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# The Ideas of Modern Physics PHYSICS 107

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This 87 page Class Notes was uploaded by Nichole Keebler on Thursday September 17, 2015. The Class Notes belongs to PHYSICS 107 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/205235/physics-107-university-of-wisconsin-madison in Physics 2 at University of Wisconsin - Madison.

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From last time Inertia tendency of body to continue in straightline motion at constant speed unless disturbed Superposition object responds independently to separate disturbances Galileo used superposition to investigate falling objects Fri Jan 26 Physics lEI7 Spring 2667 i Galileo measured this But falling motion too fast for accurate measurement Galileo was able to measure a different aspect that let him determine the time In this way he made extremely accurate measurements Fri Jan 26 Physics lEI7 Spring 2667 2 Used principle of superposition and principle of inertia Ball leaves ramp with gt constant horizontal speed After leaving ramp it continues horizontal motion at some constant speed 5 no horizontal disturbances But gravitational disturbance causes change in vertical motion the ball falls downward For every second of fall it moves to the right 5 meterssecondx1 second 5 meters Determine falling time by measuring horizontal distance Hi Jan lb l h U j ySlES iLll Spring 26 An equation From this Galileo determined that the falling time varied proportional to the square root of the falling distance Falling time 0 1lFalling distance Falling time 139 Falling distance d d 0C ZL2 dat2 Fri Jan 26 Physics 67 Spring 2667 4 Back to falling objects I drop two balls one from twice the height of the other The time it takes the higher ball to fall is how much longer than the lower ball Same Three time Four times E None of the above Fri Jan 26 Physics lEI7 Spring 2667 5 A B Twice C D Falling speed After an object is dropped it s speed is A constant I B increasing proportional to time I C increasing proportional to time squared Fri Jan 26 Physics lEI7 Spring 2667 6 Quantifying motion Distance and Time A movingobject gt Need to understand these concepts ges its position e Position speed velocity amp acceleration averse instantaneous Myposition at aII times tompieteiy describes my motion can use this inf rma n The average speed is the same to find the speed I I move 5 meters in 5 seconds Then i meters in each second 5 divided by 5 1 meterper second eg could walk 7 1 meter in the first second and Magma F Igt 1 meter in the next second etc ould also write den BUT maybe I walked 0 meters in the first second and 50 knowing average speed t en 5 meters in 4 second lets us find distance traveled Instantaneous speed Think about this one l ast a a total of 800 it Voo sit at a stoplight for in seconds then you acce erate asf s you can for in seconds going Instantaneous speed is the average velocity over an in nitesimal very short time Voor average speed over the entire 20 seconds is inteml A in feetsec i7 mihr This iswhat yourspeedometer reads B 20 EEUSEC M Wm c 40 feetsec 27 mihr Instantaneous speed gives you a better understanding of the motion D 5 IEEUSEC 34 Wm E Need to know acceleration we in an m n has Acceleration ii rains quotshim simian ii Understanding acceleration HER Reg Major points position coordinates ofabody velocity rate of change of position a avera enangeintine r inim ta e uii average velocity over a very mall time interval acceleration rate of change of velocity aymge mangeinvelaoty anaeeinnne r lrllarltarleuu average acceleration over a very mall time interval ii rains Wain enigma is Just to check Acar position on a nignway i plotted verm tirne lttum nut to be a Lraight ll ei Wnicn or mee tatement i true7 Position m A ll acceleration i negative WHEN B it acceleration i poitive c it acceleration i zero D ltvellxltylzeru Irvlumiqung is Why a0 Position Vs time is a straight line xvt Pnsltm lrn Constant velao tv Time 1 onange in position changel ntlme onstant velocity means zero acceleration means constant velocity ii rains quotshim simian ii What about constant acceleration cnange in velocity change in tirne Acceleration onstant acceleratio 7 Fur every time interval ay i ecund tne velocitychnage by tne ame amount 7 agt0 give a onirorinly intreacing velocity 7 ml give a onirorinly detreao ng velocity Velocity rns We in em m Time 1 Question You are traveling at 60 miles per hour You apply the brakes resulting in a constant negative acceleration of 10 mph second How many seconds does it take to stop A 10 seconds Velocity change is 10 mph for every second Takes six seconds to decrease the velocity to zero C 3 seconds n ma Wm 107 spmmm Back to Galileo Use position velocity acceleration to quantify the motion of a falling object m mza VivSmle swam m Distance vs time for falling ball From analyzing the Falling Ball video frame by frame we find the position vs lime m This completely describes the motion Distance proportional to lime square DISTANCE melevs o d49 m32t2 n ma Wm 107 synnvmm Average speed for falling ball Falling Ball Total time0735 Total distance26m m a melevs m Avg speed 26m073535 ms DISTANCE a 8 o a n ma VivSmle swam 22 Instantaneous speed Falling Ball peedM55mls 073570538 25 2 2 peed 45mm 04887040 peedm21ms 02757018 n ma Instantaneous speed vs time Instantaneous speed proportional to time 7 So instantaneous speed 5 increases 1 5 at a constant rate E Thi means constant K 4 acceleration 5 3 sat 9 Lu 2 gt change in speed accel change In lime 1 7 68 ms 0 069 0 01 02 03 04 05 06 07 985 m9 85 ms1 TIME s n ma 7mm swam 24 Uniform acceleration from rest Acceleration constan 98 msZ Velocity accelerationgtlttime f a Uniformly increasing velocity Distance average velgtlttime at2 x t 12atZ n ma Wm m7 spmmm 25 Falling object