Sir Isaac and Universal Gravitation... so what really happened?
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- Blaise Hines
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1 Sir Isaac and Universal Gravitation... so what really happened? okay, probably no! There's always been man made beliefs about what is! 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. Human sacrifice was required for him to move through the sky. 1
2 The Story Thus Far! Claudius Ptolemaeus Second Century AD Ptolemy ish Copernicus Brahe Galileo Kepler Newton
3 8.1 notes.notebook 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. He took as the basic starting points of his theory: The earth is not the center of the universe. The center of the universe is near the sun. The earth sun distance is negligible compared to the distance to the stars. Earth s rotation on its own axis accounts for the apparent daily rotation of the stars. The apparent annual cycle of movements of the sun is seen because the earth is orbiting it. The apparent retrograde motion of the planets is caused by the fact we observe them from a moving location, because the earth is orbiting the sun.
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. Tycho Brahe Danish At the time, astronomers held to the idea that the heavens were composed of separate individual spheres, with everything revolving around the Earth. In 157, Brahe observed a supernova in the constellation of Cassiopeia. Brighter than Venus, the new star remained visible for a year and a half. In 1577, he observed a comet. Current theory taught that both were disturbances in the atmosphere. However, Brahe's precise measurements revealed differently. He proved that the supernova never changed with regard to the surrounding stars, and that the comet orbited beyond the path of the moon, contradicting the idea that the heavens never changed. Although Brahe's observations revealed the flaws of the current system, he did not embrace Nicolaus Copernicus' newly proposed sun centered model. Instead, he offered a model that combined the two, setting the moon and sun in orbit around the Earth even as the other five known planets circled the sun. The model became popular among those who wanted to leave the older view behind but weren't ready to embrace the idea of the sun at the center of the solar system. tycho brahe biography.html 4
5 Summary of Brahe's Contributions Among the important contributions of Brahe: 1. He made the most precise observations that had yet been made by devising the best instruments available before the invention of the telescope.. His observations of planetary motion, particularly that of Mars, provided the crucial data for later astronomers like Kepler to construct our present model of the solar system.. He made observations of a supernova (literally: nova= "new star") in 157 (we now know that a supernova is an exploding star, not a new star). This was a "star" that appeared suddenly where none had been seen before, and was visible for about 18 months before fading from view. Since this clearly represented a change in the sky, prevailing opinion held that the supernova was not really a star but some local phenomenon in the atmosphere (remember: the heavens were supposed to be unchanging in the Aristotelian view). Brahe's meticulous observations showed that the supernova did not change positions with respect to the other stars (no parallax). Therefore, it was a real star, not a local object. This was early evidence against the immutable nature of the heavens, although Brahe did not interpret the absence of parallax for stars correctly, as we discuss below. 4. Brahe made careful observations of a comet in By measuring the parallax for the comet, he was able to show that the comet was further away than the Moon. This contradicted the teachings of Aristotle, who had held that comets were atmospheric phenomena ("gases burning in the atmosphere" was a common explanation among Aristotelians). As for the case of the supernova, comets represented an obvious change in a celestial sphere that was supposed to be unchanging; furthermore, it was very difficult to ascribe uniform circular motion to a comet. 5. He made the best measurements that had yet been made in the search for stellar parallax. Upon finding no parallax for the stars, he (correctly) concluded that either «the earth was motionless at the center of the Universe, or «the stars were so far away that their parallax was too small to measure. 