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MECHANICS UNIT 9
Special Theory of Relativity
Author of this note Mr. K. Prasad BSc from University of Calicut, MSc from university of Delhi, and M.B.A from IGNOU Delhi, These notes were prepared during my teaching session for under graduate students (11 Th and 12 Th class) of my school physics department. This note is helpful for under graduate students and junior level graduates. I declare that these notes are my original works based on my knowledge in physics and the books mentioned below are the reference books I used for preparing these notes. Table of contents
(1) Michelson-Morley Experiment (2) Michelson Morley Experiment details (3) Procedure of Michelson Morley Experiment (4) Postulates of Special Theory of Relativity (5) Lorentz Transformations (6) Theory of Simultaneity (7) Length contraction (8) Time dilation (9) Relativistic transformation of velocity (10)Relativistic transformation of acceleration (11) Wavenumber definition (12)Variation of mass with velocity (13) Massless particles definition (14) Mass-energy Equivalence (15) Mass – Energy formula (16)Relativistic Doppler Effect (17)The Doppler Effect (18)Doppler Effect Examples (19)Doppler Effect Formula (20)Uses of Doppler Effect (21) Transformation of Energy and Momentum. Reference Books: An Introduction to Mechanics, Daniel Kleppner& Robert Kolenkow, 2007, TataMcGrawHill Mechanics, DS Mathur, PS Hemne, 2012, S. Chand University Physics, FW Sears, MW Zemansky& HD Young 13/e, 1986, AddisonWesley
at various velocities in ether comparative with that in a vacuum. What's more, any medium having a thickness will redirect light going through it because of the peculiarity of refraction. He had additionally fostered an interferometer to probe the showing up light beams and demonstrate his hypothesis. The Michelson Morley Experiment Sound waves require some medium through which these waves can travel. Maxwell in 1864 showed that light is an electromagnetic wave and consequently was assumed that there is ether that proliferates light beams. By seeing how light proliferates through the ether , one can decide a flat out reference outline. Consequently, the Michelson Morley explore was achieved to identify ether that was thought to be the transporter of light waves. The reason for the Michelson and Morley try was to recognize the speed of the Earth to ether. The system depended on the optical gadget named interferometer that thinks about the way lengths for light beams going in perpendicular directions. Michelson Morley Experiment details As per Michelson's examination hypothesis, the light ought to go at various paces through ether. The speed at which light moves relies upon the overall movement through space. Michelson Morley planned an interferometer to recognize minute contrasts in the appearance season of light beams. Out of this multitude of beams, one can consume a large chunk of the day to arrive at the sensor while going through ether. The examination contrasted the speed of light with notice the general movement of Earth through ether. In any case, the finish of the Michelson Morley analyze emerges to be negative. It implies that they tracked down no contrast between the speeds of light while going through ether. Michelson Morley interferometer sent white light for the real perceptions and yellow light from a sodium fire through a half-transparent mirror. The mirror was utilized to part the approaching light pillar into two separate shafts heading out opposite to one another. In the wake of leaving this mirror, radiates moved out to the long arms end where they looked back reflection into the center. These two bars then recombine to deliver an example of helpful and destructive impedance. Procedure of Michelson Morley Experiment
That's what Michelson asserted assuming the speed of light was consistent concerning the ether medium through which the Earth moves, then that movement can be recognized. It tends to be detected by contrasting the speed of light opposite with and toward the Earth's movement. The subtleties of Michelson explore set-up are: The light emission gets occurrence to a half-silvered glass plate. This plate goes about as a beam splitter, what parts the light beam into two rational beams. One beam transmits, and the other reflects. The beam sent strikes the mirror, say, M1, and gets reflected. The beam reflected strikes the mirror, say, M2, which again gets reflected. The returned beam arrives at the telescope, which is utilized for interference patterns delivered by these two beams. The partition between the plate and two mirrors is something similar, which alludes to the manageable distance. The light reflected from two mirrors interferes with the mirror. Presently, from the Michelson Morley analyze notes, it tends to be seen that the device and light both are moving in a similar course. Hence, the general speed will be c - v. After reflection, the mechanical assembly, and light both move the other way. Thus, for this situation, relative velocity will become c + v. Let Us Calculate the Time Taken by the Transmitted Ray to Travel to the Mirror T1 = L/c-v T2 = L/ c+v Tl =T1+T Tl= L / c-v +L / c+v Tl=(L×(c+v))+(L×(c−v))/c^2 −v^2 Tl=Lc+Lv+Lc−Lv/c^2 −v^2 Tl=2Lc/c^2 −v^2 Tl=L[2c / c^2 −v^2 ] Tl=L/c^2 [2c/1− v^2 /c^2 ] Tl=2L / c[1− v^2 /c^2 ]− Applying Binomial Theorem on the above equation and neglecting higher power terms gives Tl=2L / c[1+c^2 /v^2 ]