constant acceleration The data show freely fallingobject has constant acceleration This is called the acceleration of gravity 98 mss 98 msZ Why does gravity result in a constant acceleration Why is this acceleration independent of mass m mza Vivswam swam 73 Tough questions These are difficult questions Maybe not completely answered even now But tied into a more basic question What causes acceleration Or how do we get an object to move A hot topic in the 17th centuw Descartes cogito ergo sum was a major player in this n ma Wm m7 spmmm 27 From Last Time Light shows both particle and wavelike properties Interference is an example of wavelike props Photons are particles of light in the particlelike description Experiments that demonstrated this are Photoelectric effect Blackbody radiation Mon Nov 1 Phy107 Lecture 23 Hour exam 2 results Average 68 Median 70 Your grade is posted at LearnUW uwmadcourseswisconsinedu Use your netid and password then go to Physics 107 course page Mon Nov 1 Phy107 Lecture 23 Blackbody radiation Blackbody radiation Object emits light with a blackbody spectrum Peak position color changes with temperature Mon Nov1 Phy107 Lecture 23 lniensny imev Waveiengih nanometersi am 400 sac 800 1000 Uitraviniel Visual Inirared White hot Object ai 7000 K 2001 BrooksCole PublishingWP omen at scum K 6000 K i r 1 Red hot 5000 K Mm 0mm ai 5000 K 200 800 10m 400 sac Waveiengih nanometersi Photoelectric effect PARTICLES Einstein says that light is made up of photons individual 39particles each with energy hf One photon collides with one electron knocks it out of metal If photon doesn t have enough energy cannot knock electron out Intensity photons sec doesn t change this Minimum frequency maximum wavelength required to eject electron Mon Nov 1 Phy107 Lecture 23 explained by quantization of light Light comprised of discrete particles photons Each photon has energy hf h Planck s constant Mon Nov 1 Phy107 Lecture 23 Intensity of light Photon of frequency f has energy hf Red light made of only red photons The intensity of the beam can be increased by increasing the photon flux Interaction with matter Photons are absorbed one at a time Interaction depends on individual photon frequency energy Mon Nov 1 Phy107 Lecture 23 Photons and Electromagnetic Waves Light has a dual nature It exhibits both wave and particle characteristics Applies to all electromagnetic radiation The photoelectric effect gives evidence for the particle nature of light Light can behave as if it were composed of particles Interference and diffraction evidence the wave nature of light Mon Nov 1 Phy107 Lecture 23 Matter waves If light waves have particlelike properties matter should have wave properties de Broglie postulated that the wavelength of matter is related to momentum as hi This is called the de Broglie wavelength Mon Nov 1 Phy107 Lecture 23 Why h p Works for photons Wave interpretation of light wavelength Speed of Light Frequency k c f Particle interpretation of light photons Energy Planck s constant x Frequency Ehf sofEh C C h Wavelength 2 f Eh Ec But photon momentum p E c h So A for a photon P Mon Nov 1 Phy107 Lecture 23 h We argue that A applies to everything I Photons and footballs both follow the same relation Everything has both wavelike and particlelike properties Mon Nov 1 Phy107 Lecture 23 Wavelengths of massive objects h deBroglie wavelength A P pmv for a nonrelativistic mv vltltc particle with mass Mon Nov 1 Phy107 Lecture 23 Wavelength of a football Make the Right Call The NFL39s Own interpretations and guidelines plus 1005 of official rulings on game situations National FootBall League Chicago 1999 short circumference 21 to 21 14 inches weight 14 to 15 ounces 043 040 kg o quotSometimes I don t know how they catch that ball because Brett wings that thing 60 70 mphquot Flanagan said 27 32 ms Momentum mv04 kg30 ms12 kg ms 34 A 55 x1035m 55 x1026nm p 12 kg mS Mon Nov 1 Phy107 Lecture 23 This is vew small 1 nm 10399 m Wavelength of red light 700 nm Spacing between atoms in solid 025 nm Wavelength of football 103926 nm What makes football wavelength so small A i gt Large mass large momentum P quotW short wavelength Mon Nov 1 Phy107 Lecture 23 Macroscopic objects don t show effects of quantum mechanics Similar to pendulum example Energy levels are quantized but discreteness is too small to be detected Mon Nov 1 Phy107 Lecture 23 Wavelength of electron Look for less massive object Electron is much less massive Mass of electron 91x103931 kg A i 6X1034J S p mv 9 x10 31kg x velocity Wavelength depends on velocity Larger velocity shorter wavelength Mon Nov 1 Phy107 Lecture 23 How do we get electrons to move Electron is a charged particle Constant electric field applies constant force accelerates electron Work done on electron is charge x voltage applied Work done change in kinetic energy 12mv2 chargexvoltage Mon Nov 1 Phy107 Lecture 23 The electronvolt New unit of energy for quantum mechanics 1 electronvolt energy gained by electron accelerating through 1 volt potential difference 1 electron volt 1 eV 16x103919C1V 16x103919J Mon Nov 1 Phy107 Lecture 23 charge Lootential eV is a small unit of energy but useful for small particles such as electrons Velocity of 1 eV electron 1 eV electron has 16x103919 J of energy It is all kinetic energy 2 E kinetic Emv 1 2 Vz V 2Ekinetic mc 2 so 2 2 c 6 me Rest energy of electron 05x106 eV Electron with 1 eV of energy is moving at 0 002C 600000 ms Mon Nov 1 Phy107 Lecture 23 Wavelength of 1 eV electron Fundamental relation is wavelength A E P Need to find momentum in terms of kinetic energy 2 0 p mv SO E kinetic 2 gt I9 lemEa nen c m h h hC A p U szkinetiC V zmc 2Ekinetic Mon Nov 1 Phy107 Lecture 23 A little complicated But look at this without calculating it constant 2 p j kinetic energy rest energy constant Vrest energyJKinetic energy Wavelength Mon Nov 1 Phy107 Lecture 23 Why is this useful