Sir Isaac and Universal Gravitation. ... so what really happened?

Transcription

1 Sir Isaac and Universal Gravitation... so what really happened? Apr 11 11:51 AM For the Aztecs, who lived in central Mexico, Tonatiuh was a Sun god. Aztecs believed that four suns had been created in four previous ages, and all of them had died at the end of each cosmic era. Tonatiuh was the fifth sun and the present era is still his. The carvings on this sunstone represent the four cycles of creation and destruction in the Aztec creation story. The skull at the center depicts the god Tonatiuh. Dec 1 9:55 AM 1

2 The artist, Peter Paul Rubens, painted this image, Neptune Calming the Tempest, in 165. Rubens is known for the style of painting termed, "baroque". Rubens ( ) combined the tradition of Flemish painting with the themes of the Renaissance Dec 1 10:40 AM Dec 1 10:41 AM

3 The Story Thus Far! Claudius Ptolemaeus Second Century AD Jan 5 7:7 AM The First Steps! Copernicus: The Foundation Nicholas Copernicus (Polish, ): Proposed the first modern heliocentric model, motivated by inaccuracies of the Ptolemaic model and based on aesthetic principles only indirectly on evidence. Features of the model: It was still based on circles and epicycles, and was not more accurate than the Ptolemaic model; However, it allowed the calculation of distances to planets in AU, and provided the correct explanation for retrograde motion. What happened: Copernicus' model was not widely read or accepted right away, because it is difficult to change established ideas (although it did spark debates, and eventually condemnation by the Catholic church), because of its limitations, and because it did not address some of the better arguments in favor of geocentric models. Jan 5 7:0 AM

4 The Data man! Tycho Brahe: The Data Tycho Brahe (Danish, ): The greatest pretelescope astronomer; Became famous after observing a (super)nova in 157 and a comet in 1577, and proving that they were farther than the Moon: Heavens can change! Was given an island on which to build his observatory. His Solar System model: Believed in a mixed model, with the Sun and Moon orbiting the Earth (no stellar parallax!); Few people ever believed in it. The observations: He made the best and most systematic naked eye observations up to that time, accurate to 1', kept extensive detailed records of them,... and hired Johannes Kepler as his assistant. Jan 5 7:0 AM Tycho Brahe Jan 5 6:46 AM 4

5 Crunching the Data! Kepler: From Observation to Model of the Solar System Johannes Kepler (German, ): Started trying to explain Tycho's data on Mars; Ended up developing the empirical model we still accept today for the solar system (basically). Laws of planetary motion: Three simple laws, 1. Ellipses: All planets move along ellipses, with the Sun at one focus;. Areas: Planets sweep out equal areas around the Sun in equal times (they move faster when closer to the Sun);. Periods: The period is proportional to the distance. [Verification: Helped by 161 observations of a Mercury transit, and 1655 observations by Giandomenico Cassini of the Sun.] Are there other laws? Is there a pattern in the spacings or periods among the planets? Not the way Kepler hoped for, but there are resonances. How good are these laws? We now know that they need small corrections, but they hold for any planetary system, any moon around a planet. Jan 5 7:06 AM The Keen Eye! Galileo: Observations and Response to Aristotle Galileo Galilei (Italian, ): Introduced the concept of inertia, with which he could address Aristotle's objection to a moving Earth. Observations: The first to use telescopes in astronomy (1609) and publish his results; Saw stars in Milky Way (so stars can be so distant that they don't show parallax), features on Moon and Sun (so not all heavenly bodies are perfect), four "little stars" around Jupiter (so another body and a moving one at that can have orbiting moons), phases of Venus (a complete set); Why is this important? Ideas: He supported the Copernican Model, but was forced to recant. Thought that planets are "worlds," not just dots of light. Other observations: He also saw that Saturn sometimes has things sticking out from its sides (like ears), and looked for stellar parallax in Mizar Jan 5 7:09 AM 5

6 The Tools of Observations! Jan 5 6:59 AM So, How does this System Work! Jan 7:06 AM 6

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8 91.5 million miles 94.5 million miles Jan 5 6:49 AM Kepler's rd Law...the ratio of the orbital radius cubed to the period squared of all inferior objects to a common superior object is constant. r o T = K What do the planets circle? the Sun, the Sun is Superior and the planets Inferior Because the rd Law is a ratio you do not have to use standard units! Jan 5 9:4 AM 8

9 earth r T = k (1.5 x ) (1 yr) = k =.4 x 10 m /yr Jupiter (7.78 x ) (11.8 yr) = k =.4 x 10 m /yr Jan 19 7:56 AM r oe T e = r oj T J earth Jupiter (1.5 x ) (1 yr) = T J = 11.8 yr (7.78 x ) T J Jupiter is 5.19 x further from the Sun than the Earth is... r oj = 5.19 r oe earth Jupiter (1 r oe ) (1 yr) = (5.19 r oe ) T J T J = 11.8 yr earth Jupiter (1.5 x ) (65.5 da) = (7.78 x ) T J T J = 410 da 11.8 yrs Jan 19 8:0 AM 9

