1Physics 107, Fall 2006 1From last time1st law: Law of inertiaEvery object continues in its state of rest, or uniform motionin a straight line, unless acted upon by a force.2nd law: F=ma, or a=F/mThe acceleration of a body along a direction is– proportional to the total force along that direction, and– inversely the mass of the body3rd law: Action and reactionFor every action there is an equal and opposite reaction.Physics 107, Fall 2006 2Gravitational forceGravitationalforce on appleby earthGravitationalforce on Earthby appleThese forces areequal and opposite,But mearth=6x1024 kg mapple=1 kgmEarthaEarth= mappleaappleaEarthaapple=mapplemEarthPhysics 107, Fall 2006 3Equal accelerations• If more massive bodies accelerate moreslowly with the same force…… why do all bodies fall the same,independent of mass?• Gravitational force on a body depends onits mass:• Therefore acceleration is independent ofmass:Fgravity= mga =Fgravitym=mgm= gPhysics 107, Fall 2006 4A fortunate coincidence• A force exactlyproportional to mass, sothat everything cancelsnicely.• But a bit unusual.• Einstein threw out thegravitational forceentirely, attributing theobserved acceleration toa distortion of space-time.Physics 107, Fall 2006 5Velocity of the moonABCWhat is the direction of thevelocity of the moon?Physics 107, Fall 2006 6ABCAcceleration of the moonWhat is the direction of theacceleration of the moon?2Physics 107, Fall 2006 7Acceleration = change in velocitychange in timeVelocity at time t1Velocity at time t2•Speed is same, butdirection has changed•Velocity has changedPhysics 107, Fall 2006 8How has the velocity changed?Velocity at time t1Velocity at time t2V(t1)V(t2)Change in velocityCentripetal acceleration = v2/r, directed towardcenter of orbit. r = radius of orbit(In this equation, v is the speed of the object,which is the same at all times)Physics 107, Fall 2006 9Earth’s pull on the moon• The moon continually accelerates toward theearth,• But because of its orbital velocity, it continuallymisses the Earth.• The orbital speed of the moon is constant, butthe direction continually changes.• Therefore the velocity changes with time.True for any body in circular motionPhysics 107, Fall 2006 10ExperimentF=m2gF=m2gAcceleration of ball m1=F/m1= m2g/ m1m1 acceleratesinward in responseto force m2gAcceleration = v2/rfor circular motionPhysics 107, Fall 2006 11Shoot the monkey• The dart gun is fired just as themonkey drops from the tree.• After the dart leaves the gun, theonly force is from gravity.• The only deviation from straight-line motion is an accelerationdirectly downward.The monkey has exactly the same acceleration downward,so that the dart hits the monkey.— another example of superpositionPhysics 107, Fall 2006 12Newton’s falling moon•FromNewton’s Principia, 1615Throwing theball fast enoughresults in orbitalmotion3Physics 107, Fall 2006 13Acceleration of moon• The moon is accelerating atdirectly toward the earth!• This acceleration is due to the Earth’s gravity.• Is this acceleration different than g,the gravitational acceleration of an object at theEarth’s surface?– Can calculate the acceleration directly frommoon’s orbital speed, and the Earth-moondistance.v2r m / s2Physics 107, Fall 2006 14Distance and diam. of moon• The diameter of the moon is thediameter of its shadow during a solareclipse. From the diameter d andangular size d/r~5 deg, infer distancer~60*r(earth).Physics 107, Fall 2006 15The radius of the earth• “Originally” fromstudy of shadows atdifferent latitudesby Eratosthenes!• R(earth)=6500 kmPhysics 107, Fall 2006 16Moon acceleration, cont• Distance to moon = 60 earth radii ~ 3.84x108 m• Speed of moon?Circumference of circular orbit =Speed =Centripetal acceleration = 0.00272 m/s22r orbital distance=2rorbital time=27.3 days= 1023 m / sThis is the acceleration of the moondue to the gravitational force of the Earth.Physics 107, Fall 2006 17Distance 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• So then the gravitational force drops as the distancesquared 9.81m/s20.00272m/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.Physics 107, Fall 2006 18Equation for force of gravity Fgravity(Mass of object 1)(Mass of object 2)square of distance between them Fm1 m2d2 F = 6.7 10-11m1 m2d2For masses in kilograms, and distance in meters,4Physics 107, Fall 2006 19Example• Find the acceleration of an apple at the surface ofthe earth Acceleration of apple = Fapplemapple= 6.7 1011mEarthd2 Force on apple = Fapple= 6.7 10-11mEarth mappled2This is also theforce on the Earthby the apple!d = distance between center of objects ~ radius of Earth= 6.67 1011N m2/ kg25.98 1024kg6.37 106m()2= 9.83 m /s2Physics 107, Fall 2006 20Gravitational force decreases withdistance 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 = 8000m— d increases from 6,370,000 m to 6,378,000 m— only about a 0.25% change! Physics 107, Fall 2006 21• 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%.Physics 107, Fall 2006 22So why is everyone floating around?Edward M. (Mike) Fincke, Expeditionscience ocerPhysics 107, Fall 2006 23The space station is falling……similar to Newton’s apple• In its circular orbit, once around the Earth every 90minutes, it is continuously accelerating toward theEarth at ~8.8 m/s2.• Everything inside it is also falling (acceleratingtoward Earth at that same rate).• The astronauts are freely falling inside a freely-falling ‘elevator’. They have the perception ofweightlessness, since their environment is fallingjust as they are.Physics 107, Fall 2006 24 Supreme Scream - 300 feetof pure adrenaline rushA freefall rided =12at2t =2da=2 300 ft32 ft /s2= 4.3 secof freefall5Physics 107, Fall 2006 25A little longer rideParabolic path of freelyfalling objectPhysics 107, Fall
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