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.

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Presentation transcript:

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 – the original party animal Johannes Kepler —the original anti-party animal

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!

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 3 rd Law of Motion The squares of the times to complete one orbit are proportional to the cubes of the average distances from the sun T 2 ~R 3 T= period of orbit, R= radius of orbit

Now for Newton

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=Gm E /R 2 Where m E = 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

Tunneling Through the Earth

Gravitational Potential Energy

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 mv 2 =GM e m/R where R=radius of earth at launch v escape = (2GM e /R) 1/2 When a particle reaches escape speed, it will not orbit but will fly out into space never to return.

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

Period of the Orbit

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= mc 2 -m 0 c 2 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/R s ) 1/2 where M=mass of the object and R s = Schwarzchild radius Mathematical construct of a sphere with radius R s is called the “event horizon”.