Particles important in quantum mechanics are characterized by their rest energy n relativity that all observers measure same rest energy l t 2 05M V e ec ron mC 9 Different for Proton mCzquot 940 Mev different particles neutron mc2 940 MeV 1 MeV 1 million electronvolts Mon Nov 1 Phy107 Lecture 23 General trends constant Wavelength o Wrest energyJKinetic energy Wavelength decreases as rest energy mass increases Wavelength decreases as kinetic energy energy of motion increases Mon Nov 1 Phy107 Lecture 23 Wavelength of 1 eV electron 0 For an EleCtron constant h w 1 123 eVlZ39m 2 4 1 NEkinetic kinetic energy rest energy 1 eV electron A123 nm 10 eV electron AO39 nm 100 eV electron AO12 nm Mon Nov 1 Phy107 Lecture 23 Question A 10 eV electron has a wavelength of 04 nm What is the wavelength of a 40 eV electron B 04 nm C 08 nm constant Jrest energyJKinetic energy Wavelength Mon Nov 1 Phy107 Lecture 23 Can this be correct f electrons are waves they should demonstrate wavelike effects eg Interference diffraction A 25 eV electron has wavelength 025 nm similar to atomic spacings in crystals Mon Nov 1 Phy107 Lecture 23 Cwstals Many solids are made up of ordered planes of atoms The facets on a diamond are particular atomic planes in the diamond crystal lattice Mon Nov 1 Phy107 Lecture 23 Diffraction from layers of atoms a Upper plane gt Lower lane A p Difference in 2003Thomsonremokscme d sin 0 length o This is interference of two electron waves 0 The waves are reflected from different planes of the crystal o Difference in path length spacing between atoms Mon Nov 1 Phy107 Lecture 23 Constructive amp Destructive Interference Interference arises when waves change their 39phase relationship Can vary phase relationship of two waves by changing physical location of speaker 5 9m 313er ii 3 Constructive Destructive Mon Nov 1 Phy107 Lecture 23 Scattered intensity DavissonGermer experiment Diffraction of electrons from a nickel single crystal 100 80 60 40 2O 0 20 40 60 80 Detector angle 4 Mon Nov 1 hy 16 7 L ec ture 23 Electron gun 54 eV electrons A 1 39Xonization 4 chamber 7 Particlewave duality Like light particles also have a dual nature Can show particlelike properties collisions etc Can show wavelike properties interference Like light they are neither particle nor wave but some new object Can describe them using quotparticle language or quotwave language whichever is most useful Mon Nov 1 Phy107 Lecture 23 Suppose an electron is a wave Here is a wave A h f ziyvv where is the electron Wave extends infinitely far in x and x direction Mon Nov 1 Phy107 Lecture 23 Analogy with sound Sound wave also has the same characteristics But we can often locate sound waves E g echoes bounce from walls Can make a sound pulse How exactly is this done One example is a 39beat frequency between two notes Two sound waves of almost same wavelength added 4 0 AM w Ml WU 15 1O 5 O 5 1O 15 J Mon Nov 1 Phy107 Lecture 23 Add more different wavelengths Two Three Four f ve of x Mon Nov 1 Phy107 Lecture 23 Adding many sound waves Six sound waves with different wavelength added together Mgtt M M105 M M110 M M115 x5 M120 M M125 Wave now resembles a particle but what is the wavelength Sound pulse is comprised of several wavelength The exact wavelength is indeterminate 8 4 8 Ax 15 1O 5 O 5 1O 15 J Mon Nov 1 Phy107 Lecture 23 Spatial extent of localized sound wave AX 15 1O 5 O 5 1O 15 J Ax spatial spread of 39wave packet Spatial extent decreases as the spread in included wavelengths increases Mon Nov 1 Phy107 Lecture 23 Same occurs for a matter wave Construct a localized particle by adding together waves with slightly different wavelengths Since de Broglie says A h p each of these components has slightly different momentum We say that there is some 39uncertainty in the momentum And still don t know exact location of the particle Wave still is spread over Ax 39uncertainty in position Can reduce Ax but at the cost of increasing the spread in wavelength giving a spread in momentum Mon Nov 1 Phy107 Lecture 23 The wavefunction Quantify this by giving a physical meaning to the wave that describing the particle This wave is called the wavefunction Cannot be experimentally measured But the square of the wavefunction is a physical quantity It s value at some point in space is the probability of finding the particle there Mon Nov 1 Phy107 Lecture 23 From Last Time The deterministic solar system Gravitational forces are apparent at a wide ran ofscales Dbeys Wquot quare uf dutance between tnern F67gtlt10 m1xzquotquot39 1 Gravitational Constant Simple Planetary motion Newtonian Determinism Newton slaws seem to determine all future motion Al I future behavior exactly known iter mew mum mmom The 39threebody problem Complicated motion of 3 bodies ta could nutsulve anypmblem pas SmglE planet arbiting tne sun Prize offered for solution of 3rbndy prablern Poincare in 18 snawed prablern net analytically solvable iter mew n ma mime Sensitivity to initial conditions in the 3body problem Flash simulation rupiiawiprir r tameimiummirmricrmi rreuaiirreeuai rmi Two planets start autwith alrnast Resulting rn Sensitive to identical pusitiuns and velucities a attractiun ta m tiuns due ta gravitatiunal e tva suns are very different lr initial canditians t 5quot ilt How can we summarize this motion u mm minnow Dynamical Systems The system evolves in time according to a set of rules The present conditions determine the future The rules are usually nonlinear There may be many interactingwriables u mu mmow Examples of Dynamical Systems c rna k The human budy heart brain lungs Ecology plant and animal pupulatiuns The electrical power grid The internet u mm minnow A double pendulum One way no drive a pendulum is to hang it from another that is swmging L httg vwwv treasure r truves cum9h sics DnubiEPEnduium ht mi u mu mmow The weather The strange behavior of nonlinear systemswas not