6. Not for the only time in human thought, a great thinker formulated a pivotal question correctly, but then made the wrong choice of possible answers: Brahe did not believe that the stars could possibly be so far away and so concluded that the Earth was the center of the Universe and that Copernicus was wrong. 7. Brahe proposed a model of the Solar System that was intermediate between the Ptolemaic and Copernican models (it had the Earth at the center). It proved to be incorrect, but was the most widely accepted model of the Solar System for a time. Thus, Brahe's ideas about his data were not always correct, but the quality of the observations themselves was central to the development of modern astronomy. Mars appears to occasionally move backwards (retrograde motion) across the sky, causing many astronomers to suggest epicycles, tiny circles within their orbit. Even Copernicus' suggestion that the planets orbited the sun in circles could not account for the red planet's strange motion. Kepler, using Brahe's detailed observations, realized that the planets moved around the sun not in circles but in stretched out circles known as ellipses. However, the problem took him almost a decade to solve, and Kepler didn't publish it until well after Brahe's death. 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. interesting dude sun minute in tycho brahe biography.html 5
6 8.1 notes.notebook 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 Galileo Galileo.mp4 constructed a hydrostatic balance for measuring small objects. presented theories on motion and falling objects, and developed the universal law of acceleration, which all objects in the universe obeyed. supported the Copernican theory that the earth and planets revolved around the sun. learned about a simple telescope built by Dutch eyeglass makers, and he soon developed one of his own. published a small booklet, The Starry Messenger, revealing his discoveries that the moon was not flat and smooth, but a sphere with mountains and craters. found Venus had phases like the moon, proving it rotated around the sun. discovered Jupiter had revolving moons, which didn t revolve around the earth. refuted the Aristotelian explanation of why objects float in water, saying that it wasn t because of their flat shape, but instead the weight of the object in relation to the water it displaced published his observations of sunspots, which further refuted Aristotelian doctrine that the sun was perfect. inventions/ 6
7 8.1 notes.notebook sun size rs = 109 re earth re =.09 cm rs = 10 cm earth distances sun rs =.15 cm. earth roe = 16 rs The Tools of Observations! 7
8 So, How does this System Work! high 8
9 91.5 million miles 94.5 million miles 9
10 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! Sun r om = orbital radius of Mercury 10
11 AU of planets r 0 AU T p Mercury 5.79 x m Venus 1.08 x m Earth 1.50 x m 1 1 Mars.8 x m Jupiter 7.78 x m Saturn 1.4 x 10 1 m Uranus.87 x 10 1 m Neptune 4.54 x 10 1 m Pluto 5.91 x 10 1 m r oe T e earth r T = k = (1.5 x ) (1 yr) = k =.4 x 10 m /yr r oj T J Jupiter (7.78 x ) = (11.8 yr) = k =.4 x 10 m /yr 11
12 earth r oe T e = r oj (1.5 x ) (1 yr) T J = Jupiter 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 (1 r oe ) (1 yr) earth = Jupiter (5.19 r oj ) T J T J = (5.19 roj ) (T e ) /r oe in AU's Jupiter T J = 11.8 yr (1.5 x ) (65.5 da) = (7.78 x ) T J T J = 410 da 11.8 yrs (book) or: r oe T e = r oj T J T J T e = r oj r oe ( ( T J ( = r oj roe T e ( T e r oj ( T J = roe ( isolate for T j T J = 1 yr( ( ) T J = 11.8 yr ( 1
13 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 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 1
14 Where do you put a satellite if you want it to circle the earth once a day? geosynchronous r os =? question: What does the satellite circle? answer: the earth! question: What else circles the earth? answer: the moon ro the The moon and satellite T = K e ro have to be the same! moon: rom =.9 x 10 8 m Tm = 7. days T Superior inferior r om = r os T m T s ros from surface (.9 x 10 8 m) = r os (7. da) (1 da) r os = 4. x 10 7 m How far above the earth's surface is that? 4. x 10 7 m 6.8 x 10 6 m =.6 x 10 7 m.6 x 10 7 m ( 1 mile 1610 m ( =,000 miles 14
15 0.007 m/s forces act in pairs = and opposite if A acts on B then, B acts on A 15
16 superior object inferior object "G" Gravitational Constant "unit strength of gravity" G = 6.67 x N m /kg 16
17 8.1 notes.notebook 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 17
