Geocentric Universe Eudoxus (409 – 356 B.C.): Model of 27 nested spheres Aristotle (384 – 322 B.C.), major authority of philosophy until the late middle.

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<ul><li> Slide 1 </li> <li> Geocentric Universe Eudoxus (409 356 B.C.): Model of 27 nested spheres Aristotle (384 322 B.C.), major authority of philosophy until the late middle ages: Universe can be divided in 2 parts: 1. Imperfect, changeable Earth, He expanded Eudoxus Model to use 55 spheres. 2. Perfect Heavens (described by spheres) </li> <li> Slide 2 </li> <li> The problem of retrograde motion </li> <li> Slide 3 </li> <li> Later refinements (2nd century B.C.) Hipparchus: Placing the Earth away from the centers of the perfect spheres Ptolemy: Further refinements, including epicycles </li> <li> Slide 4 </li> <li> Claudius Ptolemy 85-165 AD Mathematical Syntaxis (Almagest) </li> <li> Slide 5 </li> <li> The Copernican Revolution Nicolaus Copernicus (1473 1543): Heliocentric Universe (Sun in the Center) </li> <li> Slide 6 </li> <li> 1. There is no one centre in the universe. 2. The Earth's centre is not the centre of the universe. 3. The centre of the universe is near the sun. 4. The distance from the Earth to the sun is imperceptible compared with the distance to the stars. 5. The rotation of the Earth accounts for the apparent daily rotation of the stars. 6. The apparent annual cycle of movements of the sun is caused by the Earth revolving round it. 7. The apparent retrograde motion of the planets is caused by the motion of the Earth from which one observes. Church cleric, but rejected a 2000-yr old paradigm Seven axioms written in a pamphlet Little Commentary (1514) Born: 19 Feb 1473 in Torun, Poland Died: 24 May 1543 in Frombork, Poland </li> <li> Slide 7 </li> <li> De revolutionibus orbium coelestium </li> <li> Slide 8 </li> <li> Copernicus new (and correct) explanation for retrograde motion of the planets This made Ptolemys epicycles unnecessary. Retrograde (westward) motion of a planet occurs when the Earth passes the planet. </li> <li> Slide 9 </li> <li> Slide 10 </li> <li> Johannes Kepler (1571 1630) </li> <li> Slide 11 </li> <li> Kepler hypothesized that a physical force moved the planets, and that the force diminished with distance. Planets closer to the sun feel a stronger force and move faster. Elliptical orbits key to the problem of the planetary motion </li> <li> Slide 12 </li> <li> Keplers Laws of Planetary Motion 1.The orbits of the planets are ellipses with the sun at one focus. Eccentricity e = c/a c </li> <li> Slide 13 </li> <li> Eccentricities of Ellipses e = 0.02 e = 0.1e = 0.2 e = 0.4e = 0.6 1)2)3) 4) 5) </li> <li> Slide 14 </li> <li> Eccentricities of Planetary Orbits Orbits of planets are virtually indistinguishable from circles: Earth: e = 0.0167 Most extreme example: Pluto: e = 0.248 </li> <li> Slide 15 </li> <li> Slide 16 </li> <li> LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time The closer to the sun, the larger the orbital velocity </li> <li> Slide 17 </li> <li> Mercury: the closest planet to the Sun Sun Mercury Perihelion = position closest to the sun Aphelion = position furthest away from the sun Perihelion: 46 million km; Aphelion: 70 million km </li> <li> Slide 18 </li> <li> Mercury's perihelion precession: 5600.73 arcseconds per century Newtonian perturbations from other planets: 5557.62 arcseconds per century Remains unexplained: 43 arcseconds/century (Le Verrier 1855) In reality the orbits deviate from elliptical: 1 degree = 3600 arcseconds </li> <li> Slide 19 </li> <li> Urbain Le Verrier 1811-1877 Predicted the presence and position of Neptune from irregularities in Uranuss orbit Neptune was found in 1846 exactly at the predicted position In 1855 Le Verrier found that the perihelion of Mercury advanced slightly more than the Newtonian theory predicted. He and others tried to explain it with a new planet Vulcan, new asteroid belt, etc. Finally, Einstein provided an explanation using General Relativity. In the eyes of all impartial men, this discovery [Neptune] will remain one of the most magnificent triumphs of theoretical astronomy Arago I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested. A contemporary </li> <li> Slide 20 </li> <li> Torque and Angular Momentum Conservation of Angular Momentum </li> <li> Slide 21 </li> <li> Suppose there were an axle at the origin with a rigid, but massless rod attached to it with bearings so that the rod could freely rotate. At the end of the rod, of length b, there is a block of mass M as shown below: b x0x0 v0v0 rod axle A bullet is fired at the block. If the bullet strikes the block and sticks, what will be the angular velocity of the block about the axle? Neglect gravity. </li> <li> Slide 22 </li> <li> A mass m 1 is going around on a string on a frictionless table and the string goes through a hole where it is attached to a hanging mass m 2. m1m1 m2m2 </li> <li> Slide 23 </li> <li> Slide 24 </li> <li> A block of mass M is cemented to a circular platform at a distance b from its center. The platform can rotate, without friction, about a vertical axle through its center with a moment of inertia, I p. If a bullet of mass m, moving horizontally with velocity of magnitude v B as shown, strikes and embeds itself in the block, find the angular velocity of the platform after the collision. b top view vBvB axle </li> <li> Slide 25 </li> <li> What is the moment of inertia of a cylinder of thickness h, radius R and total mass M about an axis through its center? </li> <li> Slide 26 </li> <li> m1 m2 R I The rope is assumed not to slip as the pulley turns. Given m 1, m 2, R, and I find the acceleration of mass m 1. Assume m 1 &gt; m 2. Find the velocity with which mass m 2 hits the ground assuming H is known. H </li> </ul>

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