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- Included in Midterm Exams

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- 1. Objectives • To become familiar with the different branches of Modern Physics • To state the postulates of the special theory of relativity • To differentiate between inertial and non-inertial reference frames
- 2. Modern physics ⇒ started around the beginning of the 20th centuryRelativity ⇒ physics of the very, very fast (speeds approaching c)
- 3. Branches of Modern Physics Atomic and Nuclear Physics – study of the composition, structure and behavior of the nucleus of the atom
- 4. Branches of Modern Physics Quantum Physics – study of the discrete nature of phenomena at the atomic and subatomic levels ⇒ its focus is on the indivisible units of energy called quanta as described by the Quantum Theory ⇒ physics of the very, very small (protons, electrons, …)
- 5. Branches of Modern Physics Relativistic Physics – study of phenomena that take place in frame of reference that is in motion with respect to an observer ⇒ physics of the very, very fast (speeds approaching c)
- 6. Branches of Modern Physics Solid State Physics – study of all the properties of solid materials, including electrical conduction in crystals of semi- conductors and metals, superconductivity and photo- conductivity
- 7. Branches of Modern PhysicsCondensed Matter Physics– study of the properties of condensed materials (solids, liquids and those intermediate between them, and dense gas) with the ultimate goal of developing new materials withbetter properties
- 8. Branches of Modern Physics Plasma Physics – study of the fourth state of matter
- 9. Branches of Modern Physics Low-Temperature Physics – study of the production and maintenance of temperatures down to almost absolute zero, and the various phenomena that occur only at such temperature
- 10. Modern Physics showed that Newton’s laws were incomplete.Newton’s laws only apply to objects of macroscopic size (bigger than protons and electrons) and relatively small speeds (much less than the speed of light)
- 11. Albert Einstein (1879 – 1955) - published three papers of extraordinary importance 1. an analysis of Brownian motion 2. photoelectric effect (Nobel Prize) 3. special theory of relativity The special theory of relativity has made wide-ranging changes in the understanding of nature.
- 12. SpecialRelativityIn 1905, Albert Einsteindescribed in his theory ofSpecial Relativity howmeasurements of timeand space are affected bythe motion between theobserver and what isbeing observed.
- 13. SpecialRelativityThe Theory of SpecialRelativity revolutionizedthe world of physics byconnecting space and time,matter and energy, andelectricity and magnetism
- 14. The Special Theory of Relativitydefies common sense!But, the results of the Special Theory ofRelativity have been extensively testednumerous times and are in fact true!
- 15. Postulates of relativity 1. The laws of Physics are the same in every inertial frame of reference. (No experiment can be done in an inertial reference frame to detect its state of motion.) 2. The speed of light (3 x 108 m/s) is the same in all inertial frames of reference and is independent of the motion of the source (The speed of light in vacuum has the same value when measured by any observer, regardless of the observer’s state of motion.)
- 16. Inertial Reference Frame An inertial reference frame is one in which no accelerations are observed in the absence of external forces (acceleration is the result of a force) that is not accelerating Newton’s laws hold in all inertial reference frames.
- 17. Inertial Reference Frameexamples1. This room Experiment: Drop a ball. It accelerates downward at 9.8 m/s2 due to the force of gravity.2. Inside of a car moving at constant speed along a straight road. Repeat the experiment: Results are the same as in #1.3. Inside of an elevator that is moving either upward or downward at constant speed. Repeat the experiment: Results are the same as in #1 and #2.
- 18. Noninertial Reference Frame A noninertial reference frame is one that is accelerating with respect to an inertial reference frame. In a noninertial reference frame, bodies have accelerations in the absence of applied forces.
- 19. Noninertial Reference Frameexamples 1. The interior of a car that is either speeding up, slowing down, or going around a curve. Experiment: Drop a ball. If the car is slowing down, the ball accelerates downward and towards the front of the car. The acceleration toward the front of the car is not due to a force on the ball. 2. The inside of an elevator that is accelerating either upward or downward. Repeat the experiment. If the elevator is accelerating upward, the ball accelerates downward faster than 9.8 m/s2. The additional downward acceleration is not due to a force on the ball.
- 20. Reference FramesPlatform at rest, tree moving—ball is Platform moving. Observer onseen by observers on platform as being the ground (inertial frame) seesdeflected, but no force acts on it. ball move in a straight line, butViolation of Newton’s second law. sees the catcher move away. Platform is accelerating Ground is the noninertial frame inertial frame
- 21. Special Relativity:Consequences• time dilates Time to moving objects appear to slow down• length shrinks Moving objects appear shorter• mass increases Moving objects appear to be massive
- 22. Time dilationIf there is relative motion between two observers(if they are moving at different velocities), they willnot agree in their measurements of space and time.However, the two observers will agree on theirmeasurement of the speed of light. Since speed equals distance divided by time, both observers will measure the same ratio of space (distance) and time Space = Space = c time time
- 23. Time dilation⇒ An observer in the rocket moving with the clock sees the light traveling straight up and down.⇒ The observer and the clock are in the same frame of reference.
