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Physics 151: Lecture 27 Today’s Agenda Today’s Topic Gravity Planetary motion Physics 151: Lecture 27, Pg 1 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 Physics 151: Lecture 27, Pg 2 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 Physics 151: Lecture 27, Pg 3 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). Physics 151: Lecture 27, Pg 4 Moon rotating about the Earth : Calculate angular velocity : w = v / REM = 2 REM / T REM = 2 / T w= 1 rot 1 day rad x x 2 2 .66 x10 6 s -1 27 .3 day 86400 s rot So w = 2.66 x 10-6 s-1. Now calculate the acceleration. a = w2R = 0.00272 m/s2 = .000278 g Physics 151: Lecture 27, Pg 5 See text: 14.1 Gravitation (Courtesy of Newton) Newton found that amoon / g = .000278 and noticed that RE2 / R2 = .000273 amoon g R RE This inspired him to propose the Universal Law of Gravitation: |FMm |= GMm / R2 G = 6.67 x 10 -11 m3 kg-1 s-2 Physics 151: Lecture 27, Pg 6 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: m1m2 F12 G 2 R12 The direction of F12 is attractive, and lies along the line connecting the centers of the masses. m1 F12 F21 m2 R12 Physics 151: Lecture 27, Pg 7 Gravity... Compact objects: R12 measures distance between objects R12 Extended objects: R12 measures distance between centers R12 Physics 151: Lecture 27, Pg 8 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. h m Fg M Em Fg G 2 RE M RE Physics 151: Lecture 27, Pg 9 See text: 14.3 Gravity... Near the earth’s surface... Fg G So |Fg| = mg = ma a=g ME M Em m G 2 2 RE RE =g All objects accelerate with acceleration g, regardless of their mass! Where: g G M E 9.81 m / s 2 RE2 Physics 151: Lecture 27, Pg 10 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 Physics 151: Lecture 27, Pg 11 Lecture 27, Act 1 Force 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) 0.75 W (b) 1.5 W X (c) 2.25 W E Physics 151: Lecture 27, Pg 12 Lecture 27, Act 1 Solution The gravitational force on a person of mass m by another object (for instance a planet) having mass M is given by: Mm F G 2 R WX FX Ratio of weights = ratio of forces: WE FE MXm G 2 2 RX M X RE M Em M E RX G 2 RE 2 WX 1 300 .75 WE 20 (A) Physics 151: Lecture 27, Pg 13 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. Physics 151: Lecture 27, Pg 14 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. Physics 151: Lecture 27, Pg 15 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. F G G M smp M smp R 2 R 2 m pv2 R m p (2R / T ) 2 R 2 3 4 2 R T GM s Physics 151: Lecture 27, Pg 16 See text: 14.4 Kepler’s Second Law This one is really a statement of conservation of angular momentum. M smp R F Rrˆ G rˆ 2 R G M smp R rˆ rˆ 0 L R p mP R v Constant Physics 151: Lecture 27, Pg 17 See text: 14.4 Kepler’s Second Law L R p mP R v Constant dA R dR 2. The radius vector drawn from the sun to a planet sweeps out equal areas in equal times. dA Constant dt Physics 151: Lecture 27, Pg 18 See text: 14.4 Kepler’s Second Law L R p mP R v Constant dA R dR 1 1 1 L dA R dR R v dt dt 2 2 2 M dA L Constant dt 2M Physics 151: Lecture 27, Pg 19 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. Mm F G r2 r2 W F(r)dr G r1 r1 s p R2 M sm p r 2 dr U 1 1 U U f Ui W GMsm p ( ) rf ri Define ri as infinity U GM s m p RE r 0 U 1 r r 151: Lecture 27, Pg 20 Physics See text: 14.7 Energy of a Satellite A planet, or a satellite, also has kinetic energy. 1 2 GM s m p E K U mv 2 r We can solve for v using Newton’s Laws, GM s m p r2 mv 2 ma r Plugging in and solving, GM s m p GM s m p GM s m p E 2r r 2r Physics 151: Lecture 27, Pg 21 Recap of today’s lecture Chapter 13 Gravity Planetary motion Physics 151: Lecture 27, Pg 22