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Gravitation

Gravitation. “On the Shoulders of Giants”. Newton once said that his success was based on the fact that he “stood on the shoulders of giants” In other words, his work was based on the great work done by his predecessors. Tycho Brahe 1546-1601– the original party animal

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Gravitation

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  1. Gravitation

  2. “On the Shoulders of Giants” • Newton once said that his success was based on the fact that he “stood on the shoulders of giants” In other words, his work was based on the great work done by his predecessors. • Tycho Brahe 1546-1601– the original party animal • Johannes Kepler 1571-1630—the original anti-party animal

  3. The Odd Couple • Kepler wanted establish that God created the universe with a perfect mathematical order; understanding the mathematics made him feel closer to God. • Brahe just wanted to have fun…however, Brahe (while having a lot of fun) was one of the best astronomical observers of history and kept meticulous data!

  4. The Odd Couple Cont’d • Using Brahe’s data, Kepler formulated 3 laws of motion but we are just going to worry about Kepler’s 3rd Law of Motion • The squares of the times to complete one orbit are proportional to the cubes of the average distances from the sun T2 ~R3 T= period of orbit, R= radius of orbit

  5. Now for Newton

  6. Big G, little g • Big G, is a constant for the entire universe • Little g, only works on earth and is derived from Big G • g=GmE/R2 • Where mE= mass of earth • R=radius of earth • Little g varies from place to place for 3 reasons • The earth is not uniform in its composition • The earth is not a uniform sphere (oblate) • The centripetal acceleration due to earth’s rotation will cause problems

  7. Tunneling Through the Earth

  8. Gravitational Potential Energy

  9. Escape Speed • A particle escapes from the earth’s gravity when its kinetic energy is larger than the earth’s gravitational potential energy • Escape speed is the speed when these two energies are equal or • ½ mv2=GMem/R where R=radius of earth at launch • vescape= (2GMe/R)1/2 • When a particle reaches escape speed, it will not orbit but will fly out into space never to return.

  10. Orbital Velocity • The speed by which an object can orbit another object is called orbital velocity and depends on the radius of the orbit. • It is found by matching the gravitational force at a particular radius with its centripetal force

  11. Period of the Orbit

  12. Black Holes • If the escape speed of a object is equal to the speed of light, the object is called a “black hole” • While you might think that you set v=c and use the escape speed formula to solve, you would be both right and wrong • Wrong from the standpoint that general relativity takes over and of course, the kinetic energy of a particle is K= mc2-m0c2 • Right from the dumb luck that the final form of the equation does look like what you would expect from the simple approach… • c=(2GM/Rs)1/2 where M=mass of the object and Rs= Schwarzchild radius • Mathematical construct of a sphere with radius Rsis called the “event horizon”.

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