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UNIVERSAL LAW OF GRAVITATION

UNIVERSAL LAW OF GRAVITATION. Before Isaac Newton discovered the Universal Law of Gravitation, objects in the “heavens” were assumed to obey different laws compared to earth. Halley’s Comet. Edmond Halley was one of the wealthiest man of his age

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UNIVERSAL LAW OF GRAVITATION

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  1. UNIVERSAL LAW OF GRAVITATION • Before Isaac Newton discovered the Universal Law of Gravitation, objects in the “heavens” were assumed to obey different laws compared to earth

  2. Halley’s Comet • Edmond Halley was one of the wealthiest man of his age • He used Newton’s Gravitational Law to predict the return of a comet in 1758

  3. Discovery of Neptune • Astronomers noticed small deviations in the orbit of the planet Uranus • Speculated an unseen planet (Neptune) & calculated its position • Found on first night of observation • Gravity: one of four fundamental forces • By far the weakest

  4. Insight came to Newton in realizing an apple falling toward the earth is the same force that keeps the moon in orbit • The moon is also falling towards the earth • But it has large horizontal velocity and the earth surface curves enough to not allow the moon to reach it

  5. Mass & Distance Relationships • Newton discovered: • All masses attract one another • The force between them is linear with respect to each mass • The force is inversely proportional to the square of the distance between the objects centers  FGα m1 m2 / r2

  6. Each mass experiences the same force & forms an action/reaction pair • “universal” because everything with mass attracts all other masses • Generally FG is not noticed unless one of the masses is very large • The force diminishes rapidly with distance, but never completely vanishes • FG has an infinite range

  7. Weighing the Earth • Newton realized a constant was needed to create an equality  FG = G m1 m2 / r2 • He did not have enough info to determine “G” – except it was very small • Left to Henry Cavendish many years later (1798) • Using heavy lead masses, he determined G • Current value:  G = 6.67 x 10-11

  8. Unit is found by solving for G & “seeing” what units there are  G = 6.67 x 10-11 N m2/ kg2 • The importance cannot by overstated • Realize: force of gravity on surface is weight • With G, we can determine mass of: Sun, planets, moons, the solar system, the Milky Way, etc. • We also can determine the average density of earth giving us clues as to its interior

  9. Gravitational Attraction of Spherical Bodies • Universal Law applies to point objects, how about objects of finite size • Divide object into small masses, then find the force on each • Difficult for arbitrary shape, not for spheres

  10. Consider a sphere and a point object brought near it • Mass near A has stronger attraction than B, but forces are along line center to center • Mass near C & D also has a net force along “x-axis” since comps. along “y” all cancel one another

  11. To find magnitude, calculus must be used • Net force is the same as if all the mass were concentrated at center • Orbits above equation with period equal to 1 day • Appears in same location in sky at all times

  12. Escape Velocity • Using energy concepts, its possible to calc the speed needed to escape earth altogether  ve = 11,200 m/s = 11.2 km/s = 25,000 mi/h

  13. The moon has much smaller ve(5320 mi/h) • Much easier to launch rockets into space • Also reason moon has no atmosphere • Individual molecules move at speeds > ve • Light molecules (H2 & He) move faster for a given temp compared to heavier N2 & O2 • Why earth’s atmo contains very little H2 & He

  14. Black Holes • What if a star were to collapse to a relatively small size? • vebecomes very large • Could exceed the speed of light • Nothing could escape star, not even light – a black hole

  15. TIDES • Tides are due to differences in gravitational force from one side of earth to the other • Why moon creates tides, but Sun does generally does not even though its gravitational force is larger

  16. KEPLER’S LAWS • Using observations of planet Mars, Kepler deduced three laws of planetary motion • Proved that planets orbit Sun (Copernican) • Later proved to be due to Gravitation

  17. Kepler’s First Law • Planets follow elliptical orbit with Sun at one focus • Tried to match circular orbit for Mars but could not • Forced to give up, even though orbit is very nearly circular • Can be circular – a special ellipse

  18. Kepler’s Second Law • Equal are in equal time • As objects (planets) approach Sun, they move faster – further = slower • The areas swept out between the Sun & the positions of the planet are constant

  19. Kepler’s Third Law • The period squared is proportional to orbital radius cubed – T2α r3 • Worked out mathematically after years of calculations • Decades later – Newton was able to prove it quite easily

  20. Consider a sun/planet system • This relationship works for all planets, satellites orbiting planets, moons, etc.

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