fully appreciated until corn uters permitted extensive numerical simulations of motions not susceptible to analytic methods 1961 Edward Lorenz discovered that a rather simple model ofatmospheric processes exhibited erratic behavior u mm minnow Lorenz model o Lorenz studied a simple model of the evolution of temperature and pressure and found a small change in initial value led to ultimately wildly different results luv1 39 NW 32p 2 mm Phynx m7 1 m7 Lorentz 39attractor in 3D 32p 2 mm Simple 39sensitive systems o Released balloon o Air hose fire hose instability 32p 2 mm Phynx m7 1 m7 The magnetic pendulum quotiv 4 mg and rud i uwln i oPendulum comes to rest above one of the stationaw magnets attractors oResult depends sensitively on point of release Two similar release poinls Different trajectory and rest point 32p 2 mm Phyn x m7 1 m7 Quantify the dependence on initial conditions Blue and white reg ons show in tial posit ons which correspond agnet CDITIII39Ig to equilibrium around either the blue or wh te fixed magnet 32p 2 mm Phynx m7 1 m7 Fractal structure of the boundary 0 If we could blow up the region around the boundaries between blue and wh te areas we would find that they are not infin tely sharp Instead we would see a complex structure which is termed a fractal Fractals have fractional dimensions and the unique property of self similarity to all levels of magnification If you magnify any part of a fractal you see a miniature copy of the overall fractal structure repeated on the small scale 32p Zn mm mm x m7 1 m7 Three magnet pendulum pmrernne e h e attractor rarthe magnet ant ependulum Releae pendulum at diff parts a seewnereiteamestarest Twrttonnoszidtnocolsmrnnxpoxmonx but thrttdlffotnt nal Winoix Basins of attraction for 3magnet pendulum Color ceding indicates nnal et an rest pasitian of magn g pendulum g Green is above green statianary magnet etc 3 Region of salid calar is called 5 a basiri of attractian39 E This shows a fractal selfr gt similar structure x release position a ma Wigwam Fractal structure in a similar problem By fractal er selfr similarY we mean similar an all length 542a es l e the picture looks the same after naming in much 1 claser rmnmulmeiseinepi umnitreneiuwm Dr1ven systems The magnet pendula were examples of systems attracted ta a fixed stable pasitian arter same time The stable positions are attracters39 arid the final pasitian depends sensitively an the initial release paint of the pendulum ln thiscase the initial matian was damped but by frictional forces But it the pendulum were cantinually driven itwauld cantinue te ascillate forever We cauld say it is attracted Le a fixed stable motion rather than a nnal pasitian a mu minnow The driven pendulum pendulum driven pushed with a particular strength and at a particular frequency a angle of pendulum Drive mechanism m angular velocity e rotation rate lfwe know both 9 instant t kno themetionof the ulum Mm nmamr Describe motion with phasespace39 plot an plot e vs n for all times instead of e vs time Makes a compact selfcontained description of the motion If we take a strobe photograph of the pendulum once perdrive cycle we will get pairs of a and n that we can plot in phase space This is a Poincare plot a an minnow Sin all drive am plitude periodic motion W mm m w m Pha espace plot Poincare piot Angularvelocity a Angle 9 Increase the drive amplitude Drive amptitude o 1755 period doubling w w mm Mm m k mu mmpm Period four oscillation Drive amptitude 0 27277 period doubles again to four times the drive period Chaotic motion Drive amptitude o 178 motion is nowchautic a g 0 1 a 5 Angle Mm w WWW More chaotic motion Drive amplitude 069 chaotic motion Mm w WWW quu m m m k mu mmpm mu M u m k mu mmpm Route to chaos in the driven pendulum ncreaxmg dnve k mu mmpm Driven Pendum39fquot baSins of Everyday chao A dripping faucet attract10n Water anvvrngrran a raueet can Shaw ehaatre hehavmr m manna we perauuln urea raster anp rate lead ta Denad aauplrrg then ehaane rallrng dmvlet if Ehgihelefl allfli SEESquot liar l arapaetaehesltle n remamvrg wat Farlaw enaugh rates these vvhratmm dve aut a each map rnaepenaent lt hvgh arap rate vlhratvan rnrluenee next arap aetaehnent a mm enmem Natural Fractals Fractal coastline amen leaner meme mmwn Fractals as 39art Many drrrerent systerns Shaw chautr c Dr rractal pehaymr They range rrurn physrcal systerns Ll purely rnathernatrcal Dnes E ln a phase space plat the calm cudmg usually rndrcates 9 the basm er attractmn at same attractur dynamrcal state p errm39lse 9 The saturatan er the culur rnrght rndrcate the rate at 39 a whrch the system rs attracted tn that dynarnrcal state Release Dmmangreenr E Fullnwmg are seyeral rractal rrnages rrarn thew Pendulum came a rest shave 3 green magnet gt m mm mm mm See sprutt physrcs Msc edurractals htrn fur rnany ether aetieen39rntral candmam ana WW PM 5W W nnal natran gtlt release posmon areth e um mmwn mmmv mum 51 in mm Gravitational force s From last time g j e ist law Law of inertia 39 by Every ubiect continues in its state of rest ur unifurrn rnutiun in a straight line unless acted upun by a fume Them tn equal an inn law Frna ora The acceleration of a may aiung a direction is r mpnmnnal tn the total force along that directlan and rce are d apposite e inversely the rnass at the body m mew 3rd law Action and reaction For Every an inn there is an equal and opposite rea iun 42 But munflaxlllz kg aw uh mm MW W mm mappi k A fortunate coincidence A force exactly proportional to mass so that everything cancels nicely But a bit unusual Einstein threw out the I Equal accelerations If more massive bodies accelerate more slowly with the same force why do all bodies fall the same independent of mass Gravitational force on a body dependson gravitat onal force its ma entirely attrib tingt e Therefore acceleration is independent of observed an era to mass a distortion