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Concept of Modern Physics A. BeiserSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 1
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The concept of relativity had been well known since the time of Galileo.It was used by Newton and Poincare developed this idea.Einstein said that he thought of the idea whilst riding his bicycle.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 2
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Special Relativity In 1905 the 26 year old Albert Einstein described in his theory of Special Relativity “how measurements of time and space are affected by the motion between the observer and what is being observed.” The theory of special relativity revolutionized the world of physics by connecting space and time, matter and energy, electricity and magnetism …Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 3
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Introduction of the chapterAfter completing the chapter you will be familiar with the1. Two basic postulates of the STR2. Frame of reference, concept of ether and Michelson- Morley Interferometer3. Galilean transformation4. Lorentz transformation5. Time dilation6. Length contraction7. Velocity transformation8. Relativistic momentum and energy
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Postulates of Special RelativityEinstein built the special theory of relativity on two postulates: 1. The Relativity Principle: The laws of motion are the same in every inertial frame of reference. 2. Constancy of the speed of light: The speed of light in a vacuum is the same independent of the speed of the source or the observer.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 5
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Motion is always measured relative to a frame ofreference i.e. there is no absolute motion Frame S Frame S’ V0 relative to frame S V ’ = V - V0 Speed measured In frame S Speed measured In frame S’Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 6
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What is an event ?• When is it happening time t• Where is it happening position (x,y,z)• What reference frame coordinate (t,x,y,z) measured with respect to a particular observer at (0,0,0,0) frame of referenceSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 7
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Measuring an event An event is something that happens, to which an observer can assign three space coordinates and one time coordinate A given event may be recorded by any number of observers, each in a different reference frame In general different observers will assign different space-time coordinates for the same event.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 8
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Inertial Reference Frames An Inertial Frame of Reference is one in which the basic laws of physics apply- e.g., a train moving at a constant velocity, in this, objects move “ normally”. v Objects obeying the Newton’s First Law.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 9
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Non-Inertial Frames An accelerating or decelerating objects. If you are sitting / walking on that, then during this period, you are in non- inertial frame.For example: If you are in a Ferris Wheelyou are always accelerating inwardsso non-inertial. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 10
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Relative Velocity: What is the black car’s velocity relative to yourframe? V= 70 km/hr V=50 km/hr We know the answer intuitively (120 km/hr) ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 11
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Same idea with velocity of light ? C= 3 x 10^(8) m/s C= 3 x 10^(8) m/sFrom the first example, we would expect the relative velocity to be 2c = 6 x 10^(8) m/s. This is in fact WRONG !!!!!Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 12
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The Luminiferous Ether The Ether was the basis of understanding, and the term was used to describe a medium for the propagation of light. It was hypothesized that the Earth moves through this medium . The Ether; Ether Was transparent Had zero density Was everywhere Was the substance which allowed light to propagate. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 13
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Airplane windIf the velocity of the wind is v1 relative to Earth, and v2 is the velocity of airplane relative to the wind, the speed of Airplane relative to the earthis (a) v1+v2 in the same direction, (b) v2-v1 in theopposite direction and (c) in theDirection perpendicular to the wind. Observer fixed on earthSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 14
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v C c+ v v c c-v 2 2 Determine the speed of light c c v under these circumstances ?In our case the ether wind is blowing through our apparatus fixed to the Earth, determine the velocity of light ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 15
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If the Sun is assumed to be at rest in the ether, then the velocity of the Ether wind would be equal to the orbital velocity of the earth around the Sun. which has a magnitude of about 3 x 10^4 m/scompared to c= 3 X 10^8 m/s. The change in the speed of light should be detectable !!!! Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 16
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The Michelson-Morley Experiment The Famous experiment designed to detect small changes in the speed of light with motion of an observer through the ether . It was performed in 1887.Albert A. Michelson The negative results of the experiment not only meant that the speed of light does not depend on the direction of light propagated but also contradicted the ether hypothesis. Light is now understood to be a phenomenon that requires no medium for its propagation. Edward W. MorleySunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 17
