Part I Special Relativity

Size: px
Start display at page:

Download "Part I Special Relativity"

Transcription

1 Part I Speial Relativity G. W. Gibbons D.A.M.T.P., Cambridge University, Wilberfore Road, Cambridge CB3 0WA, U.K. February 14, 2008 The views of spae and time whih I wish to lay before you have sprung from the soil of experimental physis, and therein lies their strength. They are radial. Heneforth spae by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union will preserve an independent reality. H Minkowski (1908). Contents 1 The Shedule Units Einstein s Theory of Speial Relativity 6 3 *Early ideas about light* Maxwell s equations *The Speed of Light* *Roemer s measurement of * *Fizeau s measurement of * *Fouault s rotating mirror* Absolute versus Relative motion 12 6 Veloity omposition formulae 14 7 Galilean Priniple of Relativity Waves and Galilean Transformations

2 8 Spaetime Example: uniform motion in 1+1 dimensions Example: uniform motion in 2+1 dimensions Example: non-uniform motion in 1+1 dimensions Minkowski s Spaetime viewpoint Einstein s Priniple of Relativity Mihelson-Morley Experiment Derivation of the Lorentz Transformation formulae Relativisti veloity omposition law *Observational for Einstein s seond postulate* Light in a medium: Fresnel Dragging Composition of Lorentz Transformations Veloity of light as an upper bound * Super-Luminal Radio soures* The two-dimensional Lorentz and Poinaré groups The invariant interval Timelike Separation Spaelike separation Time Dilation Muon Deay Length Contration The Twin Paradox: Reverse Triangle Inequality *Hafele -Keating Experiment* Aelerating world lines Doppler shift in one spae dimension *Hubble s Law* The Minkowski metri Composition of Lorentz Transformations Lorentz Transformations in spaetime dimensions The isotropy of spae Some properties of Lorentz transformations Composition of non-aligned veloities Aberration of Light * Aberration of Starlight* Water filled telesopes Headlight effet Solid Angles *Celestial Spheres and onformal transformations* *The visual appearane of rapidly moving bodies* Transverse Doppler effet

3 15.9 *The Cosmi Mirowave Bakground* * Kinemati Relativity and the Milne Universe* *The Foundations of Geometry* The Milne metri and Hubble s Law *Relativisti omposition of veloities and trigonometry in Lobahevsky spae* Parallax in Lobahevsky spae *Rotating referene frames* Transverse Doppler effet and time dilation *The Sagna Effet* Length Contration Mah s Priniple and the Rotation of the Universe General 4-vetors and Lorentz-invariants veloity and 4-momentum veloity momentum and Energy Non-relativisti limit Justifiation for the name energy *Hamiltonian and Lagrangian* Partiles with vanishing rest mass Equality of photon and neutrino speeds Partile deays ollisions and prodution Radioative Deays Impossibility of Deay of massless partiles Some useful Inequalities Impossibility of emission without reoil Deay of a massive partile into one massive and one massless partile Deay of a massive partile into two massless partiles Collisions, entre of mass Compton sattering Prodution of pions Creation of anti-protons Head on ollisions Example: Relativisti Billiards Mandelstam Variables

4 22 Mirrors and Refletions *The Fermi mehanism* *Relativisti Mirrors* *Corner Refletors on the Moon* Time reversal Anti-partiles and the CPT Theorem aeleration and 4-fore Relativisti form of Newton s seond law Energy and work done Example: relativisti rokets The Lorentz Fore Example: partile in a uniform magneti field Uniform eletro-magneti field and uniform aeleration vetors, tensors and index notation Contravariant vetors Covariant vetors Example: Wave vetors and Doppler shift Contravariant and ovariant seond rank tensors The musial isomorphism De Broglie s Wave Partile Duality * Wave and Group Veloity: Legendre Duality* The Lorentz equation Uniformly Aelerating referene frames: Event Horizons Causality and The Lorentz Group Causal Struture The Alexandrov-Zeeman theorem Minkowski Spaetime and Hermitian matries Spinning Partiles and Gyrosopes Fermi-Walker Transport Spinning partiles and Thomas preession Bargmann-Mihel-Telegdi Equations

5 1 The Shedule Read as follows: INTRODUCTION TO SPECIAL RELATIVITY 8 letures, Easter and Lent terms [Leturers should use the signature onvention (+ ).] Spae and time The priniple of relativity. Relativity and simultaneity. The invariant interval. Lorentz transformations in (1 + 1)-dimensional spaetime. Time dilation and muon deay. Length ontration. The Minkowski metri for (1 + 1)-dimensional spaetime.[4] 4 vetors Lorentz transformations in (3 + 1) dimensions. 4 vetors and Lorentz invariants. Proper time. 4 veloity and 4 momentum. Conservation of 4 momentum in radioative deay.[4] BOOKS G.F.R. Ellis and R.M. Williams Flat and Curved Spae-times Oxford University Press paperbak W. Rindler Introdution to Speial Relativity Oxford University Press paperbak W. Rindler Relativity: speial, general and osmologial OUP paperbak E.F. Taylor and J.A. Wheeler Spaetime Physis: introdution to speial relativity Freeman paperbak 1.1 Units When quoting the values of physial quantities, units in whih = and h=1, will frequently be used. Thus, at times for example, distanes may be expressed in terms of light year. Astronomers frequently use parses whih is the distane at whih is short for paralax seond. It is the distane at whih the radius of the earth subtends one seond of ar. One parse works out to be Km or 3.3 light years. A frequently used unit of energy, momentum or mass is the eletron volt or ev whih is the work or energy required to move an eletron through a potential differene of one Volt. Physial units, masses and properties of elementary partiles are tabulated by the Partile Data Group and may be looked up at Although not neessary in order to follow the ourse, it is a frequently illuminating and often amusing exerise to go bak to the original soures. Many of the original papers quoted here may be onsulted on line. For papers in the Physial Review, bak to its ineption in the late nineteenth entury go to 5

6 . For many others, inluding Siene and Philosophial Transations of the Royal Soiety (going bak its beginning in the to seventeenth entury) go to 2 Einstein s Theory of Speial Relativity is onerned with the motion of bodies or partiles whose relative veloities are omparable with that of light = 299, 792, 458 ms 1. (1) In a nutshell, Newton s Seond Law remains unhanged in the form dp dt = F (2) where F the fore ating on a partile of momentum p and mass m 1, but while aording to Newton s Theory p = mv (3) in Einstein s Theory p = mv 1 v2 2. (4) If this were all there is in it, relativity would, perhaps, not be espeially interesting. What makes Relativity important is that it entails a radial revision of our elementary ideas of spae and time and in doing so leads to the even more radial theory of General Relativity whih omes into play when gravity is important. In this ourse we shall ignore gravity and onfine our attention to Speial Relativity. For matters gravitational the reader is direted to[34]. To see why relativity has suh a profound impat on ideas about spae and time, note that we are asserting that there atually is suh a thing as the veloity of light. For the benefit of those who have not studied Physis at A-level, or who did, but have now forgotten all they ever knew, the next setion ontains a review of the elementary physis of light. 3 *Early ideas about light* Experiments with shadows and mirrors lead to the idea that light is a form of energy that propagates along straight lines alled light rays. On refletion at a smooth surfae S at rest, it is found that 1 properly speaking rest-mass 6

7 (i) The inident ray, the refleted ray and the normal to the surfae at the point of refletion are o-planar (ii) The inident and refleted rays make equal angles with the normal. Hero of Alexander showed that these laws may be summarized by the statement that if A is a point on the inident ray, B on refleted ray and x S the point at whih the refletion takes plae, then x is suh that the distane d(a,x) + d(x, B) (5) is extremized among all paths from AyB, y S to the surfae and from the surfae to B. When light is refrated at a smooth surfae S it is found that (i) The inident ray, the refrated ray and the normal to the surfae at the point of refration are o-planar (ii) The inident and refrated rays make angles θ i and θ r with the normal suh that sin θ i = n r (6) sinθ r n i where the quantities n i and n r are harateristi of the medium and may depend upon the olour of the light and are alled its refrative index. By onvention one sets n = 1 for the vauum. Pierre Fermat showed that these laws, first learly enuniated in about 1621 by the Leyden mathematiian Willebrod Snellius or Snell in work whih was unpublished before his death in 1626, and later by Desartes, although probably known earlier to Thomas Harriot, may be summarized by the statement that if A is a point on the inident ray, B on refrated ray and x the point at whih the refration takes plae, then x is suh that the optial distane n i d(a,x) + n r d(x, B) (7) is extremized among all paths AyB, y S from A to surfae and from the surfae to B. In other words the differential equations for light rays may be obtained by varying the ation funtional nds (8) where ds is the element Eulidean distane. By the time of Galilei its was widely thought that light had a finite speed, and attempts were made to measure it. Broadly speaking there were two views about the signifiane of this speed. The Emission or Ballisti Theory held by Isaa Newton and his followers aording to whih light onsisted of very small partiles or orpusles with mass m speed and momentum p = m, the speed varying depending upon the 7

