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jw Fundamentals of Physics 1 GRAVITY

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jw Fundamentals of Physics 2 Chapter 13: Newton, Einstein, and Gravity Isaac Newton 1642 - 1727Albert Einstein 1872 - 1955

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jw Fundamentals of Physics 3 Chapter 13 Gravitation 1.Our Galaxy & the Gravitational Force 2.Newton’s Law of Gravitation 3.Gravitation & the Principle of Superposition 4.Gravitation Near Earth’s Surface 5.Gravitation Inside Earth 6.Gravitational Potential Energy Path Independence Potential Energy & Force Escape Speed 7.Planets &Satellites: Kepler’s Laws 8.Satellites: Orbits & Energy 9.Einstein & Gravitation Principle of Equivalence Curvature of Space Review & Summary Exercises & Problems

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jw Fundamentals of Physics 4 The World & the Gravitational Force –Allowed matter to spread out after the Big Bang. –Eventually pulled large amounts of mass together: –Clouds of gas –Brown dwarfs –Stars –Galaxies (The Milky Way, Andromeda) –Clusters of Galaxies (“The Local Group”) –Super Clusters “billions & billions” of galaxies and stars –Big Bang - hydrogen, helium, lithium, & very little else –Big Stars - burned out & exploded creating heavier elements - carbon, oxygen, iron, gold,... –Gravity then pulled the sun & earth together Gravity is a very weak force, but essential for us to exist!

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jw Fundamentals of Physics 5 Physics ~1680 By now the world knew: Bodies of different weights fall at the same speed Bodies in motion did not necessarily come to rest Moons could orbit different planets Planets moved around the Sun in ellipses with the Sun at one focus The orbital speeds of the planets obeyed “Kepler’s Laws” But why??? Isaac Newton put it all together.

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jw Fundamentals of Physics 6 Newton’s Law of Gravitation Every massive body in the universe attracts every other massive body through the gravitational force! Newton (1665): Gravity Force is weak! G = 6.67 x 10 -11 N m 2 / kg 2 Principle of superposition: net effect is the sum of the individual forces. (added vectorially).

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jw Fundamentals of Physics 7 Newton’s Law of Gravitation

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jw Fundamentals of Physics 8 Measurement of the Gravitational Force If two masses are brought very close together in the laboratory, the gravitational attraction between them can be detected. G = 6.673 x 10 -11 N m 2 / kg 2 (relatively poorly known: 1 part in 10,000) Cavendish Experiment(~1760):

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jw Fundamentals of Physics 9 Gravitation Near Earth’s Surface The Earth is not uniform. The Earth is not a sphere. –An ellipsoid ~0.3% Earth is rotating. –Centripetal acceleration F N -ma g = m(- 2 R) mg = ma g -m( 2 R) (measured wt.) = (mag. of gravitational force) – (mass x centripetal acceleration) g = a g – 2 R on equator difference ~ 0.034 m/s 2

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jw Fundamentals of Physics 10 Gravitation Near Earth’s Surface r is the distance between the centers of the two bodies Newton: The force exerted by any spherically symmetric object on a point mass is the same as if all the mass were concentrated at its center. Newton had to invent calculus to prove it! r

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jw Fundamentals of Physics 11 Newton’s Law of Gravitation Newton: The gravitational force provides the centripetal acceleration to hold the earth in its orbit around the sun:

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jw Fundamentals of Physics 12 Newton: The gravitational attraction between the Earth and the Moon causes the Moon to orbit around the Earth rather than moving in a straight line. Orbital Motion

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jw Fundamentals of Physics 13 Gravitation Inside A Shell Newton: gravitational force inside a uniform spherical shell of matter: The forces on m 0 due to m 1 and m 2 vary as ~1/r 2, but the masses of m 1 and m 2 grow as r 2, hence the two forces cancel out! A uniform spherical shell of matter exerts no net gravitational force on a particle located anywhere inside it.

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jw Fundamentals of Physics 14 Gravitation Outside A Shell Newton invents calculus!!! Gravitational acceleration at a distance r from a point particle! Newton: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s matter were concentrated at its center.

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jw Fundamentals of Physics 15 Gravitation Inside & Outside a Uniform Sphere Newton: Consider the sphere as a set of concentric shells: The gravitational force inside a uniform spherical shell of matter is zero. Only the matter inside radius r attracts an object at that radius.

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jw Fundamentals of Physics 16 Gravitation Inside Earth Newton: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s matter were concentrated at its center. Hooke’s Law m oscillates back & forth! Newton: The gravitational force inside a uniform spherical shell of matter is zero.

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jw Fundamentals of Physics 17 Gravitational Potential Energy of a 2-Particle System Gravity is a conservative force. The work done by the gravitational force on a particle moving from an initial point A to a final point G is independent of the path taken between the points. Gravity is a conservative force. Potential Energy: (attractive force) Only U is important; the location of U = 0 is arbitrary. Choose U = 0 to be the point at which the masses are far apart.

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jw Fundamentals of Physics 18 Escape Velocity Consider a projectile of mass m, leaving the surface of a planet. The loses kinetic energy and gains gravitational potential energy. Does not depend on the mass of the projectile nor on its initial direction (i.e. escape speed not escape velocity). Escape Speed: When the projectile reaches infinity, it has no potential energy (U = 0) and no kinetic energy (stopped: v = 0): the total mechanical energy is zero.