of space time Velocity of the moon Acceleration of the moon What is the direction of the acceleration o t nioon7 9quot ll I g I win uri mm in mi mm I l what is the direction of the velocityofthe I 7 moon l I Acceleration W How has the velocity changed Velocity at time t v t i Velocity at time t W11 kVelocity at time 1 VElDCmattlmE It I I l l Change in yelocity in this equation y is the speed or the obiect which is the same atall times room won in Valle 7 Valle Earth s pull on the moon Experiment The moon continually accelerates toward the Fm g 11 v 2 7 ml Acceleration but because of its orbital velocity it continually ofballm1Fm1 M m llllSSeS the Eart The orbital speed of the moon isconstant but l l ni1 accelerates the direction continually changes l m2 inward in response i to force ng Therefore the velocity changes with time T Fng J Acceleration Vzlr for circular motion won in Valle v won in Valle 5h t the monkey Newton s falling moon 7 another example of soperposition rhedaneonisiireaiostasthe mmteydmpsiioi the tree ihe only demation iioi weights iinemotionis oaeieiation results in orbital aieoiyaayniZS motion From Newton39s Principia ms The monkey has exactly the same acceleration downward so that the dart hits the monkey room mime ii Macaw mime n Acceleration of moon 1 v The mean is accelerating at m1 r directly toward the earth ls tnis acceleratian different man 3 tne gravitational acceleratian af an apiect at tne Earth39s surface7 e acceleratian directly frarn rnaan s arpital speed and tne Earthrmuun tance rem in mine The radius of the earth Originally from udy of shadows at different latitudes by Eratosthenes Rearth6500 km rem in mine Distance dependence of Gravity The gravitational farce depends an distance Maori acceleratian is 1 mSleasoo times smaller tnan me 0 00272 ms acceleratian af gravity an the Earth39s surface moon is 170 times farther away and 35001701 5a men the gravitational farce draps as the distance d square Distance and diam of moon The diameter of the moon is the diameter of its shadow during a solar ec lpse mm the iameter angular size dr5 deg inferdistance r60 rearth rem inimiae Moon acceleration cont Distance ta moon to earm radii 3 mm m Speed af rnaan7 circumference af circular arbit 2m 5pm urnital distance2m 11023 M urnital tirne 27 3 days Centripetal acceleratian oi00272 rnsz Equation for force of gravity Mas uf apiect l XWBXX uf apiectzi uare uf distance between them xmZ dz For masses in ldlagmms and distance in meters F67gtlt10 12quot rem in mine Example Gravitational force decreases with distance from Earth 1nd the acceleration of an apple at the surface of the earth Force on apple Fm 67x10quot 9 1 mE Am x m m x m This 1 arm the dz m Eallh app4 force on the Earth Force 0 apple 39 Fapph 57X10 dz bythe applel So movmg farther from the Earth should reduce the force of ravit d distance between center of objects radius of Earth g y F Acceleration of apple amt 67x1041m5mh o Typical airplane cruises at 5 mi 8000 m mapptt dz d increases from 6370000 m to 6378000 m only about a 025 change 667x1039 N m2kg2xM 6 37x106m2 gamma Fauzuus 19 Phygcs m7 FallZEIEIE 2 So why is everyone floating around o International space station orbits at 350 km 350000 m o d 6370000 m 350000 m 6720000 m o Again d has changed only a little so thatg is decreased by only about 10 EmmgmggwmtksWmquot 502nm Physics m7 Fall was 21 Phygcs m7 Fall was 22 A 4122 The space station is falling 43 similar to Newton s apple 4 13 G39F u reme Scream 7 300 reet ofpure adrenalme rush a g 239 In its circular orbit once around the Earth every 90 minutes it is continuously accelerating toward the Earth at 88msZ 4 Va ELIE A freefall rlde g 1 2 d Ear Eventhing inside it is also falling accelerating M toward Earth at that same rate r 7 The astronauts are freely falling inside a freely 2x 300fr falling elevator They have th erception of Z 32 my weightlessness since their environment is falling just as they are if smart Phygcs m7 Fall znns 2t Physics m7 Fall was 23 A little longer ride Humvuwr Til Parabolic path of freely falling object pm in Fall was 25 Acceleration of gravity on moon o On the moon an apple feels gravitational force from the moon o Earth is too far away mmoon xma Force on apple on moon Fem 67x10 Vi 7 F Accel of apple on moon 67x10quot mg m V we mm Compare to accel on Earth 67x10quot mfg rEanh accel on moon mmnmgm m rm HWY Physics m1 Fall was 27 Gravitational force at large distances Stars orbiting our black hole o At the center of our galaxy is a collection of stars found to be in motion about an invisible object rmquot mm Physics m7 FallZEIEIE Accel of gravity on moon accel on moon mMMm 5W accel on Earth CHMgm 74 x 10 kg60 x10ukg 17x105m64x105mz 00123 Q 775 my Physics m7 FallZEIEIE Orbits obey Newton s gravity orbiting around some central mass in the central parsec of our galaxy Movie at right summarizes 14 years of obsenations Stars are in orbital motion about some massive central object httpwwwmpempgdewww irGCigtrohtml Physics m7 FallZEIEIE an What is the central mass One star swings by the hole at a minimum distance b of 17 light hours 120 AU or close to three times the distance to Pluto at speed v5000 kms period 15years From the orbit we can derive the mass The mass is 26 million solar masses It is mostly likely a black hole at the center of ourMilky Way galaxy Phy m m7 Panama 31 From last time o Theories are tested by observations o Different theories can predict equivalent behavior within experimental accuracy o Simplicity or symmetly of a theory may be hints of its truth o In some cases a new theow forced by observations can require acceptance of radical nonintuitive ideas physics m7 FaJZEIEI5 Aristotle s views on motion Aristotle s observations VERTICAL MOTION T e element earth moves down toward its natural restingplace Air rises to its natural place in the atmosphere Fire leaps upwards to its