8 medium. On this theory, Snell s law is just onservation of momentum parallel to the surfae. p i sin θ i = p r sin θ r, (9) whene, assuming that the mass is independent of the medium sin θ i sin θ r = r i. (10) The Wave Theory proposed by the duth physiist Christian Huygens in 1678, aording to whih light is a wave phenomenon having a speed and suh that eah point on the wave front gives rise to a seondary spherial wave of radius t whose forward envelope gives the wavefront at a time t later. On this theory, Snell s law arises beause the wavelength λ i of the inident wave and the wavelength of the refrated wave λ r differ. Applying Huygen s onstrution gives sin θ i = λ i. (11) sinθ r λ r Sine, for any wave of frequeny f, λf = and sine the frequeny of the wave does not hange on refration, we have aording to the wave theory: sin θ i sin θ r = i r. (12) The two theories gave the opposite predition for the speed of light in a medium. Sine refrative indies are never found to be less than unity, aording to the emission theory the speed of light in a medium is always greater than in vauo, while aording to the wave theory it is always smaller than in vauo. One way to distinguish between the two theories was to measure the speed of light in vauo and in a medium. This was first done by Fouault in 1850, and more aurately by Mihelson in 1883 using the rotating mirror method of the former, whih will be desribed shortly. By interposing a tube filled with water in the path of the light, they showed that the speed of light in water was slower than in vauo 2. It follows that Hero and Fermat s variational properties may be summarized by the statement that the time taken for light to traverse the physial path is extremized. Another way to distinguish the theories is by their ability to aount for the diffration of light by very small obstales as observed by Grimaldi in 1665 or by experiments on slits, suh as were performed by the polymath Thomas Young 3 in Following a large number of subsequent experiments, notably by Fresnel, by Fouault s time, some form of wave theory was aepted by almost 2 The argument is in fat slightly indiret sine these experiments atually measure the group veloity of light while refration depends on the phase veloity. The distintion is desribed later. Given one, and information about the dispersion, i.e. how the refrative index varies with wavelength, one may alulate the other. 3 Young played an equal role with Champillon in the translation of the Egyptian hieroglyphis on the Rosetta stone. 8

9 all physiists. In its simplest form, this postulated that in vauo, some quantity satisfies the salar wave equation φ t 2 = 2 φ, (13) whih, if is onstant, is easily seen to admit wavelike solutions of the form or more generally φ = Asin ( 2π( x λ ft)), (14) φ = f(k ωt), (15) where f() is an arbitrary C 2 funtion of its argument and ω k =. Sine equation (13) is a linear equation, the Priniple of Superposition holds and solutions with arbitrray profiles, moving in arbitrary diretions may be superposed. A fat whih not only explains many opetial phenomena but also led to the idea of Fourier Analysis.Note that solutions (13) are non-dispersive, the speed is independent of the wavelength λ or frequeny f. Until Einstein s work, almost all physiists believed that wave propagation required some form of material medium and that light was no exeption. The medium was alled the luminiferous aether (or ether) and many remarkable properties were asribed to it. Many physiist, inorretly as it turned out, believed ed that it was inextriably linked with the nature of gravitation. Others, like Lord Kelvin, postulated that atoms ould be thought of as knotted vortex rings. This seemed to require that the ether was some sort of fluid. A key question beame: what is the speed of the earth relative to the aether? The properties of the ether beame even harder to understand when it was established that light ould be polarized. This was first notied by Huygens who was studying the refration of light through a rystal of alite also known as Ieland spar. In 1808 Malus disovered that light ould be polarized by in the proess of refletion.these observations led diretly to the idea that light due to some sort of motion transverse to the diretion of propagation, and so the quantity φ should be some sort of vetor rather than a salar. They also suggested to many that the aether should be some sort of solid. The realization that light was an eletromagneti phenomenon and the great ahievement of the Sottish physiist James Clerk Maxwell ( ) in providing in 1873 a omplete, unified and onsistent set of equations to desribe eletromagnetism, whih moreover predited the existene of eletromagneti waves moving at the speed of light and the subsequent experimental verifiation by the German physiist Heinrih Hertz ( ) around did nothing to dispel the wide-spread onfusion about the aether. Elaborate mehanial models of the aether were onstruted and all the while, it and the earth s motion through it, eluded all experimental attempts at detetion. The general frustration at this time is perhaps refleted in the words of the president of the 4 In fat it seems lear that Hertz had been antiipated by the English Eletriian D E Hughes in 1879, but the signifiane of his work was not appreiated until muh later [49]. 9

10 British Assoiation, Lord Salisbury who is reported to have proposed, at one it its meeting held at Oxford, a definition of the aether as the nominative of the verb to undulate. In a similar vein. disussing the various allegedly physial interpretations, Hertz delared that To the question What is Maxwell s Theory, I know of no shorter or more definite answer than the following: Maxwell s theory is Maxwell s system of equations. Every theory whih leads to the same system of equations, and therefore omprises the same possible phenomena, I would onsider a form of Maxwell s theory. Maxwell s equations have many beautiful and remarkable properties, not the least important of whih is invariane not under Galilei transformations as might have been expeted if the aether theory was orret, but rather under what we now all Lorentz transformations. This fat was notied for the salar wave equation (13) in 1887, long before Einstein s paper of 1905, by Woldemar Voigt( ) and both Lorentz and Poinaré were aware of the Lorentz invariane of Maxwell s equations but they regarded this as a purely formal property of the equations. As we shall see, Einstein s insight was in effet to see that it is perhaps the single most important mathematial fat about the equations. From it flows all of Speial Relativity and muh of General Relativity. 3.1 Maxwell s equations These split into two sets. The first set always holds, in vauo or in any material medium and independently of whether any eletri harges or urrents are present. They deny the existene of magneti monopoles and asserts the validity of Mihael Faraday s law of indution. divb = 0, urle = B t. (16) The seond set desribe the response of the fields to the pressene of eletri harges, harge density ρ and urrents, urrent density j. At the expense of introduing two additional fields they may also be ast in a form whih is always orret. They assert the validity of Coulomb s law, and Ampére s law, provided it is supplemented by the last, ruial, additional term, alled the displaement urrent due to Maxwell himself. divd = ρ urlh = j + D t. (17) It follows from the identity div url = 0, that eletri harge is onserved ρ + divj = 0. (18) t 10

11 In order to lose the system one requires onstitutive relations relating D and H to E and B. In vauo these are linear relations D = ǫ 0 E, H = 1 µ 0 B, (19) where µ 0 and ǫ 0 are two universal physial onstants onstants alled respetively the permeability and permittivity of free spae.thus, in vauuo, Maxwell s equations are linear and the priniple of superposition holds for their solutions. Thus, in vauo dive = ρ ǫ 0 url 1 µ 0 B = j + ǫ 0 E t. (20) If there are no harges or urrents present, use of the identity urlurl = graddiv 2 gives ǫ 0 µ 0 2 E t 2 = 2 E, (21) ǫ 0 µ 0 2 B t 2 = 2 B. (22) Thus eah omponent of the eletri and magneti field travels non-dispersiveley with veloity = 1 ǫ0 µ 0. (23) The divergene free onditions imply that solutions of the form E = E 0 f(k ωt), B = B 0 f(k ωt), are transversely (plane) polarized and moreover sine k.e = 0, k.b = 0 (24) B 0 = n E 0, E 0 = B 0 n, (25) with n = k k, the vetors (E 0,B 0,n) form a right handed normal but not orthonormal triad. Physially the diretion of the polarization is usually taken to be that of the eletri field, sine this is easier to detet. Thus for any given propagation diretion n there are two orthogonal polarization states. in the sense that one may hoose the solutions suh that E 1.E 2. = 0 and thus B 1.B 2 = 0. 4 *The Speed of Light* 4.1 *Roemer s measurement of * That light does indeed have a finite speed was first demonstrated, and the speed estimated by the Danish astronomer Olaus Roemer ( ) in 1676 [1]. He observed the phases of Io, the innermost of the four larger satellites or moons 11

