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This lecture covers topics such as the universal law of gravitation, gravitational potential energy, and stable circular orbits of gravitational attracted objects. It also discusses the physical basis for stable circular orbits and the potential energy associated with gravitational force.

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Announcements

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  1. Announcements • Second exam scheduled for Oct. 28th -- practice exams now available -- http://www.wfu.edu/~natalie/f03phy113/extrapractice/ • Thursday – review of Chapters 9-14 • Today’s lecture – • Universal law of gravitation • Gravity near the planet’s surface • Gravitational potential energy • Planetary and satelite motion PHY 113 -- Lecture 14

  2. Newton’s law of gravitation: m2attracts m1 according to: y m1 r2-r1 G=6.67 x 10-11 N m2/kg2 m2 r1 r2 x PHY 113 -- Lecture 14

  3. Vector nature of Gravitational law: y m3 d x m1 m2 d PHY 113 -- Lecture 14

  4. Gravitational force of the Earth RE m PHY 113 -- Lecture 14

  5. Question: Suppose you are flying in an airplane at an altitude of 35000ft~11km above the Earth’s surface. What is the acceleration due to Earth’s gravity? a/g = 0.997 PHY 113 -- Lecture 14

  6. Attraction of moon to the Earth: Acceleration of moon toward the Earth: F = MM a  a = 1.99x2020 N/7.36x1022kg =0.0027 m/s2 PHY 113 -- Lecture 14

  7. REM F Stable circular orbit of two gravitationally attracted objects (such as the moon and the Earth) v a PHY 113 -- Lecture 14

  8. Peer instruction question In the previous discussion, we saw how the moon orbits the Earth in a stable circular orbit because of the radial gravitational attraction of the moon and Newton’s second law: F=ma, where a is the centripetal acceleration of the moon in its circular orbit. Is this the same mechanism which stabilizes airplane travel? Assume that a typical cruising altitude of an airplane is 11 km above the Earth’s surface and that the Earth’s radius is 6370 km. (a) Yes (b) No PHY 113 -- Lecture 14

  9. Stable (??) circular orbit of two gravitationally attracted objects (such as the airplane and the Earth) X REa F v a PHY 113 -- Lecture 14

  10. RE m REM F v a MM Newton’s law of gravitation: Earth’s gravity: Stable circular orbits of gravitational attracted objects: PHY 113 -- Lecture 14

  11. More details If we examine the circular orbit more carefully, we find that the correct analysis is that the stable circular orbit of two gravitationally attracted masses is about their center of mass. x PHY 113 -- Lecture 14

  12. m2 R2 R1 m1 Radial forces on m1: T2 ? Tangential forces ? PHY 113 -- Lecture 14

  13. m2 R2 R1 m1 Peer instruction question What is the relationship between the periods T1 and T2 of the two gravitationally attracted objects rotating about their center of mass? (Assume that m1 < m2.) (A) T1=T2 (B) T1<T2 (C) T1>T2 PHY 113 -- Lecture 14

  14. m2 R2 R1 m1 PHY 113 -- Lecture 14

  15. What is the physical basis for stable circular orbits? • Newton’s second law? • Conservation of mechanical energy? E = K + U = (const) • Conservation of linear momentum? p = (const) • Torqued motion? t = I a ? • Conservation of angular momentum? L = (const) PHY 113 -- Lecture 14

  16. m2 R2 R1 m1 v2 L1=m1v1R1 L2=m2v2R2 v1 Question: How are the magnitudes of L1 and L2 related? PHY 113 -- Lecture 14

  17. The potential energy associated with the gravitational force. r PHY 113 -- Lecture 14

  18. Total mechanical energy for circular orbits: (assume M >> m) E2 E1 U(J) h/RE PHY 113 -- Lecture 14

  19. Peer instruction question • What is wrong with the previous analysis? • Nothing is wrong. (The description of circular motion due to gravitational attraction is complete.) • E depends on r and therefore must not be constant. • E can only be constant if r is constant, but it is not obvious why r is constant. • Conservation of angular moment will come to the rescue. PHY 113 -- Lecture 14

  20. REM F v a MM Angular momentum: L = r x mv For circular orbit: L = RMEMMv PHY 113 -- Lecture 14

  21. E for elliptical orbit r E for circular orbit U(r) PHY 113 -- Lecture 14

  22. Circular orbit: y x Elliptical orbit: y x PHY 113 -- Lecture 14

  23. Satellites orbiting earth (approximately circular orbits): RE ~ 6370 km Examples: *Link: http://liftoff.msfc.nasa.gov/temp/StationLoc.html PHY 113 -- Lecture 14

  24. Sample question: Suppose that the space shuttle (m=105kg) was initially in the same orbit as the International space station (hi=390km) and the engines are fired to give it exactly the amount of energy DW to raise it to the same orbit as the Hubble space telescope (hf= 600km). What is the energy DW? You can show that the energy of a satellite of mass m in a circular orbit of height h above the Earth’s surface is given by: PHY 113 -- Lecture 14

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