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)
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 ADMathematical Syntaxis (Almagest)
The Copernican Revolution
Nicolaus Copernicus (1473 – 1543):
Heliocentric Universe (Sun in the Center)
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, PolandDied: 24 May 1543 in Frombork, Poland
De revolutionibus orbium coelestium
Copernicus’ new (and correct) explanation for retrograde motion of the planets
This made Ptolemy’s epicycles unnecessary.
Retrograde (westward) motion of a planet occurs when the Earth passes the planet.
Johannes Kepler (1571 – 1630)
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
1.The orbits of the planets are ellipses with the sun at one focus.
Eccentricity e = c/a
c
Eccentricities of Ellipses
e = 0.02 e = 0.1 e = 0.2
e = 0.4 e = 0.6
1) 2) 3)
4) 5)
Eccentricities of Planetary Orbits
Orbits of planets are virtually indistinguishable from circles:
Earth: e = 0.0167Most extreme example:
Pluto: e = 0.248
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
Sun
MercuryPerihelion = position closest to the sun
Aphelion = position furthest
away from the sun
Perihelion: 46 million km; Aphelion: 70 million km
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
Urbain Le Verrier 1811-1877
Predicted the presence and position of Neptunefrom irregularities in Uranus’s 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
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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:
b
x0
v0
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.
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
top view
vB
axle
What is the moment of inertia of a cylinder of thickness h, radius R and total mass M about an axis through its center?
m1
m2
R I
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.
H