12 of Jupiter (Io, Europa, Ganymede and Callisto in order outward) whih had been disovered in 1610 by Galileo Galilei ( ) using the newly invented telesope 5 and of Io s motion around Jupiter is about 1.77 days an be dedued by observing its phases, when it is elipsed by Jupiter whose orbital period is years. In 1688 G D Cassini had published a set of preditions for these but Roemer observed that they were inaurate by about 15 minutes. The periods are shorter when Jupiter is moving toward the earth than when it is moving away from the earth. Roemer explained this and obtained a value for the speed of light by arguing that when Jupiter is moving toward the earth Io the time between elipses is shorter than when Jupiter is moving away from us beause in the former ase light the total distane light has to travel is shorter than in the latter ase. He obtained a value of 192,000 miles per seond or roughly 310,000 Km per se. If we think anahronistially 6, we might say if we think of Io as a lok, its period is Doppler shifted. 4.2 *Fizeau s measurement of * The first aurate terrestrial measurement of the speed of light was by the Frenh physiist Fizeau who, in 1849 [6], passed a beam of light through a rotating toothed wheel with 720 teeth, refleted it off a plane mirror Km away and sent the light bak toward the toothed dis. For a rotation speed of 12.6 turns per seond the light was elipsed giving a speed of about 315,000 Km per seond. 4.3 *Fouault s rotating mirror* In 1850 another Frenh physiist, Fouault [7] refleted light off a mirror whih was rotating about an axis parallel to its plane. The refleted light was then sent bak in the same diretion. If the rate of rotation of the mirror was hosen suitably the light arrived bak at its point of departure. From this Fouault dedued a value for the veloity of light of 298, 000 Kms 1. As mentioned above, he was also able to establish that the speed of light in water is less than in vauo. 5 Absolute versus Relative motion Newton based his theory on the assumption that spae was uniform and desribed by the usual laws of Eulidean geometry. There then arose the issue of whether motion with respet to that bakground was observable. If it was, then one would have a notion of absolute as opposed to relative motion. Newton argued, using the idea of a suspended buket of water, that one does have an 5 The true inventor of the telesope is not known. It seems to have been known to the English osmologist Thomas Digges and the Oxford mathematiian and explorer of Virginia, Thomas Harriot( The Doppler effet was proposed by the Austrian physiist C.J. Doppler in

13 idea of absolute rotational motion. However aording to his laws of motion there is no obvious dynamial way of deteting absolute translational motion. Sine his laws imply that the entre of mass of an isolated system of bodies one ould define an absolute frame of rest as that in whih the entre of mass of for, example the visible stars, is at rest. One later suggestion was that one ould take the entre of the Milky way. Lambert had suggested that it was the loation of a dark regent or massive body, a suggestion also made by Mädler. Interestingly we now know that at the entre of the Milky Way there is a massive Blak Hole of mass around a million times the mass of the sun, 10 6 M. The mass of the Milky way is about M. However any suh entre of mass frame an only be determined by astronomial observations. It ould not be found using purely dynamial experiments beneath loudy skies here on earth. Later, physiists, like Ernst Mah[50], began to worry about the logial foundations of Newton s laws. Exatly what was meant by the statement that a partile ontinues in a state of uniform motion if unaffeted by an external fore? Wasn t Newton s seond law in effet a tautology? et. L. Lange in 1885 [52] and others [50] had realized that an operational meaning ould be given to Newton s laws if one introdues the idea of an inertial frame of referene. This Lange thought of as a oordinate system for R 3 whih ould be determined by the free, mutually non-parallel, motion of three partiles. Then the first law ould be formulated as the non-trivial and empirially verifiable statement that any fourth free partile would move in this frame with uniform motion. In effet we are to use use the straight line motion of partiles to build up what is sometimes alled an inertial oordinate system or inertial referene system. In fat this onstrution losely resembles various onstrutions in projetive geometry, espeially if one adds in time as an extra oordinate. We will disuss this in more detail later. Mah pointed out that even if one used astronomial observations to determine a fundamental inertial frame of referene whih is at rest with respet to the fixed stars, that is stars so distant that their proper motions are negligible, this raises a puzzle. For example, in priniple we an define a non-rotating frame in two different ways, (i) Using gyrosopes for example whih, if they are subjet to no external torque will point in a onstant diretion in an inertial frame of referene, in other words using what has ome to be alled the ompass of inertia. (ii) Using the fixed stars. Nowadays we use quasi stellar radio soures quasars. It is then a remarkable oinidene that, as we shall see in detail later, to very high auray these two definitions agree. Mah had some, not very speifi, suggestions about a possible explanation. Mah s ideas strongly influened those of Einstein, espeially when he was formulating his General Theory of Relativity. They really annot be pursued in detail without General Relativity and without some understanding of Relativisti Cosmology. It was against this bakground that the question of the aether beame so important. If it really existed, it would provide an alternative frame of referene, whih might,or might not, oinide with the astronomially determined or 13

14 dynamially determined frames of referene. It ould, for example, remove the ambiguity about translational motion. The obvious guess was that it all three frames agreed. But if this was true, then the earth should be moving through the ether and this motion should be detetable. 6 Veloity omposition formulae Given that the speed of light is finite and presumably well defined on would ask, on the basis of Newtonian theory, in what frame? If the there is suh a thing as the veloity of light, independent of referene frame then the standard veloity addition formula in Newtonian Theory v v + u (26) annot be right. In fat, as we shall see later, one has a veloity omposition (rather than addition) formula. In Einstein s Theory v v+u 1+ uv 2 (27) so that if v =, v + u 1 + u 2 =. (28) Exerise Show that if u and v then u+v 1+ uv 2. Exerise Using the formula dw dt = F.v, for the rate of doing work W by a fore F ating on a point moving with veloity v, show that the work done in m aelerating a partile of mass m from rest to a final veloity v is 1 2 m 2. v2 The theory is alled the Theory of Relativity beause it deals with 2 relative veloities and what is alled the Priniple of Relativity. This idea began, at least in modern times, with Galileo and we shall begin with his version of it. 7 Galilean Priniple of Relativity Suppose a boat is moving with uniform veloity along a anal and we are looking at it. We are asked the following Question The lookout is in the row s nest and drops a heavy weight onto the dek. Will it hit the aptain below? Answer Yes. Reason We pass to a frame of referene S moving with the boat. The frame at rest with respet to the anal is an inertial frame of referene. Galileo assumed that 14

15 The laws of dynamis are the same in all frames of referene whih are in uniform motion with respet to an inertial frame of referene Now if we drop something from rest in frame S it will fall vertially down, Therefore if we drop something from rest in frame S it will fall vertially down, The boxed statement is Galilean Relativity follows in Isaa Newton s (1642- aount of dynamis beause But to transform to frame S we set and hene In frame S m d2 x = F(x, t). (29) dt2 x = x ut, (30) m d2 x dt 2 = F( x + ut) in frame S. (31) Note that Galileo assumed that the passengers in the boat would use the same oordinate t. In priniple one might have thought that one would also have to hange the time oordinate to a new oordinate t for this equivalene to work out but both Galileo and Newton agreed that Time is an absolute oordinate (32) that is, it takes the same value in all inertial frames of referene Formulae (30,33) onstitute a t = t (33) Gallean Transformation t = t, x = x ut. (34) We have just shown that Newton s equations of motion are invariant under Galilean Transformations. We shall now use Galilean transformations to dedue the Non-relativisti Veloity Addition Formulae. If a partile moves with respet to a frame S suh that then Thus and hene x = ṽt + x 0 (35) x ut = ṽ + x 0 (36) x = (u + ṽ)t + x 0. (37) v = u = ṽ, (38) gives the veloity with respet to S. Later we will imitate this simple alulation to obtain the veloity addition formula in speial relativity. 15

