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Physics 151: Lecture 27 Today’s Agenda

This lecture discusses the concept of gravity, its role in planetary motion, Sir Isaac Newton's laws of motion, and Kepler's laws of planetary motion. It covers topics such as angular velocity, acceleration, and the Universal Law of Gravitation.

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Physics 151: Lecture 27 Today’s Agenda

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  1. Physics 151: Lecture 27 Today’s Agenda • Today’s Topic • Gravity • Planetary motion

  2. See text: 14 New Topic - Gravity • Sir Isaac developed his laws of motion largely to explain observations that had already been made of planetary motion. Earth Sun Moon Note : Not to scale

  3. See text: 14.1 Gravitation(Courtesy of Newton) Things Newton Knew, • 1. The moon rotated about the earth with a period of ~28 days. • 2. Uniform circular motion says, a = w2R • 4. Acceleration due to gravity at the surface of the earth is g ~ 10 m/s2 • 5. RE = 6.37 x 106 • 6. REM = 3.8 x 108 m

  4. See text: 14.1 Gravitation(Courtesy of Newton) Things Newton Figured out, • 1. The same thing that causes an apple to fall from a tree to the ground is what causes the moon to circle around the earth rather than fly off into space. (i.e. the force accelerating the apple provides centripetal force for the moon) • 2. Second Law, F = ma So,acceleration of the apple (g) should have some relation to the centripetal acceleration of the moon (v2/REM).

  5. Moon rotating about the Earth : • Calculate angular velocity : • So  = 2.66 x 10-6 s-1. • Now calculate the acceleration. • a = 2R= 0.00272 m/s2 = .000278 g  = v / REM = 2 p REM / T REM = 2 p / T  =

  6. See text: 14.1 Gravitation(Courtesy of Newton) • Newton found that amoon/ g= .000278 • and noticed that RE2 / R2= .000273 • This inspired him to propose the Universal Law of Gravitation: |FMm|= GMm / R2 amoon g R RE G = 6.67 x 10 -11 m3 kg-1 s-2

  7. See text: 14.1 Gravity... • The magnitude of the gravitational force F12 exerted on an object having mass m1 by another object having mass m2 a distance R12 away is: • The direction of F12 is attractive, and lies along the line connecting the centers of the masses. m1 m2 F12 F21 R12

  8. Gravity... • Compact objects: • R12 measures distance between objects • Extended objects: • R12 measures distance between centers R12 R12

  9. See text: 14.1 Gravity... • Near the earth’s surface: • R12 = RE • Won’t change much if we stay near the earth's surface. • i.e. since RE >> h, RE + h ~ RE. m Fg h M RE

  10. See text: 14.3 Gravity... • Near the earth’s surface...  =g • So |Fg|= mg=ma • a = g All objects accelerate with acceleration g, regardless of their mass! Where:

  11. Example gravity problem: • What is the force of gravity exerted by the earth on a typical physics student? • Typical student mass m = 55kg • g = 9.8 m/s2. • Fg = mg = (55 kg)x(9.8 m/s2 ) • Fg= 539 N Fg • The force that gravity exerts on any object is called its Weight W= 539 N

  12. Lecture 27, Act 1Force and acceleration • Suppose you are standing on a bathroom scale in Physics 203 and it says that your weight is W. What will the same scale say your weight is on the surface of the mysterious Planet X ? • You are told that RX ~ 20 REarth and MX ~ 300 MEarth. (a)0.75W (b)1.5 W(c)2.25 W X E

  13. Ratio of weights = ratio of forces: Lecture 27, Act 1Solution • The gravitational force on a person of mass m by another object (for instance a planet) having mass M is given by: (A)

  14. See text: 14.3 Kepler’s Laws • Much of Sir Isaac’s motivation to deduce the laws of gravity was to explain Kepler’s laws of the motions of the planets about our sun. • Ptolemy, a Greek in Roman times, famously described a model that said all planets and stars orbit about the earth. This was believed for a long time. • Copernicus (1543) said no, the planets orbit in circles about the sun. • Brahe (~1600) measured the motions of all of the planets and 777 stars (ouch !) • Kepler, his student, tried to organize all of this. He came up with his famous three laws of planetary motion.

  15. See text: 14.4 Kepler’s Laws 1st All planets move in elliptical orbits with the sun at one focal point. 2nd The radius vector drawn from the sun to a planet sweeps out equal areas in equal times. 3rd The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit. • It was later shown that all three of these laws are a result of Newton’s laws of gravity and motion.

  16. See text: 14.4 Kepler’s Third Law Let’s start with Newton’s law of gravity and take the special case of a circular orbit. This is pretty good for most planets.

  17. See text: 14.4 Kepler’s Second Law This one is really a statement of conservation of angular momentum.

  18. See text: 14.4 Kepler’s Second Law dA R dR 2. The radius vector drawn from the sun to a planet sweeps out equal areas in equal times.

  19. See text: 14.4 Kepler’s Second Law dA R dR

  20. U RE r 0 See text: 14.7 Energy of Planetary Motion A planet, or a satellite, in orbit has some energy associated with that motion. Let’s consider the potential energy due to gravity in general. Define ri as infinity

  21. We can solve for v using Newton’s Laws, Plugging in and solving, See text: 14.7 Energy of a Satellite A planet, or a satellite, also has kinetic energy.

  22. Recap of today’s lecture • Chapter 13 • Gravity • Planetary motion

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