Transcript LECTURE # 1
PHYS 420-SPRING 2006 Dennis Papadopoulos LECTURE # 1 RELATIVITY I ARISTOTLE-NEWTON-GALILEO-EINSTEIN Acknowledge contributions from Chris Reynolds http://www.astro.umd.edu/~chris/Teaching/teaching.html and Nick Strobel's Astronomy Notes I : UN-AIDED OBSERVERS • Imagine a time before satellites, planes, cars, and telescopes. • What would you deduce about the world using just your own senses? – Earth is at rest (i.e., motionless) – Earth is flat – Sun, Moon, planets, stars move in the sky (from East to West) – Occasional bizarre things happen (comets, meteors) II : GREEK COSMOLOGY • First culture to look at world in the “modern scientific way” • They… – – – – Understood the idea of cause and effect Applied logic to try to understand the world Assumed that the Universe is fundamentally knowable Understood the importance of comparing theory with data. The scientific method The scientific method: Prediction (deduction) General Theory The real world Observation (+induction) Greeks and modern scientists put different emphasis on the importance of theory and observations. The spherical Earth • Greeks knew the Earth was a sphere! – Observations of ships sailing over the horizon – Observations of the Earth’s shadow on the Moon during lunar eclipses • Eratosthenes (276-195 B.C.) – Measured the radius of the Earth – Measured altitude of Sun at two different points on the Earth (Alexandria & Syene) Eudoxus and Aristotle • Earth is motionless • Sun, Moon, planets and stars go around the Earth • Eudoxus (408-355 B.C.) & Aristotle (384-322 B.C.) – Proposed that all heavenly bodies are embedded in giant spheres that revolve around the Earth. – Needed a complex set of interlocking spheres to explain observed planetary motions Ptolemy (100-170 A.D.) • Two things could not be explained by Aristotle’s model: – Planets vary in brightness – Retrograde motion • Ptolemy’s system introduces “epicycles” – Smaller sphere not centered on the Earth Dept. of Physics and Astronomy, Univ. of Tennesse – Needed more and more spheres to fit the observations of planets – Earth-Moon distance changed by factor of 2 – clearly contradicted by the simplest observations! Dept. of Physics and Astronomy, Univ. of Tennesse Aristarchos (310-230 B.C.) • Tried to measure relative distance between Sun and Moon – found that the Sun was much further away! • Deduced that Sun must be MUCH bigger than both the Earth and the Moon! • Proposed the first heliocentric model – Sun is the center of the Universe – Everything goes around the Sun • Never accepted by other philosophers of his time. III : MODERN SCIENCE Copernicus (1473-1543) • Nicholas Copernicus was modern founder of the heliocentric (Sun centered) model for the solar system • Probably based on work of Aristarchos • The Copernican Principle : The Earth is not at a special location in the Universe. • Predated the modern concept known as Generalized Copernican Principle: There is no special place in the universe, i.e., the universe has no center. Johannes Kepler (1571-1630) • Kepler examined the extremely accurate observations of Tycho Brahe • Suggested that planets orbit the Sun in ellipses rather than circles. • Major breakthrough – Allowed a very simple (and correct!) model of the solar system to be constructed – Swept away thousands of years of prejudice – Kepler let the data drive the conclusion. • Formulated the three laws of planetary motion. KEPLER’S LAWS The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. Kepler’s first law • Planets move around the Sun in ellipses, with the Sun at one focus. Kepler’s second law • The line connecting the Sun and a given planet sweeps out equal areas in equal times. – Therefore, planets move faster when they are nearer the Sun – Consequence of angular momentum conservation. Kepler’s third law • The square of the period of the orbit is proportional to the cube of the semimajor axis • Period (P) = time it takes for planet to complete one orbit • Semi-major axis (D) = half of the length of the “long” (i.e. major) axis of the ellipse. P R 2 3 • For the Earth, we know that: – P=1 year – R=150 million km (1 Astronomical Unit, A.U.) • Kepler’s 3rd law says that, for other planets, 2 P R yr AU 3 Galileo Galilei (1564-1642) • First to use a telescope to explore the sky • Found “imperfections” on the Sun and Moon (I.e. they are not perfect celestial spheres!). • Found that Jupiter clearly had its own “planetary system” – 4 objects seen orbiting Jupiter – Now called the Galilean Moons – Yet more evidence that an Earth centered model is wrong. Isaac Newton (1643-1727) • Formulated a theory of mechanics and gravity that explained the solar system with remarkable accuracy! • Newton realized that gravity is responsible for the motion of the Moon and planets. • Newton’s law of universal gravitation – Every mass attracts every other mass – Force drops off with the square of the distance – Kepler’s laws are a direct consequence of Newton’s law of gravity Newton’s laws • Newton’s laws of motion • Frames of reference • Newton’s law of Gravitation IV: NEWTON’S LAWS OF MOTION Begin by stating Newton’s laws: Newton’s first law (N1) – If a body is not acted upon by any forces, then its vector velocity, v, remains constant Note: – N1 sweeps away the idea of “being at rest” as a natural state. ACCELERATION Newton’s 2nd law (N2) – If a body of mass M is acted upon by a force F, then its acceleration a is given by F=Ma Note –N2 defines “inertial mass” as the degree by which a body resists being accelerated by a force. –Another way of saying this is that force = rate of change of momentum (rate of change of mv). Newton’s 3rd law (N3) - If body A exerts force F on body B, the body B exerts a force –F on body A. Mv1=mv2 ICE V: NEWTON’S LAW OF UNIVERSAL GRAVITATION Newton’s law of Gravitation: A particle with mass m1 will attract another particle with mass m2 and distance r with a force F given by Gm1m2 F Notes: r2 1. “G” is called the Gravitational constant (G=6.6710-11 N m2 kg-2) 2. This is a universal attraction. Every particle in the universe attracts every other particle! 3. Defines “gravitational mass” – mass vs. weight 4. Using calculus, it can be shown that a spherical object with mass M (e.g. Sun, Earth) gravitates like a particle of mass M at the sphere’s center. F GMm r2 KEPLER’S LAWS EXPLAINED • Kepler’s laws of planetary motion – Can be derived from Newton’s laws – Just need to assume that planets are attracted to the Sun by gravity (Newton’s breakthrough). – Full proof a straightforward calculus exercise – Planets natural state is to move in a straight line at constant velocity – But, gravitational attraction by Sun is always making it swerve off course – Newton’s law (1/r2) is exactly what’s needed to make this path be a perfect ellipse – hence Kepler’s 1st law – The fact that force is always directed towards Sun gives Kepler’s 2nd law (conservation of angular momentum) – Newton’s law gives formula for period of orbit 2 4 2 3 P R G ( M sun M planet )