10 Sun r om = orbital radius of Mercury Jan 1: PM 1) Mercury, Venus, and the Earth all circle the Sun ), they all that the same ratio of r /T ) Compare Mercury and Venus to the Earth Venus: r ov = 1.08 x m r oe = r ov T e T v Mercury: r om = 5.79 x m r oe = r om T e T m (1.50 x m) = (1.08 x m) (1 yr) (T v ) (1.50 x m) = ( x 10 m) (1 yr) (T m ) T v = yr or, days T m =?.4 yrs Jan 1: PM 10

11 r oe = r op T e Tp What is the period of Pluto? r oe = 1.5 x m r op = 5.91 x 10 1 m Pluto is 9.4 x further from the Sun than the Earth is... or, 9.4 x r oe, or 9.4 r oe T e is one year 1r oe = 9.4 r oe 1 yr T p T p = (9.4 r oe) x 1 yr 1r oe T p = 47 yrs Jan 11 11:0 AM Where do you put a satellite if you want it to circle the earth once a day? r os =? question: What does the satellite circle? answer: the earth! question: What else circles the earth? answer: the moon ro the of the moon and satellite T = K and moon have to be the same! moon: rom =.9 x 10 8 m Tm = 7. days Superior inferior r om = r os T m T s ros from surface (.9 x 10 8 ) = r os (7. da) (1 da) r os = 4. x 10 7 m How far above the earth's surface is that? Jan 11 11:0 AM 11

12 Jan 8 10:7 AM m/s Jan 8 10:9 AM 1

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14 Jan 11:0 AM Law of Universal Gravitation Every object in the universe attracts every other object in the universe with a force that directly proportionate to the product of the masses and inversely proportionate to with the square of the distance between Jan 5 6:41 AM 14

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17 Gravity pulls the milkman to the earth. We call this pull (force) the milkman's weight. So, the force of gravity on an object is its weight. Or, F = Gm 1 m /r and F w = mg F = F w the "m" is the mass of the or, inferior object "m " Gm 1 m /r = m g g = Gm 1 /r F w = m g Jan 5 6:51 AM g = Gm 1 /r You can find acceleration (g) due to gravity on any object if know its mass and radius. or, If you can measure "g" and know the radius you can find its mass! Example: You're on the moon and with your trusty 1.00 m pendulum you measure its period to be 4.97 s. What is the mass of the moon? a = 4π r/t a = 4π (1.00 m)/(4.97 s) a = 1.60 m/s g = Gm 1 /r m 1 = gr /G m 1 = 1.60 m/s /(1.785 x 10 6 m) /6.67 x Nm /kg m 1 = 7.64 x 10 kg Jan 7 7:1 AM 17

18 Gravity is the force that causes the moon to circle the earth. That means that gravity is acting as a centripetal force. F = Gm 1 m /r F c = mv /r F = Gm 1 m /r F c = m v /r F = F c Gm 1 m /r = m v /r v = Gm 1 /r the "m" is the mass of the or, inferior object "m " F c = m v /r note that the velocity an object orbits at in determined only by the mass and distance from the superior object. Orbiting velocity has nothing to do with the inferior objects mass! Jan 7 7: AM v = Gm 1 /r Example: What velocity does the moon orbit the earth with? r om =.9 x 10 8 m m 1 = 5.98 x 10 4 kg (earth is the superior object) v = Gm 1 /r v = 6.67 x Nm /kg (5.98 x 10 4 kg)/.9 x 10 8 kg v = 1010 m/s Jan 8 7:5 AM 18

19 Gravity is the force that causes the moon to circle the earth. That means that gravity is acting as a centripetal force. F = F Gm = Gm 1 m /r F c = mv 1 m /r F c = m4π r/t /r the "m" is the mass of the or, inferior object "m " F = Gm 1 m /r and F c = m 4π r/t F = F c F c = m 4π r/t Gm 1 m /r = m 4π r/t multiply by "r " and divide by m divide both sides by "4π " r /T = Gm 1 /4π Jan 7 7:8 AM r /T = Gm 1 /4π Example: What is the period of the moon? r om =.9 x 10 8 m m 1 = 5.98 x 10 4 kg (the earth is superior) r /T = Gm 1 /4π T m = 4π r om /Gm e T m = [4π (.9 x 10 8 m) ]/[(6.67 x nm /kg ) 5.98 x 10 4 kg] T m =.4 x 10 6 sec. = 8 days Jan 8 8:6 AM 19

20 Attachments Universal Gravitation.pptx