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, 2. Perfect Heavens (described by spheres) He expanded Eudoxus’ Model to use 55 spheres.

The problem of retrograde motion

Later refinements (2nd century B.C.) Hipparchus: Placing the Earth away from the centers of the “perfect spheres” Ptolemy: Further refinements, including epicycles

Claudius Ptolemy 85-165 AD Mathematical Syntaxis (Almagest)

The Copernican Revolution Nicolaus Copernicus (1473 – 1543): Heliocentric Universe (Sun in the Center)

Church cleric, but rejected a 2000-yr old paradigm Born: 19 Feb 1473 in Torun, Poland Died: 24 May 1543 in Frombork, Poland Church cleric, but rejected a 2000-yr old paradigm Seven axioms written in a pamphlet “Little Commentary” (1514) 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.

De revolutionibus orbium coelestium Figure 4.4: (a) The Copernican universe as reproduced in De Revolutionibus. Earth and all the known planets revolve in separate circular orbits about the sun (Sol) at the center. The outermost sphere carries the immobile stars of the celestial sphere. Notice the orbit of the moon around Earth (Terra). (Courtesy Yerkes Observatory) De revolutionibus orbium coelestium

Copernicus’ new (and correct) explanation for retrograde motion of the planets Retrograde (westward) motion of a planet occurs when the Earth passes the planet. This made Ptolemy’s epicycles unnecessary.

Figure 4.5: Tycho Brahe (1546–1601) was, during his lifetime, the most famous astronomer in the world. Proud of his noble rank, he wears the elephant medal awarded him by the king of Denmark. His artificial nose is suggested in this engraving.

Johannes Kepler (1571 – 1630) Figure 4.8: Johannes Kepler (1571–1630) derived three laws of planetary motion from Tycho Brahe’s observations of the positions of the planets. This Romanian stamp commemorates the 400th anniversary of Kepler’s birth. Ironically, it contains an astronomical error—the horns of the crescent moon should be inclined upward from the horizon.

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

Kepler’s Laws of Planetary Motion The orbits of the planets are ellipses with the sun at one focus. c Eccentricity e = c/a

Eccentricities of Ellipses 1) 2) 3) e = 0.02 e = 0.1 e = 0.2 5) 4) e = 0.4 e = 0.6

Eccentricities of Planetary Orbits Orbits of planets are virtually indistinguishable from circles: Most extreme example: Pluto: e = 0.248 Earth: e = 0.0167

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

Mercury: the closest planet to the Sun Perihelion = position closest to the sun Mercury Sun Perihelion: 46 million km; Aphelion: 70 million km Aphelion = position furthest away from the sun

In reality the orbits deviate from elliptical: 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) 1 degree = 3600 arcseconds

Predicted the presence and position of Neptune from irregularities in Uranus’s orbit Neptune was found in 1846 exactly at the predicted position Urbain Le Verrier 1811-1877 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 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.

Torque and Angular Momentum Conservation of Angular Momentum

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: v0 rod b axle x0 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.

A mass m1 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 m2. m1 m2

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, Ip. If a bullet of mass m, moving horizontally with velocity of magnitude vB as shown, strikes and embeds itself in the block, find the angular velocity of the platform after the collision. b vB axle top view

What is the moment of inertia of a cylinder of thickness h, radius R and total mass M about an axis through its center?

The rope is assumed not to slip as the pulley turns The rope is assumed not to slip as the pulley turns. Given m1, m2, R, and I find the acceleration of mass m1. Assume m1 > m2. Find the velocity with which mass m2 hits the ground assuming H is known. I m1 m2 R H