natural place above the atmosphere HORIZONTAL MOTION contrary to their natural motion o ies seem to need push orpull to maintain horizontal motion physics m7 paiizuus More Aristotle Heavier objects should fall vertically faster than lighter ones Why Theoretically Heavier ones contain more of the earth element Experimental y igJ1t objects often observe to fall slowly Harder to lift heavierobjects How much faster Aristotle saysproportional to their weight Unclearwhy but is simplest relation Clearly doesn t work in some cases eg lat papervs crumpled paper physics m7 FaJZEIEI5 Galileo Objects move downward because gravity disturbs their mo ion Claimed that heavy and light objects drop in the same way Seems counterintuitive Clearly doesn t work in some ca 5 eg Feather vspenny physics m7 paiizuus Which falls faster A Heavier mass hits first B Lighter mass hits first C Both masses hit at the same time physics m7 FaJZEIEI5 o Release two objects of We can do the experiment different masses physics m7 paiizuus Why doesn t it seem exactly right7 onfused by air resistance Air exerts a force on the fallingbody Would be clearerifwe could do it in wcu um May allow us to tell which theory is correct Apollo 15 on the moon Mum mam Just how does the object fall Galileo showed mat the falling motion is independent of mass but How long does it take to fall How fastis it going Does the speed change during the fall Or What makes something move Galileo s ideas about motion Principle aflntrtia object movingon level surface moves in unchanging d irection at constant speed unless disturbed Principle of superposition An object sub 39ect to two se arate influences disturbances responds to each without modifying its response to the other mm in mi mm in Inertia maintain horizontal r obiect retains constant speed pussiblyzeru unless pushed or pulled No continued pushingpulling required to motion Direct contradiction to previous views inertia describes degree Ln which an ubiectwill maintain its state or motion whether moving or at rest targe inertia gt difficult to change state of motion of ubiect More on motion Hittingball with hammerdisturbs it from rest changing its motion Afterthe hammer hit there is no more disturbance Motion no longer changes The ball moves at constant speed We measure speed in metersper second in s r 2 ms gt For every second the ball muves twu meters 7 E g afterZ secunds h e ya l39lmt inas traveled 4 meters n Superposition Hit the ball with two hammers Both these disturbances act on the ball causing it to change it s motion Net effect on the ball is the superposition or adding up of the two disturbances hammer hits Physis m7 humus 13 Average speed As an equation Dilance traveled d Travelmgume r gt 3 1 Average peed E The instantaneous speed is die speed over a short time interval If the speed is constant ue average and instantaneous speed are the same Plums m7 humus n Instantaneous speed Instantaneous speed is the average velocity over an infinitesimal yew short time intenal This is what your speedometer reads Instantaneous speed gives you a better understanding of the motion Physis m7 humus 15 More on Superposition Q Hit ball off end of table Ball falls downward because gravity now disturbs it We know that the gravity disturbance causes a motion straight downward The hammer hit caused a motion to the right These two motions are superposed the ball moves to the right at 2 ms and aLso moves downward muss m7 humus is Back to falling objects I drop two balls one from twice the height of the other The time it takes the higher ball to fall is how much longer than the lower ball A Twice B Three time C Fourtimes D None of the above Physis m7 humus n Details of a falling object Just how does the object fall Apparently independent of mass but how fast Starts at rest zero speed ends moving fast IHence speed is not constant Final speed increases with height Falling time increases with height was m7 humus lE Galileo measured this o But falling motion too fast for accurate measurement o Galileo was able to measure a different aspect that let him determine the time o In this way he made extremely accurate measurements Physic m7 FallZEIEIE 19 Used principle of superposition and principle of inertia Ball leaves ramp with constant horizontal speed After leaving ramp it continues horizontal motion at some constant speed 5 no horizontal disturbances But gravitational disturbance causes change in vertical motion the ball falls downward For even second of fall it moves to the right 5 meterssecondgtlt1 second 5 meters Determine falling time by measuring horizontal distance An equation From this Galileo determined that the falling time varied proportional to the square root of the falling distance Fallingtime 0 qualling distance Falling time t Falling distance d d 0c t2 d022 Physic lc ma WW Zl How much longer does it take I drop two balls one from twice the height of the other The time it takes the higher ball to fall is how much longer than the lower ball A Two times longer B Three times longer C Four times loner Physic m7 Fall was 22 Slow motion in 1632 o The inclined plane Redirects the motion of the ball Slows the motion down But character of motion remains the same assume that the speed acquired by the same movable object over di erem inclinations ofthe plane are equal whemer the heights 0fth0se plane are equal Physic m7 FallZEIEIE 23 How can we show this o Focus on the speed at end of the ramp o Galileo claimed this speed independent of ramp angle as long as height is the same Physic m7 Fall znns u Using the inclined plane u i motion i tretched out o everything happen rnore lovity Speed oftheeare quot tne ame Physis m7 Fallz us 25 Measure gravitational dropping motion with inclined plane o Have argued motion is the same o Difficult to measure velocity o but can now measure distance and time from marks on the inclined plane