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Light source Mirror Compensating plate Semi-silvered plateTelescope Actual Experimental set-upSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 18
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The Michelson interferometer produces interference fringesby splitting a beam of monochromatic light so that one beamstrikes a fixed mirror and the other a movable mirror. Whenthe reflected beams are brought back together, aninterference pattern results.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 20
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M1 C V Ether wind Arm two L Light velocity -c Source B A Arm one M2 L A = Semi silvered plate B= Mirror total C = Mirror total ObserverSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 21
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Michelson Interferometer Precise distance measurements can be made with the Michelson interferometer by moving the mirror and counting the interference fringes which move by a reference point. The distance d associated with m fringes isSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 22
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Michelson-Morley Interferometer has two arms of equal length L. First, the beam traveling parallel to the direction of the ether wind Velocity of light beam moves to the right, with respect to the Earth is = c - v Velocity of light beam moves to the left, with respect to the Earth is = c + v The total time of travel for the round –trip along the horizontal path is 2 1 L L 2L v t1 1 2 c v c v c cSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 23
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Now, the light beam traveling perpendicular to the wind, so in this case the speed of the beam relative to the 1 Earth is 2 2 2 c vTotal time of travel for the round-trip is 2 12 2L 2L v t2 1 2 2 12 c c2 c vThus the time difference between the right beam travelinghorizontally and the beam traveling vertically is 2 1 2 12 2L v v t t1 t2 1 2 1 2 c c c Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 24
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2 Lv After simplification , t t1 t 2 3 c Because After rotating the interferometer through 90 degree v2 1.The path difference corresponding to this time difference is c2 2 Lv 2 d c2 t c2The corresponding fringe shift is equal to this path difference divided by the wavelength of light, lembda, because a changein path of 1 wavelength corresponds to a shift of 1 fringe, 2 Lv 2 Shift c2Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 25
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Change in path of the order of lambda = one fringe shiftIf change is one then fringe shift is equal to 1 over lambdaIf change is d in path then change in shift would be equal to dinto one upon lambda.
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The speed of the Earth about the Sun, gives a path difference of 2(11)(3 10 4 m / s ) 2 7 d 2.2 10 m (3 10 8 m / s ) 2 7 d 2.2 10 m Shif t 7 0.40 5.0 10 mConclusion: 1. No detection of fringe shift in the pattern 2. No motion of Earth with respect to Ether. 3. The speed of light is same for all observers“ Most famous negative result in the History of Physics”.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 27
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Questions/ doubts 1) How Michelson able detect change in the speed of light? 2) Concept of aether ? 3) Concept of frame of reference ? 4) How shifting of fringes decides whether the speed of light is same or not ? 5) In theory of relativity fourth coordinate Is of time , why we take it so? because to define position of particles only distance from 3 co-ordinates is needed? 6) How we come to know light is electromagnetic wave through Moreley Experiment? 7) In Morley experiment is source of light is moveable or mirror is moveable? 8) The basic points how Morely had thought before doing experiment ? 9) Is theory of relativity and special theory of relativity same? 9) Conclusion about the persences of aether from M-M expt.?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 28
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d11) Why there is change in velocity of light when shift = 1 ? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 29
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12) Why do hypothetical concept of ether was needed? 13) Mathematics of M-M Expt? 14) When Interferometer is rotated through 90 degree what happens then? 15) Derivation of M-M expt. ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 30
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What Time Dilation means…………? • A moving clock ticks more slowly than a clock at restWhen two events are occurs at the same location in an inertial reference frame , the time interval between them, measured in that frame, is called the proper time interval or the “ proper time”Measurements of the same time interval from any other inertial reference frame are always greater.The amount by which a measured time interval is greater than the corresponding proper time interval is called time dilation. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 31
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vTime DilationOne clock is at rest in a laboratory on the ground andthe other is in a spacecraft that moves at the speed vrelative to the ground. An observer in the laboratorywatches both clocks; Does he/she find that they tickat the same rate ? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 32
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moving clocks run slow. This means that if twoevents occur at the same place, such as the ticks ofa clock, a moving observer will measure the timebetween the events to be longer. The relationbetween a time measured by a stationary observert0 to the time t measured by an observer movingwith velocity v is:The gamma factor is common in relativity, and wewill use it often. It is always greater than unity. If thevelocity were greater than c, it would be undefined.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 33
