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Discuss the law of universal gravitation and satellite motion.
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  1. 1. Copyright Sautter 2015
  2. 2. Gravitation • The Law of Universal Gravitation is based on the observed fact that all masses attract all other masses. The force of attraction decreases as the distance between the masses increases. • This relationship is called an inverse square law since the decrease in attraction between objects is relative to the square of that distance. • If the distance between the masses doubles, the force of gravitational attraction becomes ¼ of the original force. If the distance triples, the force becomes 1/9 of the original, as so on. • For example, 22 = 4, 32 = 9, etc. 2
  3. 3. Weight & Mass • Weight and mass measure different things. • Mass measures the quantity of matter which is present. It represents the inertia property of matter meaning its ability of resist changes in motion. Mass is measured in grams, kilograms or slugs. • Weight is a force resulting from the effect of gravity on a mass. A mass without gravity is weightless. Weight is measured in dynes, newtons or pounds. • Gravity on the Earth’s surface is measured as – 980 cm/s2, - 9.8 m/s2 or – 32 ft/s2. (The negative sign means that gravity always acts downward) • As we move above the Earth’s surface or to other planets, the strength of the gravitational field changes and so does the weight. 3
  4. 4. Planet Force of gravity Force of gravity (weight) at the Earth’s surface Fearth = G m1 me re 2 Force of gravity (weight) at a point (P) above the Earth’s surface Fpoint P = G m1 me rp 2 w = m1g = G m1 me r2 g = G me r2 g r2 = Gme gearth r2 earth = Gmearth gpoint p r2 point P = Gmearth Therefore gearth r2 earth =gpoint p r2 point P 4
  5. 5. Radius of Earth = 4000 miles scale150 lbs Two Radius of Earth = 8000 miles scale37.5 lbs Three Radius of Earth = 12000 miles scale16.7 lbs ¼ wt 1/9 wt Normal wt 5
  6. 6. scale 150 lbs scale 25.6 lbs scale 406 lbs g = 9.81 m/s2 g = 1.67 m/s2 g = 26.6 m/s2 6
  7. 7. Satellites • When satellites orbit a planet the force which supplies the centripetal force, and thereby the circular motion, is the pull of gravity of the planet which is orbited. • Without the force of gravity, the satellite would move in a straight line due to inertia. • When the satellite orbits, the force of gravity must equal the centripetal force. If the force of gravity exceeded the centripetal force the satellite would spiral into the planet. If the centripetal force exceeded the force of gravity, the satellite would seek a wider orbit or move off in a straight line. 7
  8. 8. Velocity Vectors Acceleration Vectors Force Vectors 8
  9. 9. Earth A satellite is a projectile shot from a very high elevation and is in free fall about the Earth. 9
  10. 10. Inertial position Centripetal force Centripetal force Centripetal force Centripetal force Gravity supplies centripetal force inward towards the center of the circular path 10
  11. 11. Fg = gravity force between m1 and m2 separated by a distance r G is the Universal Gravitational Constant The weight of an object is its mass times g’, the gravity value at location r 11
  12. 12. Planet Force of gravity Fg Centripetal force Fc Fg = Fc Fg = G m1 m2 r2 Fc = m v2 r G m1 m2 = m1 v2 r2 r Canceling m1 & r on both sides V2 = G m2 r 12
  13. 13. (1) V2 = G m2 r (2) V2 r= G m2 (3) V = ωr (4) ω= 2πf (5) T = 1/f (6) ω = 2π / T (7) (ωr)2r = Gm2 (8) ω2 r3 = Gm2 (9) ( 2π / T)2 r3 = Gm2 (10) 4π2 r3 / T2 = Gm2 (11) T2 / r3 = 4π2/ Gm2 = a constant T2 / r3 = a constant Kepler’s Third Law 13
  14. 14. Kepler’s Laws of Satellite Motion • (1) Satellites travel in elliptical paths. (The Earth and the inner planets as well as the moon travel in nearly circular orbits. The orbits of the outer planets are more ellipsoid. Comets orbits are very elliptical.) • (2) Areas swept out in equal times are equal even though the speed of the satellite varies. Satellite velocity is least when it is furthest from the central body (apogee) and greatest when it is nearest (perigee). • (3) The period of motion squared divided by the average orbital radius cubed gives a constant for all satellites orbiting the same body. ( T2 1/ r3 1 = T2 2/ r3 2) 14
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