Phy s m7 Panama 25 Falling speed As an object falls it s speed is Constant C Increasing proportional to time squared Physis m7 Fallz us 27 Constant acceleration In fact the speed of afalling object increases uniformly with time We say that the acceleration is constant Acceleration Change in speed A change in time At Units are men meters per secondsecond mss abbreviated ms2 Phy s m7 Panama 22 Galileo s experiment A piece of wooden rnoulding or mntling aoout 12 Eubil aoout 7rn long half a cuoit aoout 30 crn wide and tnree fingerrbreadth aoout 5 crn thick was taken on it edge was cut a cnannel a little more tnan one finger in oreadtn naving rnade tni groove verv traight smooth and and nming lined it witn parcnrnent also m smooth and polixhed a poxxible we rolled along it a nard smooth and verv round oronzeoall 5 r p a For tne meaxuremenl of tirne we ernploved a large vessel of water placed in an elevated poxilion to tne oottorn of tni vessel was soldered a pipe of small diarneter giving a tnin iet of water wnicn we collected in a small grim during tne tirne of eacn descent tne water thus collected was weighed a ter eacn descent on a verv accurate oalance tne difference and ratio of these Weight gave us tne difference and ratio of the tin es Next time Have uncovered some quantitative relations regarding motions o 39 But need to clearly define the concepts to describe this motion Next time we will investigate position velocity acceleration momentum Phy s m7 Fanzuns do From last time In tia tendency of body to continue in straightrlirie motion at constant speed unless disturbed Superposition abiect responds independently to separate disturbances i g e Fallirgtmie Dmvamarialtaquareaffallirgdistance autwriw Gravity disturbs the abiect leading to falling matian Buthuw daes mislead to Galileo39s results7 in mi new sex Position speed velocity amp acce eration Need to understand these concepts n position 7 distance e speed velvety aveiage instantaneous e acceleration aveiage instantaneous u iaa when 3pm Quantifying motion Distance and Time Amoving object 6 changes its position with time x pos at time t1 x2 pos at time t2 n Harpl My position at all times tornpletely destnoes my motion in iaai inseam see i Can use this information to find the speed Ifl move 5 meters in 5 seconds Then i meters in each second 5 divided by 5 1 meter persecond eg could walk 1 meter in the first second and 1 meter in the next second etc our maybe lwalked 0 meters in the first second and t en 5 meters in 4 seconds a ia aquot tea The average speed is the same As an equation Distaneetraveie Travelingtiiii lveageaaee ould also write 1131 So knowingaverage speed lets us find distance traveled p16 Instantaneous speed Instantaneous speed is the average velocity over an infinitesimal very short time interval This is what your speedometer reads Instantaneous speed gives you a better understandingor the motion an in 3pm Think about this one mph This takes 60 seco Your average speed is A 10 mph 40 mph D so mph in van meson sex You increase your speed uniformly from o to 60 nds Acceleration u we Err 1m 3sz Understanding acceleration Major points position coordinatesora body velocity rate orchange orposition r aver n as n r instantaneous average velocity ever a very small UNE interval acceleration rate orchange orvelocity mmga ahangemveiaatv ahangem the r instantaneous average acceleration ever a very small UNE interval u we Err 1m 3sz Just to check A car39s position an a highway ntted versus time he hich of these statements is true7 Position m A its acceleration is negative B H5 acceleration is positive ci lts acceleration is zero Di its velocity is zero mywsim 3px Why a0 Position VS time is a straight line at x vt anxtmnoxmon Position m Canxtantveiamly W2 5 chargeiri paetian means constant velocity W onstant velocity means zero acceleration a 1m 3sz What about constant acceleration h 39 l 39t Acceleration c ange nve c y change In tIme Constant acceleration For every time interval say 1 second die velocity chnages by the same amount agt0 gives a uniformly increasing velocity Question You are traveling at 60 miles per hour You apply the brakes resulting in a constant negative acceleration of 10 mph second How many seconds does it take to stop w e E mmmmamy j C 3 seconds Velocity change is 10 mph for t39 every second Takes six seconds to mm v u r K nag aaam 78 decrease the velocity to zero M M m Time s M M W m Questions How far does the car go during that time A 01 mile E 02 mile C 005 mile Since speed changes uniformly with time from 60 mph to 0 mph so average speed is 30 mph Distance average speed x time 30 mileshour x 6 seconds 30 mileshour x 1600 hr 120 mile 5 M mm Physmm p s Back to Galileo Use position velocity acceleration to quantify the motion of a falling o 39ect M 11 m2 Phy s m7 spans Distance vs time for falling ball From analyzing the Falling Ball video frame by frame 3 we find the position vs 25 time A This completely 2 descri hes the motion 5 Distance proportional to lime square 1 Dl STANCE meleis o 5 d49 m32t2 M mm Physmm p s Average speed for falling ball Falling Ball Total time0735 Total distance26m m a melevs m Avg speed 26m073535 ms DlSTANCE a 8 o a M 11 m2 Phy s m7 spans 12 Instantaneous speed Falling Ball 25177201quot peed755m 073570538 5 E peed 45mm 04887040 peedm21ms 02757018 M mm mmm p s Instantaneous speed vs time Instantaneous speed proportional to time 1 Soinstantaneousspeed 5 increases 1 5 at a constant rate E This meansconstant t 4 acceleration 5 3 sat 9 han eins eed g 2 accel change in lime 68 m 0 0 01 02 03 04 05 06 07 1 TIME s 069 985 m985 m M 11 m2 F39th 107 spans 20 Uniform acceleration from rest Acceleration constant Velocity accelerationgtlttime at Uniformly increasing velocity Distance average velgtlttime 1Zatgtlt t 12atZ M mm Physmm Sims 2 Falling object constant acceleration The data show that a falling