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Show Me The Derivation?For our derivation, we will consider twomeasurements. One taken by a rider with theclock and the other measurement for theclock will be made by a stationary observer(referred to as the stationary for the mover). Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 34
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A light pulse clock at rest On the ground as seen by An observer on the ground. mirror The light travels the total Distance 2L at speed c, therefore Recording device The time for entire trip is 2L t cLo Meter stick Light pulse mirror Photosensitive surface Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 35
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Clock is moving with v velocity in spacecraft. What observer notice who is in theRest with respect to spacecraft (seen from the ground). The time interval betweenticks is t. v 0 t/2 t B ct 2 Lo A C D vt 2 Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 36
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Observer in same inertial frame and notice the events to = 2L/cSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 37
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Now, observer in another frame and seen eventsfrom the ground …………Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 38
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The light ray travels the path AB and mirror AD with velocity v. The distance AB= ct/2 and AD=vt/2 2 2 ct 2 vt L0 2 2 2 L0 / c t0 t 1 v2 c2 1 v2 c2 To= time interval onclock at rest relative to an observer= proper time T= time interval on clock in motion relative to an observer v-= speed of relative motion C= speed of light.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 39
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Example 1: If you were to board a craft and travel at 0.2 c, how long would 1 hour be? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 40
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Example 2: If you were to board a craft and travel at 0.8 c, how long would 1 hour be? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 41
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Example 3: If you were to board a craft and travel c, how long would 1 hour be? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 42
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Example 4: If you were to board a craft and travel at 300 m/s, how long would 1 hour be? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 43
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Question:As we watch, a spaceship passes us in time t. The crew of thespaceship measures the passage time and finds it to be t.Which of the following statements is true?A) t is the proper time for the passage and it is smaller tB) t is the proper time for the passage and it is greater than tC) t is the proper time for the passage and it is smaller than tD) t is the proper time for the passage and it is greater than tE) None of the above statements are true.Ans. C Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 45
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QuestionSpaceship A, traveling past us at 0.7c, sends a messagecapsule to spaceship B, which is in front of A and istraveling in the same direction as A at 0.8c relative to us.The capsule travels at 0.95c relative to us. A clock thatmeasures the proper time between the sending andreceiving of the capsule travels:A) in the same direction as the spaceships at 0.7c relative to usB) in the opposite direction from the spaceships at 0.7c relative to usC) in the same direction as the spaceships at 0.8c relative to usD) in the same direction as the spaceships at 0.95c relative to usE) in the opposite direction from the spaceships at 0.95c relative to usAns. D Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 46
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ProblemYou wish to make a round trip from Earth in a spaceship, traveling atconstant speed in a straight line for 6 months and then returning at thesame constant speed. You wish further, on your return, to find Earth as itwill be 1000 years in the future.(a)How fast must you travel?(b) Does it matter whether you travel in a straight line on your journey?If, for example, you traveled in a circle for 1 year, would it still find 1000years had elapsed by Earth clock when you returned? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 47
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QuestionA millionairess was told in 1992 that she had exactly 15years to live. However, if she travels away from the Earthat 0.8 c and then returns at the same speed, the last NewYears day the doctors expect her to celebrate is:A) 2001B) 2003C) 2007D) 2010E) 2017Ans. E
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The Time Dilation equation for Relativity is:What is t, to v, and c? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 49
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The relativity of length and Length ContractionThe Relativity of LengthThe length L0 of an object measured in the rest frame of theobject is its proper length or rest length. Measurements of thelength from any reference frame that is in relative motionparallel to that length are always less that the proper length 2 v L L0 1 2 c Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 50
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Speed of Spaceship Observed Length Observed HeightAt rest 200 ft 40 ft10 % the speed of light 199 ft 40 ft86.5 % the speed of light 100 ft 40 ft99 % the speed of light 28 ft 40 ft99.99 % the speed of light 3 ft 40 ftSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 51
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QuestionA measurement of the length of an object that is movingrelative to the laboratory consists of noting the coordinatesof the front and back:A) at different times according to clocks at rest in thelaboratoryB) at the same time according to clocks that move withthe objectC) at the same time according to clocks at rest in thelaboratoryD) at the same time according to clocks at rest withrespect to the fixed starsE) none of the aboveAns. C Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 52
