times t A and t B (which are only a few ns) and we are less sensitive to uncertainties in the speed of light.

32 Michelson and Morley Mirrors -v L -v Detector L Semi-transparent mirror Interferometer Light source The detector measures differences in the position of the maxima or minima of the light-waves of each of the two beams. (Yes, light is a wave!)

33 Intermezzo: Interferometers 1881 Michelson invented a device now known as the Michelson Interferometer he received the Nobel prize for it! We will see it in action in the famous Michelson-Morley experiment, which will lead us to the special relativity theory. à So the interferometer had a huge impact!! Such interferometers are nowadays widely used for various precision measurements. State-of-the-art visible-light interferometers achieve resolutions of ~100pm! (X-ray interferometers are ~1pm). (100pm = 1Å = diameter of a Hydrogen atom.)

34 The Michelson interferometer Mirrors Light source Semi-transparent mirror Detector

35 Electromagnetic waves E-field (for a single color): E(x,t) = E 0 sin[ ( 2πx / λ ) ωt +φ] Light source E λ λ = 2πc/ω, ω = 2πf = 2π/T E φ E 0 Wavelength λ of visible light is: λ ~ 350 nm 750 nm. x B x

36 Free physics simulations! Radio Waves.jar Wave interference.jar

37 EM-Waves in an interferometer L Mirrors Light source L Semi-transparent mirror

38 Constructive interference E sum (x,t) = ½ E 0 sin(ωt+2πx/λ +φ) + ½ E 0 sin(ωt+2πx/λ +φ) =? = E 0 sin(ωt+2πx/λ +φ) = E light source (x,t) + =? Screen:

39 Unequal arm lengths L Light source L+ ΔL / 2 ΔL / 2 ΔL

40 Destructive interference E sum (x,t) = ½ E 0 sin(ωt+2πx/λ +φ) + ½ E 0 sin(ωt+2π(x+δx)/λ +φ) = ½ E 0 sin(ωt+2πx/λ +φ) - ½ E 0 sin(ωt+2πx/λ +φ) = 0 if Δx = λ / 2: sin(x+π) = - sin(x) + =? Screen:

41 Moving mirror: What do you see? Light source ΔL Screen Intensity λ / 2 ΔL

42 Tilted mirror: What Fringes! do you see? Light source Screen

43 Application: Flatness measurement Michelson interferometer Ultrahigh power laser (>3.5kW average and 250MW peak power)

44 Interference in daily life:

45 Do you want a bigger interferometer? There you go

46 Gravitational wave detectors

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48 Physics Review Letter, vol 116, (2016)

49 Do you remember this guy? The blue color originates from constructive interference of blue light and destructive interference of all other colors.

50 Summary for Interferometers Michelson interferometers allow us to measure tiny displacements. Displacements of less than 100 nm are made visible to the eye! Interferometers find many applications in precision metrology such as for displacement, distance and mechanical stress measurements, as well as flatness measurements. Interferometers have played an important role in physics: Michelson-Morley experiment à special relativity Testing general relativity: Gravitational wave detection Global survey of groundwater (GRACE satellites)

51 End of our little interferometer intermezzo