- 24. Time dilation⇒ An observer on the ground (who is not in the same reference frame as the clock) sees the light traveling in a diagonal path.⇒ In the frame of reference of the observer on Earth, the light travels a longer distance.
- 25. Time dilation⇒ Since the speed of light is the same in all reference frames the light must travel for a longer time in the Earth than in a reference frame of the rocket.⇒ The stretching out of time is called time dilation.
- 26. Time dilation⇒ Moving clocks run slow.⇒ Time dilation has nothing to do with the mechanics of clocks but with the nature of time itself.⇒ Time passes more slowly in a reference frame that is moving than in a reference frame that is at rest.
- 27. Time dilation⇒ is given as Δt’ Δt = √ 1 – v2/c2 Δt time interval in the moving frame Δt’ time interval in the frame at rest v speed of relative motion c speed of light
- 28. Length shrinks⇒ The lengths of objects appear to be contracted (shortened) when they move at relativistic speeds.⇒ This length contraction is really a contraction of space.⇒ As the speed increases, length in the direction of motion decreases. Lengths in the perpendicular direction do not change.
- 29. Length shrinks⇒ Length contraction is given as l = l’√1 – v2/c2 where l length measurement of the moving frame l’ length measurement of the frame at rest v speed of the moving frame c speed of light
- 30. Mass increases⇒ The mass of an object moving at a speed v relative to the observer is larger than its mass when at rest relative to the observer⇒ the relativistic mass is given as m = m’ √1 – v2/c2 where m relativistic mass m’ mass of the object at rest v speed of the moving frame c speed of light
- 31. General Theory of Relativity Relativity refers to the observation of the motion of a body by two different observers in relative motion to each other General Theory of Relativity is a geometrical theory of gravitation published by Albert Einstein in 1915
- 32. General Theory of Relativity unifies special relativity and Sir Isaac Newtons law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time, with this curvature being produced by the mass-energy and momentum content of the spacetime.
- 33. General Theory of Relativity has three parts: – equivalence of inertial and gravitational mass (Galileo’s principle) – laws of physics same in freely falling lab as in lab at rest far from any mass – physical laws in accelerating lab same as in stationary lab in gravitational field
- 34. The Equivalence Principle New ton Einstein This compartment is This compartment is at rest in the Earth’s moving in a gravity-free gravitational field. environment The apple hits the floor of The apple hits the floor the compartment because of the compartment the Earth’s gravity because the compart- accelerates the apple ment accelerates. downward
- 35. General Theory of Relativity “Equivalence Principle”: Observers cannot distinguish between inertial forces due to acceleration and uniform gravitational forces due to a massive body. • Consequence: Gravity, inertia, and acceleration are related to the curvature of space-time
- 36. General Theory of Relativity “Mass tells space how to curve. Curvature tells mass how to accelerate”.In the context of the Theory of General Relativity,gravitation was redefined as a property of thespace-time continuum. The force was replaced bythe strength of the curvature of the space whichis depending on the mass and the size of anobject, i.e., its ability to bend space.
- 37. General Theory of Relativity Representation of the warping of space and time due to large mass
- 38. General Theory of Relativity currently the most successful gravitational theory, being almost universally accepted and well confirmed by observations such as: • gravitational redshift • deflection of light by mass • bending of light by gravitation • perihelion precession of Mercury Perihelion is the point in the path of a celestial body (as a planet) that is nearest to the sun.
- 39. General Theory of Relativity Gravitational redshift- the effect when light or other forms of electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational field (and which could be said to have climbed "uphill" out of a gravity well) will be found to be of longer wavelength when received by an observer in a region of weaker gravitational field. The gravitational redshift of a light wave as it moves upwardsIf applied to optical wave-lengths this manifests against a gravitational field (caused by the yellow star below).itself as a change in the color of the light as the http://en.wikipedia.org/wiki/Gra vitational_redshiftwavelength is shifted toward the red (making itless energetic, longer in wavelength, and lower infrequency) part of the spectrum
- 40. General Theory of Relativity Gravitational redshift Light leaving a region where the gravitational force is large will be shifted towards the red (its wavelength increases; similarly, light falling into a region where the gravitational pull is larger will be shifted The gravitational redshift towards the blue. http://physics.ucr.edu/~wudka /Physics7/Notes_www/node89.h tml#fig:redshift
- 41. General Theory of Relativity Deflection of light by mass One immediate consequence of the curvature of the space- time is that light must also be subject to gravity http://library.thinkquest.org/C0116043/generaltheory.htmFigure above shows a beam of light from a star passing by the Sunand continuing on to the Earth. Because the light ray is bent, thestar appears to be shifted from its actual location.This prediction was first tested in 1919 during a total solar eclipse.
- 42. General Theory of Relativity Deflection of light by mass A light ray arriving from the left would be bent inwards such that its apparent direction of origin, when viewed from the right, would differ by an angle (α, the deflection angle, see diagram) whose size is inversely proportional to the distance (d) of the closest approach of the ray path to the center of mass.