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QuestionA certain automobile is 6 m long if at rest. If it ismeasured to be 4/5 as long, its speed is:A) 0.1cB) 0.3cC) 0.6cD) 0.8cE) > 0.95cAns. C Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 53
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ProblemA cubical box is 0.50 m on a side.(a) What are the dimensions of the box as measured by anobserver moving with a speed of 0.88c parallel to one of theedges of the box?(b) What is the volume of the box as measured by thisobserver? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 54
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2 v L L0 1 2 c L= 0.5 (1-(0.88)**2)1/2 =0.24m The observed dimension are= 0.24*0.5*0.5=0.059m**3Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 55
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An Important general question: If we know the co-ordinate x, y, z and time t of anevent, as measured in a frame S, How can we findthe coordinates x’, y ’ , z ’ and t’ of the same event asmeasured in a second frame S? Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 56
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Galilean TransformationSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 57
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To convert velocity components measured in theS frame to their equivalents in the S’ frame accordingto the Galilean Transformation, we simply differentiatex’, y’, and z’ with respect to time: Galilean transformation dx v x vx v violate both of the dt postulates of special dy relativity. v y vy 1). If we measure the speed dt of light in the x-direction in the S-system to be c, dz however, in the S’ system it vz vz dt will be c’= c-v Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 58
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Lorentz’s Transformation The primed frame moves with velocity v in the x direction with respect to the fixed reference frame. The reference frames coincide at t=t=0. The point x is moving with the primed frame. The reverse transformation is: Much of the literature of relativity uses the symbols β and γ as defined here to simplify the writing of relativistic relationshipsSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 59
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Relativistic Velocity TransformationNo two objects can have a relative velocity greater than c! But what if I observea spacecraft traveling at 0.8c and it fires a projectile which it observes to bemoving at 0.7c with respect to it!? Velocities must transform according to theLorentz transformation, and that leads to a very non-intuitive result calledEinstein velocity addition. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 60
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Just taking the differentials of these quantities leads to the velocity transformation. Taking the differentials of the Lorentz transformation expressions for x and t above gives Putting this in the notation introduced in the illustration above:Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 61
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The reverse transformation is obtained by just solving for u in the above expression. Doing that givesApplying this transformation to the spacecraft traveling at 0.8c which fires aprojectile which it observes to be moving at 0.7c with respect to it, we obtain avelocity of 1.5c/1.56 = 0.96c rather than the 1.5c which seems to be thecommon sense answer. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 62
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When ux and v are both much smaller than c (the non relativistic case) , the denominator of equation approaches unity and so u’x = ux –v. This corresponds to the Galilean velocity transformation, In the other Extreme, when ux =c ; the equation becomes U’x = c , From this result, we see that an object moving with a speed c relative to an observer in S also has a speed c relative to an observer in S’- independent of the relative motion of the S and S’.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 63
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Question: Imagine a motorcycle rider moving with a speed of0.800c past a stationary observer, as shown in figurebelow, If the rider tosses a ball in the forward directionwith a speed of 0.700c with respect to himself, what isthe speed of the ball as seen by the stationary observer? 0.700c 0.800c
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In this situation, the velocity of the motorcycle with respect to the stationary observer is v=0.800c. The velocity of the ball in the frame of reference of the motorcyclist is ux’=0.700c. Therefore, the velocity, ux, of the ball relative to the stationary observer is u x v ux uxv 1 c2 0.700c 0.800c 0.9615c 1 0.700c 0.800c 1 2 cSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 65
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The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation. The length is maximum in the frame in which the object is at rest.Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 66
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Questions/ Problems/ Doubts:1). I know the derivation but I have problem in basic concept, I did not understand the application of Galilean and Lorentz transformation?2.) I know the concept, but I face problem when I try to solve the numerical problem?3.) No proper notes/material of subject?4.) Problem in length contraction, when observer is moving or object is moving?5.) Why Galilean transformation do not obey laws of physics?6.) Transformation equation?7.) When we move away from a building it becomes smaller and smaller, is it the case of length contraction or some other physical process?8.) If object is in moving frame of reference, is there any change of dimension?9.) Why there is need of transformation, what we get from it?10.) How a large building contract, when seen in the glass/mirror of moving vehicle?