16 7.1 Waves and Galilean Transformations If, in a frame S at rest with respet to the aether, we have a wave of the form φ = sin ( k.x ωt ) (39) Its speed is = ω 2π k, its wavelength λ = k and frequeny f = ω 2π. If we submit it to a Galilei transformation it beomes φ = sin ( k. x (ω u.k)t ). (40) In the frame S, the wave has the same wavelength but the frequeny f = ω is hanged f = f(1 u osθ) (41) and the speed is, where = u osθ. (42) The formula for veloity in the moving frame S is very muh what one expets on the basis of a partile viewpoint but note that the angle θ is the angle between the diretion of the wave n = k k and the relative veloity u of the two frames S and S. Both frames agree on this as do they on the diretion of motion of the wave. In other words, Galilei s transformation formulae predit that there is no aberration. Later, we will obtain the physially orret results using the same method as above, but instead of a Galilei transformation we shall substitute using a Lorentz transformation. 8 Spaetime Before proeeding, we will pause to develop a way of thinking about kinematis that in fat goes bak to Lagrange and D Alembert. The latter wrote, in his artile on dimension in the Enylopédie ou Ditionaire raisonée des sienes, des arts et des metiers in 1764 A lever aquaintane of mine believes that it is possible to think of time as a fourth dimension, so that the produt of time and solidity would in some sense be the produt of four dimensions; it seems to me that this idea, while debatable, has ertain merits-at least the merit of novelty. The German pioneer of psyho-physis Gustav Theodor Fehner ( ) wrote a popular artile entitled Der Raum hat vier Dimension whih disusses related ideas. By that time the study of extra spatial dimensions was quiet advaned and the German Astronomer Johann C F Zollner ( ) gained notoriety for laiming that the alleged ability of self-laimed spiritualists to 16

17 untie knots sealed in glass jars was only expliable if they had been moved into a fourth spatial dimension. By 1880 s the Frenh railway engineer Ibry was using spaetime diagrams in a pratial way to onstrut railway time tables (see illustration on p 55 of [34]) The following examples illustrate the power of the view point in solving this type of mundane problem. 8.1 Example: uniform motion in 1+1 dimensions A ommuter is usually piked up by his/her spouse who drives at onstant speed from their house to meet the ommuter at 5 o lok. One day the ommuter arrives on an earlier train at 4 o lok and deides to walk. After a while the ommuter is piked up by his/her spouse who has driven to meet him as usual. They arrive bak at their house 10 minutes earlier than usual. For how long did the ommuter walk? 8.2 Example: uniform motion in 2+1 dimensions Four ships, A, B, C, D are sailing in a fog with onstant and different speeds and onstant and different ourses. The five pairs A and B, B and C, C and A, B and D, C and D have eah had near ollisions; all them ollisions. Show that A and D neessarily ollide. Hint Consider the triangle in the three-dimensional spaetime diagram formed by the world-lines of A, B and C. 8.3 Example: non-uniform motion in 1+1 dimensions A mountain hiker sets off at 8.00 am one morning and walks up to a hut where he/she stays the night. Next morning he/she sets of at 8.00 am and walks bak down the same trak. Show that, independently of how fast or slowly he/she walks there is at least one time on the two days when he/she is at the same point on the trak. For an interesting history of ideas of the fourth dimension before Einstein in art and popular ulture,inluding H G Wells s ideas about time travel, one may onsult the interesting book [23]. 9 Minkowski s Spaetime viewpoint In what follows we shall initially be onerned with the simplified situation in whih all motion is restrited to one spae dimensions. Thus the position vetors x have just one omponent. In this ase, it is onvenient to adopt a graphial representation, we draw a spaetime diagram onsisting of points we all events with spaetime oordinates (t, x). The two-dimensional spae with these oordinates is alled spaetime. 17

18 Passing to another frame of referene is like using oblique oordinates in spaetime. However, aording to Galilei, all observers use the same time oordinate. Geometrially while the lines of onstant x have different slopes in different frames, the lines of onstant time are all parallel to eah other. This means that two events (t 1,x 1 ) and (t 2,x 2 ) whih are simultaneous in frame S must be simultaneous in frame S, that is thus ( t 1, x 1 ) = (t 1, 1 ut 1 ), (43) ( t 2, x 2 ) = (t 1,x 2 ut 2 ) (44) t 1 t 2 = 0 t 1 t 2 = 0. (45) We say that in Newtonian theory simultaneity is absolute, that frame independent. Let s summarize (i) The Laws of Newtonian dynamis are invariant under Galilei transformations t = t, x = x ut. (46) (ii) veloities add v = ṽ + u. (47) (iii) Time is absolute. (iv) Simultaneity is absolute. 10 Einstein s Priniple of Relativity We have disovered that no purely dynamial experiment an determine our absolute veloity. If we are in a losed railway arriage moving uniformly we annot tell, by dropping partiles et, how fast we are traveling. The natural question to ask is whether we an tell using experiments involving light. If this has speed relative to some privileged inertial frame S, (identified before Einstein with the mysterious Aether or Ether ), it should, aording to Galileo, have speed u relative to a frame S moving with respet to the aether. By measuring this speed it should be possible to determine u. This was tried in the 10.1 Mihelson-Morley Experiment This is desribed learly and in detail in Mihelson s own words in [2]. Therefore the present desription will be brief. The light travel times T and T of light moving in diretions restively perpendiular and parallel to the motion along 18

19 arms of an interferometer of lengths L and L are measured. It was argued that in the parallel diretion (working in frame S) T = L [ u + 1 ] T = 2L u 1. (48) 1 u2 2 On the other hand (working in frame S) it was argued that the total distane the perpendiularly moving light has to travel is, by Pythagoras, L 2 + (ut 2 )2 = 2T 2L T = 1 u2 2 (49) Thus, for example, if L = L and T T we should be able to measure u. However in 1887 the experiment arried out by the Amerian Physiist MIhelson and Morley [3] revealed that T = T! Einstein drew the onlusion that no experiment, inluding those using light, should allow one to measure one s absolute veloity, that is he assumed. The Invariane of the Speed of Light The veloity of light is the same in all frames of referene whih are in uniform motion with respet to an inertial frame. 19

20 In Einstein s own words the same laws of eletrodynamis and optis will be valid for all frames of referene for whih the equations of mehanis hold good. We will raise this onjeture (the purport of whih will hereafter be alled the Priniple of Relativity ) to the status of a postulate and also introdue another postulate, whih is only apparently irreonilable with the former, namely that light is always propagated in empty spae with a definite veloity whih is independent of the state of motion of the emitting body. If Einstein is orret, then Galilei s transformations annot be orret. We need a new transformations alled Lorentz Transformations. They turn out to be (proof shortly) Lorentz Transformations x = x ut, t = 1 u2 2 t u 2 x 1 u2 2. (50) Note that (i) the time t gets transformed to t as well as x to x. (ii) Simultaneity is no longer absolute t 1 t 2 = t 1 t 2 u 1 2 (x 1 x 2 ) (51) u2 2 and hene t 1 t 2 t 1 t 2, if x 1 x 2. (52) (iii) If we take the non-relativisti limit in whih the speed of light is infinite we Lorentz transformations (50) we reover the Galilei transformations (34) Derivation of the Lorentz Transformation formulae We assume (i) ( t, x) are linear funtions of (t, x) (ii) 2 t 2 x 2 = 2 t 2 x 2 and hene the speed of light is invariant beause x = t x = ± t. In this first look at the subjet we assume (ii) but in more sophistiated treatments one makes onsiderably weaker assumptions. A preise statement will be made later. Even at this point it should be lear that we are ignoring trivial dilations or homotheties x = λx, t = λt, for λ 0 whih obviously leave the speed of light invariant. However we do not usually inlude these in the set of Galilei transformations. We shall also treat spae and time translations 20

21 t t + t 0, x x + x 0 as trivial Thus it is suffiient to onsider light rays through the origin of spaetime (t, x) = (0, 0) We shall also regard as trivial spae reversal x = x, t = t and time reversal x = x, t = t. Clearly (50) satisfy (i) and (ii). The onverse is obtained by setting ( t x ) ( A B = C D ) ( t x ), (53) with A > 0, D > 0 beause we are exluding time reversal and spae reversal. Substitution gives (At B) 2 (Ct Dx) 2 2 t 2 x 2 = 0. (54) Thus equating oeffiients of t 2 and x 2 to zero, we get (i) A 2 C 2 = 1 A = oshθ 1, C = sinhθ 1 (55) (ii) D 2 B 2 = 1 D = oshθ 2, B = sinhθ 2 (56) For some θ 1 and θ 2. Now equating the oeffiient of xt to zero gives (iii) AB = CD oshθ 1 sinhθ 2 = oshθ 2 sinhθ 1 θ 1 = θ 2. (57) Thus ( ) t = x ( oshθ sinhθ sinhθ oshθ ) ( ) t. (58) x Setting x = 0 allows us to see that the origin of the S frame satisfies xoshθ = t sinhθ. But if this is to agree with x = ut, where u is the relative veloity, we must have u = tanhθ := β, (59) where θ is alled the rapidity. It follows that and 1 oshθ = := γ (60) 1 u2 2 sinhθ = u 1 u2 2 = βγ. (61) The quantities β and γ do not, as far as I am aware, have individual names, and perhaps for that reason γ is often, rather inelegantly, referred to as the relativisti gamma fator. The use of the symbols β and γ is both traditional and almost universal in the subjet. A Lorentz transformation of the form (50) is often alled a boost whih is analogous to a an ordinary rotation. The analogue of the verb rotating is, unsurprisingly, boosting. A useful relation, partiularly in Tripos questions, is γ 2 (1 β 2 ) = 1. (62) 21

