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UW-Madison PHYSICS 107 - Moon acceleration

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1Phy107 Spr 07, Lecture 7 1Last Time…HW: shift due date to Tuesdays.Next assignment due next weekHour Exam 1: next week Wed (Valentine’s day) in classDifference between mass and weightCircular motion: centripetal accelerationNewton’s gravity:attractive force depends on separationPhy107 Spr 07, Lecture 7 2Velocity and accelerationCentripetal accelerationdirected inwardAccelVelocitya=v2/rTrue forany bodyin circularmotion.r=orbitradiusPhy107 Spr 07, Lecture 7 3ExperimentF=m2gF=m2gAcceleration of ball m1=F/m1= m2g/ m1m1 acceleratesinward in responseto force m2gAcceleration = v2/rfor circular motionPhy107 Spr 07, Lecture 7 4Superposition againIf gravity weresuddenly turned off,what wouldthe moon do?ABCDPhy107 Spr 07, Lecture 7 5Moon acceleration• Distance to moon = 60 earth radii~3.84x108 m• Speed of moon?Circumference of circular orbit =Speed =Centripetal acceleration = v2/r= 0.00272 m/s2! 2"r ! orbital distance = 2"rorbital time = 27.3 days= 1023 m / sThis is the acceleration of the moondue to the gravitational force of the Earth.Phy107 Spr 07, Lecture 7 6Distance dependence of Gravity• The gravitational force depends on distance.• Moon acceleration is times smaller than theacceleration of gravity on the Earth’s surface.• The moon is 60 times farther away, and 3600=602• Gravitational force drops as the distance squared ! 9.81!m/s20.00272!m/s2" 3600Newton: I thereby compared the force requisite to keep the Moon inher orb with the force of gravity at the surface of the Earth, andfound them answer pretty nearly.2Phy107 Spr 07, Lecture 7 7Equation for force of gravity ! Fgravity"(Mass of object 1)#(Mass of object 2)square of distance between them ! F "m1# m2d2 ! F = 6.7 "10-11m1" m2d2For masses in kilograms, and distance in meters,Phy107 Spr 07, Lecture 7 8Example• Find the acceleration of an apple at thesurface of the earth ! Acceleration of apple = Fapplemapple= 6.7 "10#11mEarthd2 ! Force on apple = Fapple= 6.7 "10-11mEarth" mappled2This is also the forceon the Earth by theapple!d = distance between center of objects ~ Earth radius! = 6.67 "10#11N # m2/ kg2"5.98 "1024kg6.37 "106m( )2= 9.83 m / s2This is the ‘acceleration of gravity’Phy107 Spr 07, Lecture 7 9Gravitational force decreaseswith distance from Earth ! Force on apple = Fapple= 6.7 "10-11mEarth" mappled2So moving farther from the Earth should reduce the force of gravity• Typical airplane cruises at ~5 mi = 8000!m— d increases from 6,370,000 m to 6,378,000 m— only about a 0.25% change! Phy107 Spr 07, Lecture 7 10• International space stationorbits at 350 km = 350,000 m• d = 6,370,000 m + 350,000 m = 6,720,000 m• Again d has changed only a little, so that g isdecreased by only about 10%.Phy107 Spr 07, Lecture 7 11So why is everyone floating around?Edward M. (Mike) Fincke, Expedition9 science officer and flight engineerJames S. Voss, Expedition 2 flight engineerPhy107 Spr 07, Lecture 7 12Why weightless?Why are the astronauts in the space stationapparently weightless?A. The moon gravity cancels the Earth’sB. The space station shields Earth’s gravityC. Space station rotation cancels gravityD. None of the above3Phy107 Spr 07, Lecture 7 13The space station is falling……similar to Newton’s apple• Both astronauts and stationaccelerating toward Earth at ~8.8 m/s2• centripetal acceleration from Earth’s gravity.• Astronauts and station are in ‘free fall’Phy107 Spr 07, Lecture 7 14 Supreme Scream - 300 feetof pure adrenaline rushA freefall ride! d =12at2t =2da=2 " 300 ft32 ft / s2= 4.3 secof freefallPhy107 Spr 07, Lecture 7 15A little longer rideParabolic path of freelyfalling objectPhy107 Spr 07, Lecture 7 16But gravitational forceis only 2% weaker -Need to move fartherfrom the Earth.Phy107 Spr 07, Lecture 7 17QuestionHalfway to the moon, what is theacceleration of an apple due to the Earth’s gravity?Moon is 60 Earth radii from Earth)A. g/2B. g/4C. g/60D. g/900Moon is 60 Earth radii from the Earth.Halfway is 30 Earth radii.So apple is 30 times farther than when on surface.Gravitational force is (30)2 times smaller = g/900Phy107 Spr 07, Lecture 7 18Acceleration of gravity on moon• On the moon, an apple feels gravitational forcefrom the moon.• Earth is too far away. ! Force on apple on moon = Fapple= 6.7 "10-11mmoon" mapplermoon2 ! Accel. of apple on moon = Fapplemapple= 6.7 "10#11mmoonrmoon2 ! Compare to accel on Earth = 6.7 "10#11mEarthrEarth2 ! accel. on moonaccel. on Earth= mmoon/ mEarthrmoon/ rEarth( )24Phy107 Spr 07, Lecture 7 19Accel. of gravity on moon ! accel. on moonaccel. on Earth= mmoon/ mEarthrmoon/ rEarth( )2=7.4 "1022kg /6.0 "1024kg1.7 "106m /6.4 "106m( )2=0.0123.265( )2= 0.175 #16So you weight only 1/6 your Earth weighton moon.Phy107 Spr 07, Lecture 7 20Gravitational force at large distances:Stars orbiting our black hole• At the center of ourgalaxy is a collection ofstars found to be inmotion about aninvisible object.Phy107 Spr 07, Lecture 7 21Orbits obey Newton’s gravity,orbiting around some central mass• Scientists at the Max Planck Institute forExtraterrestrische Physik has used infraredimaging to study star motion in the central parsecof our galaxy.• Movie at right summarizes 14 years ofobservations.• Stars are in orbital motion about some massivecentral objecthttp://www.mpe.mpg.de/www_ir/GC/intro.htmlPhy107 Spr 07, Lecture 7 22What is the central mass?• One star swings by the hole at a minimum distance b of17 light hours (120 A.U. or close to three times thedistance to Pluto) at speedv=5000 km/s, period 15 years.• From the orbit we can derive the mass.• The mass is 2.6 million solar masses.• It is mostly likely a black hole at the center of our MilkyWay galaxy!Phy107 Spr 07, Lecture 7 23Summary of Newtonian gravity• Gravitational forces are apparent at a wide range ofscales.• Obeys ! Fgravity"(Mass of object 1)#(Mass of object 2)square of distance between them ! F = 6.7 "10-11m1" m2d2Gravitational ConstantPhy107 Spr 07, Lecture 7 24Discussion so far…• So far we have talked about– Velocity– Acceleration– Momentum conservation of momentum– Momentum transfer changing the velocity of an object– Force changing the velocity of an object– Newton’s relation: Acceleration = Force / mass5Phy107 Spr 07, Lecture 7 25Something missing• We these tools, can think about the world in many ways.– Collisions resulting in a momentum transfer– Gravitational forces resulting in acceleration of falling


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UW-Madison PHYSICS 107 - Moon acceleration

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