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Questions/doubts: 11. WHY THERE IS CHANGE IN VELOCITY OF LIGHT WHEN FRINGE SHIFT = 1 ? 12. When an observer and object of length L is moving with velocity v1 and v2 respectively, what will be the length contraction, v2> v1 ? 13. Not able to understand why the length seems less while moving……..Length contraction? 14. How length contraction takes place in MUON’S DECAY ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 68
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Questions/ doubts……..417 We consider earth an inertial frame of reference, and there are hardly any motions comparable to speed of light. Why do we refer to the relativity then ? does time, length and velocity all relativistic quantities?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 69
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Questions/ doubts How does time change in S’ as per Lorentz transformation & Physically why is this change of time? Explain the use of telescope and compensating plate in the setup of experiment? How does the height of the object changes when observer is along x-axis and object is moving along y- axis ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 70
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Questions/ doubts/ problems How to implement these formulas in typical questions? Galilean & Lorentz transformation with numericals? Tried to understand the topic ‘ Length contraction’ using text book but was unable to understand, also tried numerical problems on the topic ‘ time dilation’ and was unable to solve them too…......!!!!! Explain length contraction of a thin road, when observer moves perpendicular to the rod ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 71
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How much Do I know? What Do I need to learn? What I have Learned ?Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 72
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Relativistic mass Mass m-0 of an object measured when it was at rest and mass m measured when it was in motion with velocity v, Relativistic Mass Increase 6 m0 m 5 4 2 1 v Mass 3 2 c 2 1 0 0 0.2 0.4 0.6 0.8 1 Speed( c = 1) Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 73
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Q. The total energy of a proton is three times its rest energy.(a) Find the proton’s rest energy in electron volts. 2 mpc 27 8 2 (1.67 10 kg)(3.0 10 m / s ) 10 19 (1.50 10 J )(1eV / 1.6 10 J) 938MeVSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 74
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Q. With what speed is the proton moving? E mc 2 mpc2 E 3m p c 2 u2 1 c2 1 3 1 u 2 c2 solv ing f or u giv es u2 1 1 c2 9 or 8 u c 2.83 108 m / s 3Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 75
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Relativistic Energy How Does the Total Energy of a Particle Depend on Speed? We have a formula for the total energy E = K.E. + rest energy, 2 2 m0c E mc 2 2 1 v /c so we can see how total energy varies with speedSunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 76
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How Does the Total Energy of a Particle Depend on Momentum? It turns out to be useful to have a formula for E in terms of p. Now m0 c 4 2 E2 m2c 4 1 v2 / c2 m 2 c 4 (1 v 2 / c 2 ) m0 c 4 2 m2c 4 m2v 2c 2 m0 c 4 2 m2c 4 E2 m0 c 4 2 m2c 2v 2 2 4 2 2 hence using p = mv we find E mc 0 c p If p is very small, this gives p2 E m0 c 2 2m0the usual classical formula.If p is very large, so c2p2 >> m02c4, the approximate formula is E = cp Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 77
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The High Kinetic Energy Limit: Rest Mass BecomesUnimportant!Notice that this high energy limit is just the energy-momentumrelationship Maxwell found to be true for light, for all p. This could onlybe true for all p if m02c4 = 0, that is, m0 = 0.Light is in fact composed of “photons”—particles having zero “restmass”, as we shall discuss later. The “rest mass” of a photon ismeaningless, since they’re never at rest—the energy of a photon m0c 2 E mc 2 1 v2 / c2is of the form 0/0, since m0 = 0 and v = c, so “m” can still be nonzero.That is to say, the mass of a photon is really all K.E. mass.For very fast electrons, such as those produced in high energyaccelerators, the additional K.E. mass can be thousands of times the restmass. For these particles, we can neglect the rest mass and take E = cp. Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 78
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