Lecture chapter 8_gravitation

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Lecture chapter 8_gravitation

  1. 1. 4/8/2012 Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force? Newton’s realization was that the force must come from the Earth. GRAVITATION He further realized that this force must be what keeps the Moon in its orbit. A force of attraction between objects that is due to their masses. Because gravity is less on the moon than on Earth, walking on the moon’s surface was a very bouncy experience for the Apollo astronauts. Compared with “all” the objects around you, Earth has a enormous mass. Any two bodies with masses can attract each  Newton’s Law of Universal Gravitation states other. This universal effect is known as that the gravitational force FG between any gravitation two bodies of mass m1 and m2, separated by The force with which one body attracts the a distance r, is described by, other due to their masses is known as gravitational force m1m2 FG  r2  It is a center-to center attraction between all forms of matter. 1
  2. 2. 4/8/2012 Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a third law >The moon is actually pair: the Earth exerts a downward force on you, and falling toward Earth but you exert an upward force on the Earth. has great enough When there is such a disparity in masses, the reaction tangential velocity to force is undetectable, but for bodies more equal in avoid hitting Earth. mass it can be significant. >If the moon did not fall, it would follow a straight-line path. 8.1.1 NEWTON’S CONFIRMATION OF 1/r2 INVERSE SQUARE LAW Inverse-square law:  relates the intensity of an effect to the inverse- square of the distance from the cause. • SMALL ‘d’ • LARGE ‘d’ • LARGE ‘F’ • SMALL ‘F’  in equation form: intensity = 1/distance2.  for increases in distance, there areThis applies to any case where the effect from a localized sourcespreads out evenly decreases in force.  even at great distances, force approachesOTHER EXAMPLES WHERE THE INVERSE SQUARE LAW IS APPLIED: Light, Radiation, Sound but never reaches zero. NOTE: The force between any two objects NEVER reaches zero, it justgets very small (Asymptotically approaches zero). 2
  3. 3. 4/8/2012 Mass The gravitational force on either one of the m1 m2 two object is proportional to both m1 and m2, m1m2 FG  r2 decrease mass  Decreased force force varies directly with masses Fg α m1 m2 Gravitational force increases as  Gravity is the weakest of four known mass increases. fundamental forces ◦ Imagine an Elephant and a Cat  With the gravitational constant G, we have ◦ Or imagine the Earth and the Moon the equation mm FG  G 1 2 r2  Universal gravitational constant: Gravitational force decreases as distance G = 6.67  10-11 Nm2/kg2 increases.  Once the value was known, the mass of ◦ Gravity between you and the Earth Earth was calculated as 6  1024 kg ◦ Gravity between you and the Sun Newton’s Law of Universal Gravitation Example 6-2: Spacecraft at 2rE. What is the force of gravity between two 3 kg blocks that are placed 4 meters apart? What is the force of gravity acting on a Gm1m2 (6.67 x1011)(3Kg )(3Kg ) Fg    3.75 x1011 N 2000-kg spacecraft when it orbits two d2 42 Earth radii from the Earth’s center (thatHow much is the force of gravity from the Earth acting on a 90 Kg man? is, a distance rE = 6380 km above the(Mass of the Earth = 6.0 x 1024 Kg ; radius of the Earth = 6400 km) Earth’s surface)? The mass of the Earth is mE = 5.98 x 1024 kg. Gm1m2 (6.67 x1011)(90 Kg )(6.0 x1024 Kg ) Fg    879 N d2 (6400000) 2 3
