Universal Gravitation and Kepler’s Laws

1 Universal Gravitation and Kepler’s LawsPhysics 12 ...
Author: Ross Robbins
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1 Universal Gravitation and Kepler’s LawsPhysics 12

2 Newton’s Universal Law of Gravitation and Kepler’s First LawNewton had shown in his original article De Motu and later in Principia that the inverse square nature of gravity would lead to elliptical not circular orbits

3 Newton’s Universal Law of Gravitation and Kepler’s Second LawEven though planets are moving in elliptical orbits, the concepts of circular motion mostly apply Determine the speed of the Earth using the following data and assume that centripetal force is equal to gravitational force: Sun’s Mass – 1.99x1030kg Earth’s Mass – 5.98x1024kg Distance Earth to Sun (aphelion) – 152,171,522 km Distance Earth to Sun (perihelion) – 147,166,462 km

4 Newton’s Universal Law of Gravitation and Kepler’s Second LawEarth’s speed: Aphelion – 2.95x104m/s Perihelion – 3.00x104m/s This would indicate that Kepler’s Second Law is also supported by Newton’s Universal Law of Gravitation

5 Newton’s Universal Law of Gravitation and Kepler’s Third LawSince Kepler’s Third Law is a ratio of the orbital radius cubed to the orbital period squared, we should be able to apply Newton’s Universal Law of Gravitation to the planetary motion to determine the value of the constant Set the centripetal force equation equal to Newton’s Universal Law of Gravitation Replace the velocity expression using orbital radius and orbital period

6 Newton’s Universal Law of Gravitation and Kepler’s Third LawThis is called Newton’s version of Kepler’s Third Law

7 Mass of the Sun and PlanetsHenry Cavendish developed a torsion balance to determine the value of the Universal Gravitation Constant (approximately 70 years after Newton’s death) Using his apparatus, he was able to determine a value of G = 6.75x10-11Nm2/kg2 which is within 1% of the currently accepted answer Using G, he was then able to determine the mass of the Sun and other planets

8 Experiment results

9 Data Outline Little bounces tell me how stiff the hanging cord isDistance between flat lines tells me how much the cord/spring was displaced Knowing stiffness and displacement from equilibrium you can calculate force

10 The Hitch Hiker's Guide to the Galaxy“There is an art, it says, or rather a knack to flying. The knack lies in learning how to throw yourself at the ground and miss. Pick a nice day, it suggests, and try it. The first part is easy. All it requires is simply the ability to throw yourself forward with all your weight, and the willingness not to mind that it's going to hurt. That is, it's going to hurt if you fail to miss the ground. Most people fail to miss the ground, and if they are really trying properly, the likelihood is that they will fail to miss it fairly hard. Clearly, it's the second point, the missing, which presents the difficulties.”

11 Hitchhikers Guide to the GalaxyDouglas Adams loved the sciences One of the most ingenious description of flight I’ve heard Oddly enough, it is also one of the best

12 Newton’s Mountain Shortly after developing the Universal Law of Gravitation, Newton began a series of thought experiment involving artificial satellites Newton’s thought was that if you had a tall enough mountain and launched a cannonball fast enough horizontally, it would fall towards the Earth at the same rate the Earth would “fall” away This would result in the cannonball orbiting the Earth

13 Newton’s Mountain How far does it fall? How far does it go?

14 Newton’s Mountain How far doe the earth “fall” away over 8km?Earth’s radius is 6400km

15 Satellite If launched at the right velocity, then an object can continuously fall and never hit the earth Lets consider gravity as a centripetal force again… this time consider the force between Earth and a satellite Everything orbiting at the same radius must be travelling at the same speed

16 More on Satellites At this speed assuming the orbit is circular (or close to) we can say the following

17 Weightlessness The idea of weightlessness should mean no gravity, only that you feel no acceleration due to gravity. “zero g” All object fall at the same rate, so they do not fall relative to each other and seem to go in straight lines

18 Weightlessness “up” is opposite gravity No net force means no “up”

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20 “Vomit Comet” Anything inside an object in free fall will appear as though there is no gravity, but that is only relative to the container in free fall. This is regularly replicated by using air craft

21 Gravitational Energy Before you have defined the gravitational potential energy as Using this idea of a gravitational field, we can refine this this to be accurate over a larger range Note: negative sign Attractive force

22 Gravitational Energy This result is a statement of how much energy an object has relative to another object

23 More Gravitational EnergyAs before, potential energy can be converted into kinetic energy This is consistent with Keplers 2nd law “Equal area law” Fast when close, slower when farther out When you are close to the sun, some of your potential energy has converted to kinetic energy

24 Example If the earth travels at km/s at the perihelion, how fast does it travel at its Aphelion? Aphelion km Perihelion km

25 Escape velocity In order to leave the earth and go into space, you must give your craft at least as much energy as is necessary to overcome the gravitational energy. What is the escape velocity on earth?

26 Sun’s Influence? What is the magnitude of the force of gravity between the earth and the sun?

27 Example What is the net force on a Trojan asteroid?Mass of Jupiter = 1.90x10^(27)kg Mass of Sun = 1.99x10^(30)kg Orbital Radius of Jupiter = 7.78x10^11m

28 Truely Zero G Is there a place between the earth and the sun at which the net force on an object is zero? Where?

29 Legrange Points Interestingly, the last two examples belong to a family of points called “Legrange points” Special places with stable orbits involving gravity from 2 objects

30 Legrange points

31 Perturbation The effect of a small change in a system

32 The insight of Einsteinvia Alice and Bob

33 Space-Time Starting with the premise of the speed of light being constant in any inertial reference frame we can get “Special relativity” time and space can no longer be considered separate from each other: space-time Moving clocks tick slower Moving rulers are shorter Nothing can go faster than the speed of light

34 Space-Time If we include the idea that an inertial reference frame in a gravitational field appears to be accelerating we get “General Relativity” Clocks tick slower in gravitational fields All objects moving in a gravitational field are moving in a straight line but space itself is curved