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jw Fundamentals of Physics 19 Escape Velocity ~ 25,000 mi/hr Escapes Bound Orbit No hydrogen & helium in the earth’s atmosphere: Escape Speed

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jw Fundamentals of Physics 20 http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.htmlww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html

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jw Fundamentals of Physics 21 Tycho Brahe (1546-1601) Johannes Kepler (1571-1630) Planets & Satellites: Kepler’s Laws Sun Path of Mars across the sky.

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jw Fundamentals of Physics 22 Planets & Satellites: Kepler’s Laws –Equal Areas in Equal Times by sun-planet line. Sun Kepler’s Laws: –Elliptical orbits with the Sun at one focus. –T 2 ~ r 3 (Period & Mean Distance from sun)

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jw Fundamentals of Physics 23 Kepler’s 1 st Law The Law of Orbits: All planets move in elliptical orbits, with the Sun at one focus. ellipse - sum of distances from 2 foci is constant (2a). eccentricity of earth = 0.0167 e = 0 circle (1 focus) Sun is very near one focus. perihelion - closest point to Sum aphelion - farthest point

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jw Fundamentals of Physics 24 Kepler’s 2 nd Law The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant. slowerfaster http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/Kepler/Kepler.html

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jw Fundamentals of Physics 25 Kepler’s 2 nd Law The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant. Conservation of Angular Momentum: Any Central Force

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jw Fundamentals of Physics 26 Kepler’s 3 rd Law The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. Consider the special case of a circular orbit: See Table 13-3. Determine the mass of a planet by measuring period and mean orbital distance of a moon orbiting it.

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jw Fundamentals of Physics 27 What did Newton know? Kepler’s 3 rd Law: acceleration of the moon: velocity of the moon: His own law! A little algebra by Newton: The force holding the moon in orbit depends on the square of its distance from earth.

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jw Fundamentals of Physics 28 Einstein & Gravitation “Science is a perfectionist” - Narlikar –Newton’s Law of Gravity works great! –But, how & why does it work? –A small problem with the planet Mercury.

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jw Fundamentals of Physics 29 Einstein & Gravitation A “Small” problem with the planet Mercury: Mercury takes 88 Earth days to orbit the sun. But, measurements show the perihelion “advancing”: –0.159 o per 100 years! (Newton’s Law says zero!) Attractions by the other planets almost explain it. –0.147 o per 100 years! 43 arc-seconds per 100 years unexplained. Einstein explains it!

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jw Fundamentals of Physics 30 Einstein & Gravitation Special Theory of Relativity –Space & Time are intimately connected. c = constant everywhere –Mass & Energy are equivalent. E = m c 2 General Theory of Relativity –A theory of gravitation space–time geometry mass (material bodies)

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jw Fundamentals of Physics 31 Einstein & Gravitation Galileo / Newton Universe –No interdependence of time, space & mass. Time flows uniformly. Space is immutable. –Euclidean geometry - “straight lines” Mass of a body is constant. (shape & dimension of rigid bodies is also constant) Einstein Universe Gravitational forces reach out to infinity. –All bodies are moving in the gravitational field of other bodies. Not moving in a straight line. Space is curved!

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jw Fundamentals of Physics 32 Einstein & Gravitation The General Theory of Relativity: The Equivalence Principle: Is it gravity or rockets that causes a? No experiment can tell the difference! Einstein’s Strong Equivalence Principle:

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jw Fundamentals of Physics 33 If light appears to follow a curved path in the elevator, gravity must also cause it to curve. Gravity & Light: Einstein’s Strong Equivalence Principle: Observe light in an accelerating elevator: Einstein: Light does not travel in a straight line!

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jw Fundamentals of Physics 34 In his General Theory of Relativity, Einstein explained the force of attraction between massive objects in this way: “Mass tells space-time how to curve, and the curvature of space- time tells masses how to accelerate.” Einstein’s View of Gravitation “space-time” refers to 4-dimensional space: x, y, z, ct Einstein: Space-Time is curved by the presence of mass!

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jw Fundamentals of Physics 35 Bending of Space Time Newton said the forces due to the mass of the Earth and the mass of the Moon kept the Moon in orbit. Einstein said the Moon was trapped in the funnel of the Earths gravity well. –A gravity well is formed when a mass bends the fabric of space- time. General Theory of Relativity –Orbit is not caused by forces but by the curvature of space-time (funnel curve)

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jw Fundamentals of Physics 36 Einstein & Gravitation Einstein: Space-Time is curved by the presence of mass!

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jw Fundamentals of Physics 37 Einstein proposed a radical experiment to test his theory: 1919: Einstein’s prediction verified by Eddington during a solar eclipse by the moon. 1915 Einstein becomes the most famous scientist of the 20th century!

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jw Fundamentals of Physics 38 Gravitational Lensing “An Einstein Ring”

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jw Fundamentals of Physics 39 Gravitational Lensing These usually involve light paths from quasars & galaxies being bent by intervening galaxies & clusters. a galaxy behind a galaxy “Einstein Ring”“Einstein Cross” multiple images

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