22 10.3 Relativisti veloity omposition law Of ourse the point is that veloities don t add. Suppose that In frame S x = ṽ t x 0, (63) then using the lorentz transformations (50) we have that In frames x uv 1 u2 2 = ṽ(t ux 2 ) 1 u2 2 + x 0. (64) Thus and hene x(1 + uv 2 ) = (ṽ + u) + t + x 0 1 u2 2, (65) Relativisti veloity omposition law ṽ = u + ṽ 1 + uṽ 2. (66) Thus, for example, ṽ = v =, whih is the invariane of the speed of light *Observational for Einstein s seond postulate* This is that the veloity of light is independent of the veloity of it s soure. Many high preision experiments give indiret evidene for it s validity. In addition, diret observational support for this inludes (i) The light urves of binary stars. De-Sitter [9] pointed out that if, for example, two stars are in orbit around eah other with orbital period T, then if light oming from that portion of the orbit when the star is moving toward us had a larger speed than when it was moving away from us, then light from an earlier part of the motion might even arrive more than half an orbital period before light oming from the intermediate portion of the orbit when it is neither moving toward us or away from us. This would lead to signifiant distortion of the plot of luminosity or of veloity against time. Consider, for example, the ase when we are in the plane a irular orbit of radius R and period P whose entre is a large distane L from us. The relation between time of emission t e and time of observation t o expeted on the basis of Newtonian theory is, sine R << L, t o = t e + L R sin 2πte P + v os 2πte P. (67) In this formula, the quantity v is the extra veloity supposed to be imparted to the photons moving toward us. Aording to the Ballisti theory of light of Newton and Galilei we would apply the usual rules for partiles of speed in 22

23 the rest frame of the moving stars. Thus we expet v = v orbital = 2πR P, but in general it ould be muh smaller. Sine L >> R and >> v we have t o t e + L R sin 2πt e P The observed Doppler shift is given by + Lv 2 sin 2πt e P. (68) dt 0 = 1 v dt e os 2πt e P + Lv2π 2 P sin 2πt e P. (69) If unless Lv2π P is small there will be signifiant distortion of the light urves. 2 Indeed t o may not be a monotoni funtion of t e, in whih ase, t e will not be a unique funtion of t o. In other words, pulses from different phases of the orbit may arrive on earth at the same time t o. Suh effets have not be seen. De Sitter himself onsidered the binary star β-aurigae. One example sometimes quoted is, the binary star Castor C. It is 45 light years away and has a period of.8 days. The stars have v orbital = 130Kms 1. The effet should be very large, but the light urves of the two stars are quite normal [11]. Using pulsating X-ray soures in binary star systems, Breher [24] was able to onlude that Einstein s seond postulate was true to better than 2 parts in a thousand million v v orbital < (70) This is ertainly an improvement on Zurhellen, who in (1914) obtained a limit of 10 6 using ordinary binary stars [29]. (ii) The time of travel over equal distanes of gamma rays emitted by a rapidly moving positron annihilating with a stationary eletron an be measured as they are found to be equal [12]. (iii) A similar measurement an be done using the deay of a rapidly moving neutral pion whih deays into two gamma rays [14] Light in a medium: Fresnel Dragging In a medium, the veloity of light is redued to n, where n 1 is alled the refrative index. In general n may depend upon wavelength λ but here we will neglet that effet. Fresnel proposed, in the 1820 s, measuring the speed of light in a stream of water moving with speed u relative to the laboratory. Naive Newtonian theory would give a speed n + u (71) but experiments by Fizeau in 1851 using the toothed wheel method did not agree with this. If we use the relativisti addition formula in the ase that u is small we get instead n + u(1 1 n 2 ) +... (72) 23

24 whih does agree with Fizeau s experiments. The fator (1 1 n 2 ) is alled Fresnel s dragging oeffiient and had in fat been proposed earlier by the Frenh physiist Fresnel around 1822 using an argument based on wave theory. The experiment was repeated after Einstein had proposed his theory by the 1904? Nobel prize winning Duthman Zeeman ( ) Composition of Lorentz Transformations We ould just multiply the matries but there is a useful trik. We define Thus x ± = x ± t. (73) x = 0 we have a rightmovinglightray (74) x + = 0 we have a leftmovinglightray (75) Now Lorentz transformations (50) take the form x + = e θ x + x + = e +θ x +, (76) x = e +θ x + x = e θ x +. (77) We immediately dedue that the inverse Lorentz transformation is given by setting u u, θ θ,i.e. the inverse of (50 ) is Inverse Lorentz Transformations x = x + u t 1 u2 2, t = t + u x 2. 1 u2 2 (78) Now onsider three frames of referene S, S and S suh that we get from S to S by boosting with veloity u 1 and from S to S by boosting with relative veloity u 2. To get from S to S we have to boost with relative veloity u3. If θ 1, θ 2, θ 3 are the assoiated rapidities, we have Thus i.e. x ± = e θ1 x ±, (79) x ± = e θ2 tx ±. (80) x ± = e θ3 x ±, (81) rapidities add θ 3 = θ 1 + θ 2. (82) Using a standard addition formula for hyperboli funtions tanhθ 3 = tanhθ 1 + tanhθ tanhθ 1 tanhθ 2. (83) That is we re-obtain the veloity omposition formula: u 3 + u2 u1 =. (84) 1 + u1u2 2 24

25 10.7 Veloity of light as an upper bound Suppose that u 1 < and u 2 <, then u 3 <. Proof Sine the hyperboli tangent funtion is a one to one map of the real line onto the open interval ( 1, +1), we have Thus u 1 < < θ 1 <, (85) u 2 < < θ 2 <. (86) < θ 1 + θ 2 < u 3 <. (87) Thus no matter how we try, we annot exeed the veloity of light * Super-Luminal Radio soures* An interesting apparent ase of super-luminal veloities but whih is perfetly expliable without invoking the existene of anything moving faster than light, has been disovered by radio astronomers. What are alled quasars or quasistellar radio soures exhibit jets of matter symmetrially expelled from a dense entral region probably assoiated with a blak hole. For the sake of a simple first look we assume that we an use the geometry of Minkowski spaetime despite the great distanes and that the entral quasar is loated a distane L away from us. We shall also assume that there is a frame in whih both the entral quasar and ourselves are at rest We assume, in the simplest ase possible, that the matter in the jets are expelled at right angles to our line of sight at time t = 0 and therefore at time t = t o the material in the jets has have travelled a distane vt e. Light or radio waves oming from the jets will arrive here at time The angle α subtended is, for small angles t o = t e + 1 L2 + v 2 t 2 e. (88) α = vt e L. (89) The rate of hange with respet to the observation time is We have Thus dα = v dt e. (90) dt 0 L dt o t e = 1 ( t o + t 2 1 v2 o + (L2 2 + t2 v2 ) 0 )(1. (91) 2 2 dt e = 1 ( 1 + dt o 1 v2 v 2 25 t o t 2 o + L 2 ( 1 v 2 1) ). (92)

26 For large t o we get α = v 1 L 1 v (93) Clearly if v is lose to, then dte dt o an be muh bigger than unity. Thus the size of the effet is muh larger than one s naive Newtonian expetations. If the jet makes an angle with the line of sight we obtain α = v sin θ L 1 1 v os θ (94) The existene of suh apparent superluminal motions was suggested by the present Astronomer Royal in 1966 while a researh student in DAMTP[22]. Just over 4 years later, in 1971, the radio astronomers Irwin Shapiro amd Marshall Cohen and Kenneth Kellerman astronomers found, using very long base line interferometry (VLBI) suh jets, hanging in apparent size over a period of months, in the quasars 3C273 and 3C279. Nowadays the observation of suh apparently super-luminal soures is ommonplae The two-dimensional Lorentz and Poinaré groups Clearly Lorentz transformations, i.e. boosts in one spae and one time dimension, satisfy the axioms for an abelian group (losure under omposition, assoiativity and existene of an inverse) whih is isomorphi to the positive reals under multipliation (one multiplies e θ ) or all the reals under addition (one adds θ). This is ompletely analogous to the group of rotations, SO(2) in two spatial dimensions. The standard notation for the group of boosts is SO(1, 1). If we add in the abelian group of time and spae translation translations t t + t 0, x x + a, (95) we get the analogue of the Eulidean group plane, E(2) whih is alled the Poinaré group and whih may be denoted E(1, 1). 11 The invariant interval Consider two spaetime events (t 1, x 1 ) and (t 2, x 2 ) in spaetime. The invariant interval τ between them is defined by τ 2 = (t 1 t 2 ) 2 (x 1 x 2 ) 2. (96) The name is hosen beause τ 2 is invariant under Lorentz transformations (50). This is beause of the linearity ( ) ( ) ( ) t 1 t 2 oshθ sinhθ t1 t = 2. (97) x 1 x 2 sinhθ oshθ x 1 x 2 26