When an object is moving, its length shrinks in similar direction as its speed. This is called length compression. This influences all articles consistently when they are moving. It affects a race car going around a track, you when you're running, and even a snail slowly inching its way along the ground. Time dilation Is the slowing back of time as per the hypothesis of relativity that happens in a system moving comparative with an external spectator and that becomes evident particularly as the speed of the system moves toward that of light — called additionally time dilatation, Relativistic transformation of velocity Relativistic velocity can be defined as the speeds of an object moving at a relativistic speed: u= v+u′1+vu′c2. A spectator of electromagnetic radiation sees relativistic Doppler impacts assuming that the source of the radiation is moving comparative with the eyewitness. Relativistic transformation of acceleration Acceleration in special relativity (SR) follows, as in Newtonian Mechanics, by differentiation of velocity with respect to time. In light of the Lorentz change and time dilation, the ideas of time and distance become more mind complex, which likewise prompts more complicated meanings of acceleration. Wavenumber definition wavenumber, likewise called wave number, a unit of frequency, frequently used in nuclear, sub-atomic, and atomic spectroscopy, equivalent to the true frequency isolated by the speed of the wave and hence equivalent to the number of waves in a unit distance. In free space, the wavenumber is equivalent to the frequency of the wave divided by the speed of light. The wavenumber and frequency are directly related to one another, high frequency waves will have high wavenumber or more number of waves.in a unit distance
Variation of mass with velocity At the point when an object is going at a high velocity, its resistance to acceleration increase doesn't change and its capacity to encounter gravity doesn't change. The mass of an article so doesn’t change when it goes at high speed At the point when we consider mass as energy, we can start to comprehend the reason why an article will expand its 'mass' as it speeds up. As an article speeds up, so does how much energy that it has, this energy is the thing we refer to as 'the expansion in mass' (simply recall, this is inertial mass). Massless particles definition Massless means particle having no mass after about 85 years of research, scientists have affirmed the presence of a massless molecule called the Weyl fermion unexpectedly. With the remarkable capacity to act as both matter and antimatter inside a crystal, this odd molecule can make electrons that have no mass the photon is a massless particle. According to theory it has energy and momentum but no mass, and this is confirmed by experiment to within strict limits. Mass-energy Equivalence This equivalence of mass and energy is a result of Einstein's hypothesis of unique relativity. This basically implies that mass can be changed over into energy and visa-versa Mass – Energy formula The law of preservation of energy expresses that energy can neither be made nor obliterated - just changed over starting with one type of energy then onto the next. This implies that a framework generally has a similar measure of energy, except if it's additional from an external perspective. This is especially befuddling on account of non-moderate powers, where energy is changed over from mechanical energy into nuclear power, however the general energy continues as before. The best way to utilize energy is to change energy starting with one structure then onto the next.
The relativistic Doppler effect can be defined as is the adjustment of frequency and wave length of light, brought about by the general movement of the source and the observer (as in the classical Doppler effect), while considering effects depicted by the unique hypothesis of relativity. The Doppler Effect Doppler Effect or Doppler shift is a peculiarity that is noticed at whatever point the wellspring of waves is moving concerning an eyewitness. For instance, an emergency vehicle crossing you with its alarm blasting is a typical actual exhibition of the Doppler Effect. In this article, let us comprehend the complexities of the Doppler impact exhaustively. Doppler Effect Explained Doppler Effect is a significant peculiarity in different logical disciplines, including planetary science. The Doppler effect or the Doppler shift portrays the progressions in the recurrence of any sound or light wave created by a moving source as for an eyewitness. Doppler effect in material science is characterized as the increment (or decrease) in the recurrence of sound, light, or different waves as the source and onlooker move towards (or away from) one another. Waves transmitted by a source going towards a spectator get packed. Interestingly, waves discharged by a source voyaging away from an eyewitness get loosened up. Christian Johann Doppler previously proposed the Doppler Effect (Doppler Shift) in 1842. Doppler Effect Examples Allow us to envision the accompanying situation: Case 1: Two individuals A and B, are remaining out and about, as displayed underneath in the image.
Which individual hears the firing up motor with a more magnitude? Individual A Individual A hears the firing up motor with a more noteworthy size than individual B. Individual B, remaining behind the vehicle, gets less waves each second (since they're fanned out), bringing about a low-pitched sound. Be that as it may, individual A who is before the vehicle, gets a greater amount of Those sound wave swells each second. Thus, the recurrence of the waves is higher, and that implies the sound has a higher pitch. Case 2: Now let us think about the accompanying circumstances: Circumstance 1: How is the example of waves shaped when you out of nowhere bounce into a lake? Circumstance 2: How is the example of waves shaped when you are strolling in a lake? The picture given beneath features the contrast between wave designs in the two circumstances The distinction in the wave design is because of the source's development in the subsequent case. This is the thing the Doppler impact is. In the Doppler impact,
Doppler Effect Limitations Doppler Effect is applicable only when the velocities of the source of the sound and the observer are much less than the velocity of sound. The motion of both source and the observer should be along the same straight line. Transformation of Energy and Momentum To put it simply, force is relative to the square root of kinetic energy since energy is directly corresponding to velocity, while kinetic energy is corresponding to velocity squared. This truly intends that on the off chance that kinetic energy becomes multiple times as large, the force will just twofold. If the Kinetic energy is zero, momentum will be zero. Hence, a body cannot have momentum without having energy.