- 43. General Theory of Relativity Bending of light by gravitation Light travels always the shortest distance in a curved space-time.
- 44. General Theory of Relativity Bending of light by gravitationThe figure above shows three different possible (mathematical)paths for a pulse of light travelling around the Sun: the path withno gravity, the path as predicted by Newtonian gravity, andthe path as predicted by Einsteins General Theory of Relativity
- 45. General Theory of Relativity Bending of light by gravitationdeflection angle df turning point R0 istells how far away the closest distancefrom a straight line that the light pulsethe path of the light gets to the Sun. f=0pulse in question was corresponds to R = R0,deflected by the Sun. No gravity, the path is a straight line. The path of a straight line in polar coordinates centered at the center of the Sun would be: 1/r = (1/R0) cos(f) To find df, look at the figure to the left and imagine the straight line path extending infinitely far to the right and left of your screen. When r = infinity, by symmetry of the coordinate system 0 = (1/R0) cos(df/2). Therefore Df = p is the total difference in angle swept out by the light pulse as it comes in from infinitely far away and travels back out infinitely far away. The deflection angle here is df = df - p = 0, as it should be for a straight line.
- 46. General Theory of Relativity Bending of light by gravitationNewtonian gravitydoesnt work well for describing theproperties of light, which can be modeledlike the propagation of a masslessparticle. But it is possible using theequation for a Newtonian hyperbolic orbit: 1/r = (G M(m/L)2)(1 + e cos(f)), e = (1 + (2E/m)(L/GMm)2))1/2where the eccentricity e is a function of the incoming particles energy E, mass m andangular momentum L. The turning point R0 = (L/m)2/(G M (1 + e)).To fake the propagation of light in Newtonian gravity,the energy E = m v2/2 = m c2/2 so that (2 E/m) = c2. The angular momentum per unitincoming mass (L/m) becomes L/m = R0 c.The total angular sweep df = p + df is given by 0 = (1/R0) cos(df/2) + (G M/c2)/R02,- cos(p/2 + df/2) = sin(df/2) ~ df/2 = (G M/c2)/R0Finally, dfN = 2 (G M/c2)/R0 is the deflection angle for light found by naively usingthe Newtonian model for a particle with velocity c.
- 47. General Theory of Relativity Bending of light by gravitationEinsteins General RelativityIn General Relativity, the path of alight pulse is described as a null geodesicsatisfying the geodesic equation for theSchwarzschild metric, the distancefunction that solves the Einstein equationsaround a massive object in outer space suchas the Sun. An approximate equation for thetrajectory is 1/r = (1/R0) cos(f) + ((G M/c2)/R02) (2 - cos2(f)).The term cos2(f) can be neglected if the deflection angle df is very small anddf/2 is close to p/2.Therefore, to lowest order in df the 0 = (1/R0) cos(df/2) + 2 (G M/c2)/R02,- cos(p/2 + df/2) = sin(df/2) ~ df/2 = 2 (G M/c2)/R0. Therefore dfE = 4 (G M/c2)/R0 =2 dfN is the deflection angle for light found by using null geodesics in theSchwarzschild metric according to General Relativity.
- 48. General Theory of Relativity Perihelion Precession of Mercury The orbit of Mercury did not behave as required by Newtons equations.(a long-standing problem in the study of the Solar System)As Mercury orbits the Sun, itfollows an ellipse...but onlyapproximately: it is found thatthe point of closest approach ofMercury to the sun does notalways occur at the same placebut that it slowly moves aroundthe sun. This rotation of theorbit is called a precession. Artist’s version of the precession of mercury’s orbit around the sun http://physics.ucr.edu/~wudka/Physics7/Notes_ www/node98.html
- 49. General Theory of Relativity Perihelion Precession of Mercury All the planetary orbits precess and Newtons theory predicts these effects, as being produced by the pull of the planets on one another. . The precession of the orbits of all planets except for Mercurys can, in fact, be understood using Newton;s equations. But Mercury seemed to be an exception. Artist’s version of the precession of mercury’s orbit around the sun http://physics.ucr.edu/~wudka/Physics7/Notes_ www/node98.html
- 50. General Theory of Relativity Perihelion Precession of Mercury As seen from Earth the precession of Mercurys orbit is measured to be 5600 seconds of arc per century (one second of arc=1/3600 degrees). Newtons equations, predicts a precession of 5557 seconds of arc per century. There is a discrepancy of 43 seconds of Artist’s version of the precession of mercury’s arc per century. orbit around the sun http://physics.ucr.edu/~wudka/Physics7/Notes_ www/node98.html
- 51. General Theory of Relativity Perihelion Precession of Mercury http://library.thinkquest.org/C0116043/generaltheory.htm Most of the effect is due to the pull from the other planets but there is a measurable effect due to the corrections to Newtons theory predicted by the General Theory of Relativity.
- 52. References Serway, Raymond A., Vuille, Chris and Faugnn, Jerry S.(2009). College Physics (Volume 2) 8th ed. Brooks/Cole Cengage Learning http://www.youtube.com/watch?v=ev9zrt__lec

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