27 Now there are three ases: Timelike separation τ 2 > 0 t 1 t 2 > x 1 x 2 In this ase a partile with v < an move between the two events. Lightlike separation τ 2 = 0 t 1 t 2 = x 1 x 2. (98). (99) In this ase a light ray or partile with v = an move between the two events. Spaelike separation τ 2 < 0 t 1 t 2 < x 1 x 2 In this ase no partile with v < an move between the two events.. (100) 11.1 Timelike Separation In this ase there exists a frame S in whih both events have the same spatial position, x 1 = x 2 τ 2 = ( t 1 t 2 ) τ = t 1 t 2, where we have fixed the sign ambiguity to make τ positive. Proof We need to solve for θ the equation ( ) ( ) oshθ sinhθ t1 t 2 = ( t sinhθ oshθ x 1 x 1 t 2 0) tanhθ = x 1 x 2. 2 t 1 t 2 (101) Clearly a real solution for θ exists. Stritly speaking, this is all we an say purely mathematially. However we an say more if we aept the physial lok postulate that a physial lok at rest in frame S would measure an elapsed time t 1 t 2. Then we an identify τ with the time measured by a lok at rest in S and passing between the two events. We all this the proper time between the two events. At this stage it may be helpful to reall the definition of the seond aording to the Bureau International des Poids et Mesures (BIPM) who are responsible for defining and maintaining the International System of Units (SI units). Traditionally 1/ of the mean solar day, it has been sine 1960 had the definition The seond is the duration of periods of the radiation orresponding to the transition between two hyperfine levels of the ground state of the aesium 133 atom. The definition of the metre is formerly defined in 1960 of the wavelength of krypton 86 radiation but in 1983 the BIPM delared that The metre is the length of the path travelled by light in vauum during a time interval of 1/ of a seond. Note that not only does the BIPM ompletely aept Einstein s Priniple of the invariane of light but also that the veloity is independent of wavelength. 27

Classical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk

Classical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples

More information

10.1 The Lorentz force law

10.1 The Lorentz force law Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been

More information

arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA

arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA LBNL-52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrene Berkeley National Laboratory

More information

Relativity in the Global Positioning System

Relativity in the Global Positioning System Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 80309-00390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop

More information

Derivation of Einstein s Equation, E = mc 2, from the Classical Force Laws

Derivation of Einstein s Equation, E = mc 2, from the Classical Force Laws Apeiron, Vol. 14, No. 4, Otober 7 435 Derivation of Einstein s Equation, E = m, from the Classial Fore Laws N. Hamdan, A.K. Hariri Department of Physis, University of Aleppo, Syria nhamdan59@hotmail.om,

More information

Chapter 1 Microeconomics of Consumer Theory

Chapter 1 Microeconomics of Consumer Theory Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deision-makers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some

More information

Physics 43 HW 3 Serway Chapter 39 & Knight Chapter 37

Physics 43 HW 3 Serway Chapter 39 & Knight Chapter 37 Physis 43 HW 3 Serway Chapter 39 & Knight Chapter 37 Serway 7 th Edition Chapter 39 Problems: 15, 1, 5, 57, 60, 65 15. Review problem. An alien ivilization oupies a brown dwarf, nearly stationary relative

More information

Relativistic Kinematics -a project in Analytical mechanics Karlstad University

Relativistic Kinematics -a project in Analytical mechanics Karlstad University Relativisti Kinematis -a projet in Analytial mehanis Karlstad University Carl Stigner 1th January 6 Abstrat The following text is a desription of some of the ontent in hapter 7 in the textbook Classial

More information

Physical and mathematical postulates behind relativity

Physical and mathematical postulates behind relativity Physial and mathematial postulates behind relativity Tuomo Suntola Physis Foundations Soiety, Finland, www.physisfoundations.org In this presentation we look for answers to questions: What was the problem

More information

Isaac Newton. Translated into English by

Isaac Newton. Translated into English by THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY (BOOK 1, SECTION 1) By Isaa Newton Translated into English by Andrew Motte Edited by David R. Wilkins 2002 NOTE ON THE TEXT Setion I in Book I of Isaa

More information

Another Look at Gaussian CGS Units

Another Look at Gaussian CGS Units Another Look at Gaussian CGS Units or, Why CGS Units Make You Cool Prashanth S. Venkataram February 24, 202 Abstrat In this paper, I ompare the merits of Gaussian CGS and SI units in a variety of different

More information

THE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT ALBANY, NY

THE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT ALBANY, NY P THE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT ALBANY, NY 4 Referene Tables for Physial Setting/PHYSICS 006 Edition List of Physial Constants Name Symbol Value Universal gravitational

More information

) ( )( ) ( ) ( )( ) ( ) ( ) (1)

) ( )( ) ( ) ( )( ) ( ) ( ) (1) OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples

More information

Chapter 5 Single Phase Systems

Chapter 5 Single Phase Systems Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude

More information

On the Notion of the Measure of Inertia in the Special Relativity Theory

On the Notion of the Measure of Inertia in the Special Relativity Theory www.senet.org/apr Applied Physis Researh Vol. 4, No. ; 1 On the Notion of the Measure of Inertia in the Speial Relativity Theory Sergey A. Vasiliev 1 1 Sientifi Researh Institute of Exploration Geophysis

More information

Computer Networks Framing

Computer Networks Framing Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial

More information

1.3 Complex Numbers; Quadratic Equations in the Complex Number System*

1.3 Complex Numbers; Quadratic Equations in the Complex Number System* 04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x - 2x - 22 x - 2 2 ; x

More information

Comay s Paradox: Do Magnetic Charges Conserve Energy?

Comay s Paradox: Do Magnetic Charges Conserve Energy? Comay s Paradox: Do Magneti Charges Conserve Energy? 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 1, 2015; updated July 16, 2015) The interation energy

More information

Chapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112

More information

How To Understand General Relativity

How To Understand General Relativity Chapter S3 Spacetime and Gravity What are the major ideas of special relativity? Spacetime Special relativity showed that space and time are not absolute Instead they are inextricably linked in a four-dimensional

More information

SHAFTS: TORSION LOADING AND DEFORMATION

SHAFTS: TORSION LOADING AND DEFORMATION ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter 3.1-3.5 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220

More information

Sebastián Bravo López

Sebastián Bravo López Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that

More information

HEAT CONDUCTION. q A q T

HEAT CONDUCTION. q A q T HEAT CONDUCTION When a temperature gradient eist in a material, heat flows from the high temperature region to the low temperature region. The heat transfer mehanism is referred to as ondution and the

More information

cos t sin t sin t cos t

cos t sin t sin t cos t Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the

More information

A novel active mass damper for vibration control of bridges

A novel active mass damper for vibration control of bridges IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural

More information

5.2 The Master Theorem

5.2 The Master Theorem 170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended

More information

Dispersion in Optical Fibres

Dispersion in Optical Fibres Introdution Optial Communiations Systems Dispersion in Optial Fibre (I) Dispersion limits available bandwidth As bit rates are inreasing, dispersion is beoming a ritial aspet of most systems Dispersion

More information

Channel Assignment Strategies for Cellular Phone Systems

Channel Assignment Strategies for Cellular Phone Systems Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious

More information

THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES

THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES Proeedings of FEDSM 98 998 ASME Fluids Engineering Division Summer Meeting June 2-25, 998 Washington DC FEDSM98-529 THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES John D. Wright Proess

More information

In order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method

In order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method BEAM DESIGN In order to be able to design beams, we need both moments and shears. 1. Moment a) From diret design method or equivalent frame method b) From loads applied diretly to beams inluding beam weight