  4. 4. 4/8/2012 Newton’s Law of Universal Gravitation  Gravitational acceleration on the moon and Example 6-3: Force on the Moon. nine planets: PLANET GRAVITATIONAL ACCELERATION Find the net force on the Moon (m/s2) (m M = 7.35 x 1022 kg) due to the Mercury 3.7 gravitational attraction of both the Venus 8.9 Earth (m E = 5.98 x 1024 kg) and the Earth 9.8 Sun (m S = 1.99 x 1030 kg), Moon 1.6 assuming they are at right angles Mars 3.7 to each other. Jupiter 26 Saturn 12 Uranus 11 Neptune 12 Pluto 2 Joe Averages mass is same everywherein the universe, but his weight at various places is not the same  For a long time, most scientists thought all satellites travel in perfectly circular orbits ◦ NOT TRUE  Using the circular orbit theory… they could not make accurate predicts of their motion  Planets, moons, etc. were not where they were supposed to be! ◦ Planets did not follow these predicted paths ◦ So something must be wrong ◦ Then…… along came Johannes Kepler First Law An Imaginary line from the sun to a planet 1) The paths of the planets are ellipses, with sweeps out equal areas in equal time intervals. the sun at one focus (the other focus is just a point in space) This means planets move faster when they are closer to the sun and slower when they are further away 4
  5. 5. 4/8/2012 (1) V 2 = G m2 r  The square of the ratio of the periods of any (2) V2 r= G m2 two planets revolving about the sun is equal (3) V= ωr to the cube of the ratio of their average (4) ω= 2πf distances from the sun. Thus, if Ta and Tb (5) T = 1/f are the planets periods, and ra and rb are their average distances from the sun we get (6) ω = 2π / T (7) (ωr)2r = Gm2  (Ta/Tb)2 =(ra/rb)3 (8) ω2 r3 = Gm2 (9) ( 2π / T)2 r3 = Gm2 (10) 4π2 r3 / T2 = Gm2 (11) T2 / r3 = 4π2/ Gm2 = a constant Kepler’s Third T2 / r3 = a constant Law  A satellite does not fall because it is moving, being given a tangential velocity by the rocket that launched it. It does not travel off in a straight line because Earth’s gravity pulls it toward the Earth.  The tangential speed of an object in a circular orbit is given by: GM v E Velocity Velocity r increases decreases (perigee) (apogee)  If the period of the orbit is known, the velocity may be determined using: 2  r v TEqual Areas  The period of a satellite can be determined by: In 2  rEqual Times T vSpeedof a a Satellite:Speed of Satellite: Mem = mv2Fg = G _________ ____ r Velocity Acceleration Vectors Vectors r² Gme v r Force Vectors 5
  6. 6. 4/8/2012 Centripetal force Inertial position Earth Centripetal force Centripetal force Gravity supplies centripetal force inward towards A satellite is a projectile shot from a very high elevation the center of the and is in free fall about the Earth. circular path Centripetal force Centripetal force Force of gravity Fc Fg 5000 km Fg = Fc F g = G m1 m2 Re r = re + h r2 me = 6 x 10 24 kg Planet Fc = m v2 r Re = 6.4 x 10 6 m a. b. Gm2 Gme v2  , v  2r 2r r r v T  G m1 m2 = m1 v 2 11 T v (6.67 10 )(6.0 10 ) 24 r2 r v 2 (11.4 106 ) Canceling m1 & r on both sides (6.4 106  5.0 106 T 5900 V 2 = G m2 v  5900 m / s r T  1.2 10 4 s  3.4 hrAt what speed must a spacecraft be injected  What is gravity in outer space?into orbit if it is to circle the Earth at treetop  Where space shuttle orbits…g = 8.7m/s 2height?  How come astronautsGiven: rearth= 6.4×106 m, mearth= 6.0×1024 kg are “floating” then? Gm v  g = F/m r 11 (6.67 10 Nm / kg )(6.0 10 kg ) 2 2 24 v 6.4 106 m v  7.9 103 ms 1 6
  7. 7. 4/8/2012Weight: Weightlessness: force an object exerts against a supporting  no support force, as in surface free fall Examples: • standing on a scale in an elevator accelerating Example: Astronauts in downward, less compression in scale springs; orbit are without support weight is less forces and are in a • standing on a scale in an elevator accelerating continual state of upward, more compression in scale springs; weightlessness. weight is greater • at constant speed in an elevator, no change in weight How you feel weight, is different than your actual weight. As long as you are near the surface of the Earth you will always have the same weight but you may “feel” like you have a different weight This can happen if you are accelerating up or down Imagine an elevator…….. 7

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