object has constant acceleration This is called the acceleration of gravity 98 mss 98 msZ But why does gravity result in a constant acceleration Why is this acceleration independent of mass M 11 m2 F39th 107 spans 22 Tough questions These are difficult questions Maybe not completely answered even now But tied into a more basic question What causes acceleration Or how do we get an object to move A hot topic in the 17th centuw Descartes cogito ergo sum was a major player in this M mm Physmm Sims 2 Descartes view Motion and rest are primitive states of a body without need of further explanation Bodies only change their state when acted upon by an external cause This is similar our concept of inertia M 11 m2 F39th 107 spans u Inertia and momentum Principle of inertia object continues at constant velocity unless disturbed So a disturbance will change the velocity This change in velocity is acceleration Could start an object moving that is at rest or stop an object that is moving An object at rest subject to a disturbance could start moving at some velocity M mm Physmm sons 25 Different types of objects Objects with lots of inertia don t change motion as much as lighter objects subject to the same disturbance Inertia measures the degree to which an object at rest will stay at rest An object with lots of inertia is difficult to accelerate acceleration change in velocity M mm Phymsim Spins demu 25 Momentum Same disturbance applied to different objects results in different velocities eg hitting bowling ball and softball whammer But the product mass x velocity is the same eg for the bowling ball and the softball Momentum massxvelocity M mm Physmm sons 27 Descartes also said That a body upon coming in contact with a stronger one loses none of its motion but that upon coming in contact with a weaker one it loses as much as it transfers to that weaker body So for Descartes the total amountof motion is always die same We call the amountof motion momentum and Descartes law as conserva on of momentum M 11 m2 F39th m7 spans 22 Momentum conservation Can easily describe interactions of objects The total momentum sum of momenta of each object of the system is always the same We say that momentum is consened Momentum can be transferred from one object to the other but it does not disappear example M m2 Physmim ms 29 Descartes was able to move beyond the complicated details of collisions to some basic governing princip es Next time look at how Newton egtlttended these ideas with his three laws of motion Builds on Galileo and Descartes but includes the concept of a force M 11 m2 F39th m7 spans an Physics 107 Ideas of Modern Physics uwphysicswiscedurzchowskiphy107 Modern Physics essentially post1900 Why 1900 Two radical developments Relativity amp Quantum Mechanics Both changed the way we think as much as did Galileo and Newton Goals of the course Learn a process for critical thinking and apply it to evaluate physical theories Use these techniques to understand the ideas underlying modern physics Implement the ideas in some basic problems Learn where physics is today and where it is going How is this done Read the textbook ltgt Physics Concepts amp Connections Come to the lectures ltgt955 MWF in 1300 Sterling Hall Participate in discussion section ltgtOne per week starting Sep 13 Do the homework ltgtAssigned each Wed due the following Wednesday Write the essay ltgtOn a physics topic of your choice due Dec 6 Take the exams ltgtThree inclass hour exams one cumulative final exam What will we cover Scientific observation and reasoning Motion and energy Relativity Quantum Mechanics Gmww Particle theory and cosmology Single atoms and quantum waves Entire galaxies Where s the math Math is a tool that can often help to clarify physics In this course we use algebra and basic geometry and trigonometry We will do calculations but also focus on written explanation and reasoning What do you get An understanding of the physical universe A grade 15 HW 15 essay 20 each for 2 of 3 hour exams lowest dropped 30 from cumulative final exam A theory of the universe o Look around what you see is the universe o What can you say about how it works What Aristotle saw Earth air water fire For terrestrial objects and aether from which celestial bodies are formed Aristotle s ideas about motion Earth moves downward Water downward air rises up fire rises above air Straightline motion Celestial bodies have a perfectly circular motion Motion of the celestial bodies E 302272 5 Apparent motion of stars Rotation about a point it every 24 hours Moon sun and planets were known to move with respect to the stars Motion of the stars over 6 hrs Viewing fr9m CDIUmbus 999 9951999 99399 PM Lucal Daily motion of sun amp planets over 1 year 92902000 000000UT Movie by R 39Po ge Oh139o Stgte L Aristotle s crystal spheres EarthWater Air Fire Prime mover 24 hrs Cristal sphere 49000 yrs quot Firmament 1000 yrs Saturn 30 years Jupiter 12 years Mars 2 years Sun 1 yr Venus 1 yr Mercury 1 yr Moon 28 days Detailed Observations of planetary motion Ptolemy Iubmn bi in cpi ctioma vinbll39sx vm 39Elb dima rcptimli 5mm 6393 m5 2 3 0 Elrgummtizunc igna 6 5 lg 12 9 7 m1 war 7 a R IE I quotLExTIL 339 51 EIEEEFEEEIIEEI 1 mnglillinlllllll39 r if I i l Eii i riigggg 3t nmneri g in u E j Bah W1quot Lu 3 le3713 391 LL l v H LALJJJlLlw Observational notes from Ptolemy s Almagest Retrograde planetary motion Retrograde motion of Mars Apparent motion not always in a straight line Mars appears brighter during the retrograde motion Epicycles deferents and equants the legacy of Ptolemy Epicycle 4 Defcrent Epicycle reproduced planetary retrograde motion Ptolemy s universe 0 In 39final form 40 epicycles and st deferents j Equants and 3 eccentrics for sun 5 gs moon and planets a a Provided detailed g planetary positions gs for 1500 years m R

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