More information

Conversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory

Conversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory PHYSICAL REVIEW B 76, 35114 007 Conversion of short optial pulses to terahertz radiation in a nonlinear medium: Experiment and theory N. N. Zinov ev* Department of Physis, University of Durham, Durham

More information

User s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities

User s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear Least-Squares Fits with a Common Interept: Determination

More information

Effects of Inter-Coaching Spacing on Aerodynamic Noise Generation Inside High-speed Trains

Effects of Inter-Coaching Spacing on Aerodynamic Noise Generation Inside High-speed Trains Effets of Inter-Coahing Spaing on Aerodynami Noise Generation Inside High-speed Trains 1 J. Ryu, 1 J. Park*, 2 C. Choi, 1 S. Song Hanyang University, Seoul, South Korea 1 ; Korea Railroad Researh Institute,

More information

Improved SOM-Based High-Dimensional Data Visualization Algorithm

Improved SOM-Based High-Dimensional Data Visualization Algorithm Computer and Information Siene; Vol. 5, No. 4; 2012 ISSN 1913-8989 E-ISSN 1913-8997 Published by Canadian Center of Siene and Eduation Improved SOM-Based High-Dimensional Data Visualization Algorithm Wang

More information

Gravity and the quantum vacuum inertia hypothesis

Gravity and the quantum vacuum inertia hypothesis Ann. Phys. (Leipzig) 14, No. 8, 479 498 (2005) / DOI 10.1002/andp.200510147 Gravity and the quantum vauum inertia hypothesis Alfonso Rueda 1, and Bernard Haish 2, 1 Department of Eletrial Engineering,

More information

Solutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 7

Solutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 7 Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid April 21, 2002 Chapter 7 Problem 7.2 Obtain the Lorentz transformation in which the velocity is at an infinitesimal angle

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007. Name:

Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007. Name: Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007 Name: Directions: Listed below are twenty (20) multiple-choice questions based on the material covered by the lectures this past week. Choose

More information

Capacity at Unsignalized Two-Stage Priority Intersections

Capacity at Unsignalized Two-Stage Priority Intersections Capaity at Unsignalized Two-Stage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minor-street traffi movements aross major divided four-lane roadways

More information

How To Fator

How To Fator CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems

More information

Let s first see how precession works in quantitative detail. The system is illustrated below: ...

Let s first see how precession works in quantitative detail. The system is illustrated below: ... lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

More information

Einstein s Theory of Special Relativity Made Relatively Simple!

Einstein s Theory of Special Relativity Made Relatively Simple! Einstein s Theory of Special Relativity Made Relatively Simple! by Christopher P. Benton, PhD Young Einstein Albert Einstein was born in 1879 and died in 1955. He didn't start talking until he was three,

More information

The Reduced van der Waals Equation of State

The Reduced van der Waals Equation of State The Redued van der Waals Equation of State The van der Waals equation of state is na + ( V nb) n (1) V where n is the mole number, a and b are onstants harateristi of a artiular gas, and R the gas onstant

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

More information

Special Theory of Relativity

Special Theory of Relativity June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition

More information

Weighting Methods in Survey Sampling

Weighting Methods in Survey Sampling Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiao-hih Chang Ferry Butar Butar Abstrat It is said that a well-designed survey an best prevent nonresponse. However, no matter

More information

Advanced Topics in Physics: Special Relativity Course Syllabus

Advanced Topics in Physics: Special Relativity Course Syllabus Advanced Topics in Physics: Special Relativity Course Syllabus Day Period What How 1. Introduction 2. Course Information 3. Math Pre-Assessment Day 1. Morning 1. Physics Pre-Assessment 2. Coordinate Systems

More information

Theory of electrons and positrons

Theory of electrons and positrons P AUL A. M. DIRAC Theory of electrons and positrons Nobel Lecture, December 12, 1933 Matter has been found by experimental physicists to be made up of small particles of various kinds, the particles of

More information

DSP-I DSP-I DSP-I DSP-I

DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I Digital Signal Proessing I (8-79) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:

More information

Static Fairness Criteria in Telecommunications

Static Fairness Criteria in Telecommunications Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat-208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, e-mail: vesatimonen@hutfi

More information

Programming Basics - FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html

Programming Basics - FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html CWCS Workshop May 2005 Programming Basis - FORTRAN 77 http://www.physis.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html Program Organization A FORTRAN program is just a sequene of lines of plain text.

More information

Deformation of the Bodies by the Result of Length Contraction: A new Approach to the Lorentz Contraction

Deformation of the Bodies by the Result of Length Contraction: A new Approach to the Lorentz Contraction 1 Deformation of the Bodies by the Result of Length Contraction: A new Approach to the Lorentz Contraction Bayram Akarsu, Ph.D Erciyes University Kayseri/ Turkiye 2 Abstract It has been more than a century

More information

ON THE ELECTRODYNAMICS OF MOVING BODIES

ON THE ELECTRODYNAMICS OF MOVING BODIES ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries

More information

Impact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities

Impact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary

More information

Revista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, 1999 469. Surface Charges and Electric Field in a Two-Wire

Revista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, 1999 469. Surface Charges and Electric Field in a Two-Wire Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, 999 469 Surfae Charges and Eletri Field in a Two-Wire Resistive Transmission Line A. K. T.Assis and A. J. Mania Instituto de Fsia Gleb Wataghin'

More information

4/3 Problem for the Gravitational Field

4/3 Problem for the Gravitational Field 4/3 Proble for the Gravitational Field Serey G. Fedosin PO bo 6488 Sviazeva str. -79 Per Russia E-ail: intelli@list.ru Abstrat The ravitational field potentials outside and inside a unifor assive ball

More information

ON THE ELECTRODYNAMICS OF MOVING BODIES

ON THE ELECTRODYNAMICS OF MOVING BODIES ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. DARK MATTER AND THE UNIVERSE

Data Provided: A formula sheet and table of physical constants is attached to this paper. DARK MATTER AND THE UNIVERSE Data Provided: A formula sheet and table of physical constants is attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester (2014-2015) DARK MATTER AND THE UNIVERSE 2 HOURS Answer question

More information

Henley Business School at Univ of Reading. Chartered Institute of Personnel and Development (CIPD)

Henley Business School at Univ of Reading. Chartered Institute of Personnel and Development (CIPD) MS in International Human Resoure Management (full-time) For students entering in 2015/6 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date

More information

Measurement of Powder Flow Properties that relate to Gravity Flow Behaviour through Industrial Processing Lines

Measurement of Powder Flow Properties that relate to Gravity Flow Behaviour through Industrial Processing Lines Measurement of Powder Flow Properties that relate to Gravity Flow ehaviour through Industrial Proessing Lines A typial industrial powder proessing line will inlude several storage vessels (e.g. bins, bunkers,

More information

Reflection and Refraction

Reflection and Refraction Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

More information

Lecture 24: Spinodal Decomposition: Part 3: kinetics of the

Lecture 24: Spinodal Decomposition: Part 3: kinetics of the Leture 4: Spinodal Deoposition: Part 3: kinetis of the oposition flutuation Today s topis Diffusion kinetis of spinodal deoposition in ters of the onentration (oposition) flutuation as a funtion of tie:

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

A Holistic Method for Selecting Web Services in Design of Composite Applications

A Holistic Method for Selecting Web Services in Design of Composite Applications A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the

More information

BENEFICIARY CHANGE REQUEST

BENEFICIARY CHANGE REQUEST Poliy/Certifiate Number(s) BENEFICIARY CHANGE REQUEST *L2402* *L2402* Setion 1: Insured First Name Middle Name Last Name Permanent Address: City, State, Zip Code Please hek if you would like the address

More information

Waveguides. 8.14 Problems. 8.14. Problems 361

Waveguides. 8.14 Problems. 8.14. Problems 361 8.4. Problems 36 improving liquid rystal displays, and other produts, suh as various optoeletroni omponents, osmetis, and hot and old mirrors for arhitetural and automotive windows. 8.4 Problems 9 Waveguides

More information

Newton s Laws. Newton s Imaginary Cannon. Michael Fowler Physics 142E Lec 6 Jan 22, 2009

Newton s Laws. Newton s Imaginary Cannon. Michael Fowler Physics 142E Lec 6 Jan 22, 2009 Newton s Laws Michael Fowler Physics 142E Lec 6 Jan 22, 2009 Newton s Imaginary Cannon Newton was familiar with Galileo s analysis of projectile motion, and decided to take it one step further. He imagined

More information

Intuitive Guide to Principles of Communications By Charan Langton www.complextoreal.com