18 m = 90. kg F w = mg F w = 90.kg( 9.8 m/s ) = 88 N F e p = Gm 1 m /r F e p = Gm e m p /r F = 6.67 x Nm /kg (5.98 x 10 4 kg)(90 kg (6.8 x 10 6 kg) F p e = 88 N F e p = 88 N a p = ΣF/m = 88 N/90 kg = 9.8 m/s F p e = +88 N a e = ΣF/m = +88 N/5.98 x 10 4 kg = 1.5 x 10 m/s 18
19 F e p = 88 N F p e = +88 N What acceleration does my force give the earth? a e = ΣF/m = +88 N/5.98 x 10 4 kg = 1.5 x 10 m/s Hey buddy, let's go for a walk! I walk about.0 m/s. How long would my force on the earth take to get it up to my walking speed? a = 1.5 x 10 m/s v =.0 m/s t =? a = v/t t = v/a = [.0 m/s]/[1.5 x 10 m/s ] t = 1.0 x 10 s 4.1 x yr There are about 7.5 billion people on earth. Let's say the average mass of each is about 50 kg, that a total mass of about 4 x kg that's a force of 4 x 10 1 N F e ap = 4 x 10 1 N F ap e = +4 x 10 1 N What acceleration does this force give the earth? a e = ΣF/m = +4 x 10 1 N /5.98 x 10 4 kg = 6.7 x 10 1 m/s I we all walk about.0 m/s. How long would this force on the earth take to get it up to our walking speed? a = v/t a = 6.7 x 10 1 m/s v =.0 m/s t =? t = v/a = [.0 m/s]/[6.7 x 10 1 m/s ] t = x x 10 4 yr 19
20 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, We all have a job to do! F = Gm 1 m /r and F w = mg or, F = F w Gm 1 m /r = m g the "m" in the nd Law always the mass of the inferior object "m " (the object being accelerated by the force) F w = m g g = Gm 1 /r This is how we find the "g" on any planet, star, or any other object!!!!! 0
21 F = F w Gm 1 m /r = F and F w = m g Gm 1 m /r = m g g = Gm 1 /r F = F c F c = m a c F = Gm 1 m /r F c = m v /r Gm 1 m /r = m v /r Gm 1 m /r = m v /r Gm 1 /r = v v = Gm 1 /r The velocity of any orbiting object is determined by the m 1 (superior objects mass) and r o!!! 1
22 F = F c F c = m a c F c = m v /r F = Gm 1 m /r and F c = m 4π r/t Gm 1 m /r = m 4π r/t Gm 1 m /r = m 4π r/t r /T = Gm 1 /4π r /T = k = Gm 1 /4π The "k" value for any superior object is only determined by its mass!!! What acceleration (g) does the milkman experience? m 1 = m e = 5.98 x 10 4 kg r = r e = 6.8 x 10 6 m F = F w Gm 1 m /r = m g g = Gm 1 /r g = 6.67 x Nm /kg (5.98 x 10 4 kg) (6.8 x 10 6 m) g = 9.8 m/s g = Nm (kg) kg (m ) [kg(m/s )]m kg = kg (m = m/s )
23 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? r m = x 10 6 m (pendulum motion obeys circular motion rules) a = 4π r/t a = 4π (1.00 m)/(4.97 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 a = 1.60 m/s 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! Saturn: m s = 5.69 x 10 6 kg r s = 6.0 x 10 7 m g s = Gm s /r s g s = 6.67 x Nm /kg (5.69 x 10 6 kg) (6.0 x 10 7 m) g s = 10.4 m/s Jupiter: m j = 1.90 x 10 7 kg r j = 7.15 x 10 7 m g j = Gm j /r j g j = 6.67 x Nm /kg (1.90 x 10 7 kg) (7.15 x 10 7 m) g j = 5 m/s.6 g's
24 Gravity is the force that causes the moon to circle the earth. That means that gravity is acting as a centripetal force. F = F c F = Gm 1 m /r F c = m v /r F = Gm 1 m /r F c = mv /r the "m" is the mass of the inferior object "m " F c = m v /r Gm 1 m /r = m v /r Gm 1 m /r = m v /r Gm 1 /r = v v = Gm 1 /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! v = Gm 1 /r Example: What velocity does the moon orbit the earth with? r om =.9 x 10 8 m m 1 = m e = 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 4
25 8.1 notes.notebook gm = 1/60re =.007 m/s g = Gm1/r g = 6.67 x Nm/kg(5.98 x 104 kg) (.9 x 108 m) g =.006 m/s a = v/r a = (1010 m/s)/.9 x 108 m a =.006 m/s Gravity is the force that causes the moon to circle the earth. That means that gravity is acting as a centripetal force. m /r = Gm F =FGm 1m1/r /r =π mv Fc =Fcm4 r/t the "m" is the mass of the or, F = Gm1m/r and Fc = m4π r/t F = Fc Gm1m/r = m4πr/t r/t = Gm1/4π this equation is known as "Newton's Variation of Kepler's rd Law" r/t = k = Gm1/4π inferior object "m" Fc = m4π r/t Fc = mm4π r/t multiply by "r" and divide by m divide both sides by "4π" Notice that you have "Kepler's rd Law on the left the right side of the equation is the value of "k" for the superior object!!!!!!! 5
26 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 sun, earth, moon grav fields Hill sphere 6
27 7
28 Cavendish Pascal bernoulli moment of inertia kirchoffs law T moon = 7.4 days m e = 5.98 x 10 4 kg m N = 1.0 x 10 6 kg r moon (oribital) =.84 x 10 8 m r e (orbital) = 1.50 x m r N =.7 x 10 7 m r moon = 1.74 x 10 6 r e = 6.8 x 10 6 m r on = 4.50 x 10 1 m m moon = 7.5 x 10 kg 8
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