Intuitive Guide to Principles of Communications By Charan Langton www.complextoreal.com Intuitive Guide to Priniples of Communiations By Charan Langton www.omplextoreal.om Understanding Frequeny Modulation (FM), Frequeny Shift Keying (FSK), Sunde s FSK and MSK and some more The proess of

More information

CHAPTER J DESIGN OF CONNECTIONS

CHAPTER J DESIGN OF CONNECTIONS J-1 CHAPTER J DESIGN OF CONNECTIONS INTRODUCTION Chapter J of the addresses the design and heking of onnetions. The hapter s primary fous is the design of welded and bolted onnetions. Design requirements

More information

AUDITING COST OVERRUN CLAIMS *

AUDITING COST OVERRUN CLAIMS * AUDITING COST OVERRUN CLAIMS * David Pérez-Castrillo # University of Copenhagen & Universitat Autònoma de Barelona Niolas Riedinger ENSAE, Paris Abstrat: We onsider a ost-reimbursement or a ost-sharing

More information

Electrician'sMathand BasicElectricalFormulas

Electrician'sMathand BasicElectricalFormulas Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In. 1.888.NEC.CODE www.mikeholt.om Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial

More information

PHYSICS FOUNDATIONS SOCIETY THE DYNAMIC UNIVERSE TOWARD A UNIFIED PICTURE OF PHYSICAL REALITY TUOMO SUNTOLA

PHYSICS FOUNDATIONS SOCIETY THE DYNAMIC UNIVERSE TOWARD A UNIFIED PICTURE OF PHYSICAL REALITY TUOMO SUNTOLA PHYSICS FOUNDATIONS SOCIETY THE DYNAMIC UNIVERSE TOWARD A UNIFIED PICTURE OF PHYSICAL REALITY TUOMO SUNTOLA Published by PHYSICS FOUNDATIONS SOCIETY Espoo, Finland www.physicsfoundations.org Printed by

More information

physics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves

physics 1/12/2016 Chapter 20 Lecture Chapter 20 Traveling Waves Chapter 20 Lecture physics FOR SCIENTISTS AND ENGINEERS a strategic approach THIRD EDITION randall d. knight Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide

More information

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

More information

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic

More information

h 2 m e (e 2 /4πɛ 0 ).

h 2 m e (e 2 /4πɛ 0 ). 111 111 Chapter 6. Dimensions 111 Now return to the original problem: determining the Bohr radius. The approximate minimization predicts the correct value. Even if the method were not so charmed, there

More information

Granular Problem Solving and Software Engineering

Granular Problem Solving and Software Engineering Granular Problem Solving and Software Engineering Haibin Zhu, Senior Member, IEEE Department of Computer Siene and Mathematis, Nipissing University, 100 College Drive, North Bay, Ontario, P1B 8L7, Canada

More information

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6 Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.

More information

Henley Business School at Univ of Reading. Pre-Experience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD)

Henley Business School at Univ of Reading. Pre-Experience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD) MS in International Human Resoure Management For students entering in 2012/3 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date of speifiation:

More information

3 Game Theory: Basic Concepts

3 Game Theory: Basic Concepts 3 Game Theory: Basi Conepts Eah disipline of the soial sienes rules omfortably ithin its on hosen domain: : : so long as it stays largely oblivious of the others. Edard O. Wilson (1998):191 3.1 and and

More information

An integrated optimization model of a Closed- Loop Supply Chain under uncertainty

An integrated optimization model of a Closed- Loop Supply Chain under uncertainty ISSN 1816-6075 (Print), 1818-0523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 9-17 An integrated optimization model of a Closed- Loop Supply Chain under unertainty Xiaoxia

More information

This paper is also taken for the relevant Examination for the Associateship. For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00

This paper is also taken for the relevant Examination for the Associateship. For Second Year Physics Students Wednesday, 4th June 2008: 14:00 to 16:00 Imperial College London BSc/MSci EXAMINATION June 2008 This paper is also taken for the relevant Examination for the Associateship SUN, STARS, PLANETS For Second Year Physics Students Wednesday, 4th June

More information

Physics 201 Homework 8

Physics 201 Homework 8 Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

More information

Intelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds

Intelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds Intelligent Measurement Proesses in 3D Optial Metrology: Produing More Aurate Point Clouds Charles Mony, Ph.D. 1 President Creaform in. mony@reaform3d.om Daniel Brown, Eng. 1 Produt Manager Creaform in.

More information

Fixed-income Securities Lecture 2: Basic Terminology and Concepts. Present value (fixed interest rate) Present value (fixed interest rate): the arb

Fixed-income Securities Lecture 2: Basic Terminology and Concepts. Present value (fixed interest rate) Present value (fixed interest rate): the arb Fixed-inome Seurities Leture 2: Basi Terminology and Conepts Philip H. Dybvig Washington University in Saint Louis Various interest rates Present value (PV) and arbitrage Forward and spot interest rates

More information

Electromagnetism Laws and Equations

Electromagnetism Laws and Equations Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2

More information

SPATIAL COORDINATE SYSTEMS AND RELATIVISTIC TRANSFORMATION EQUATIONS

SPATIAL COORDINATE SYSTEMS AND RELATIVISTIC TRANSFORMATION EQUATIONS Fundamental Journal of Modern Physics Vol. 7, Issue, 014, Pages 53-6 Published online at http://www.frdint.com/ SPATIAL COORDINATE SYSTEMS AND RELATIVISTIC TRANSFORMATION EQUATIONS J. H. FIELD Departement

More information

INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS

INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS Virginia Department of Taxation INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS www.tax.virginia.gov 2614086 Rev. 07/14 * Table of Contents Introdution... 1 Important... 1 Where to Get Assistane... 1 Online

More information

PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS

PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS 1. Photons 2. Photoelectric Effect 3. Experimental Set-up to study Photoelectric Effect 4. Effect of Intensity, Frequency, Potential on P.E.

More information

Physics 53. Gravity. Nature and Nature's law lay hid in night: God said, "Let Newton be!" and all was light. Alexander Pope

Physics 53. Gravity. Nature and Nature's law lay hid in night: God said, Let Newton be! and all was light. Alexander Pope Physics 53 Gravity Nature and Nature's law lay hid in night: God said, "Let Newton be!" and all was light. Alexander Pope Kepler s laws Explanations of the motion of the celestial bodies sun, moon, planets

More information

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc. Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

More information

protection p1ann1ng report

protection p1ann1ng report f1re~~ protetion p1ann1ng report BUILDING CONSTRUCTION INFORMATION FROM THE CONCRETE AND MASONRY INDUSTRIES Signifiane of Fire Ratings for Building Constrution NO. 3 OF A SERIES The use of fire-resistive

More information

INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS

INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS Virginia Department of Taxation INCOME TAX WITHHOLDING GUIDE FOR EMPLOYERS www.tax.virginia.gov 2614086 Rev. 01/16 Table of Contents Introdution... 1 Important... 1 Where to Get Assistane... 1 Online File

More information

A Comparison of Default and Reduced Bandwidth MR Imaging of the Spine at 1.5 T

A Comparison of Default and Reduced Bandwidth MR Imaging of the Spine at 1.5 T 9 A Comparison of efault and Redued Bandwidth MR Imaging of the Spine at 1.5 T L. Ketonen 1 S. Totterman 1 J. H. Simon 1 T. H. Foster 2. K. Kido 1 J. Szumowski 1 S. E. Joy1 The value of a redued bandwidth

More information

Analysis of micro-doppler signatures

Analysis of micro-doppler signatures Analysis of miro-doppler signatures V.C. Chen, F. Li, S.-S. Ho and H. Wehsler Abstrat: Mehanial vibration or rotation of a target or strutures on the target may indue additional frequeny modulations on

More information

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc. Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

More information

Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

More information

Performance Analysis of IEEE 802.11 in Multi-hop Wireless Networks

Performance Analysis of IEEE 802.11 in Multi-hop Wireless Networks Performane Analysis of IEEE 80.11 in Multi-hop Wireless Networks Lan Tien Nguyen 1, Razvan Beuran,1, Yoihi Shinoda 1, 1 Japan Advaned Institute of Siene and Tehnology, 1-1 Asahidai, Nomi, Ishikawa, 93-19

More information

Convergence of c k f(kx) and the Lip α class

Convergence of c k f(kx) and the Lip α class Convergene of and the Lip α lass Christoph Aistleitner Abstrat By Carleson s theorem a trigonometri series k osπkx or k sin πkx is ae onvergent if < (1) Gaposhkin generalized this result to series of the

More information