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f10_sep16 - Circular Velocity Thurs Sep 16 2010 Gravity...

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Thurs., Sep . 16, 2010 Gravity, Tidal Force, & Light Newton’s Law of Gravity Tidal Force: a difference in gravity Light as a wave and as a particle Given: (1) speed in a circular orbit is: Circular Velocity v circ = 2 " R P = K R P # $ % & ' ( where K is a constant, and (2) Kepler’s Third Law, P 2 = K ! (R 3 ) where the value of K ! depends on the units (= 1 for P in years and R in A.U.), and we replace a with R (it’s a circle). Find the relationship between v circ and R , by eliminating P . That is, express v circ as some power of R , the orbit’s radius (you may combine or drop the constants). Relating velocity to orbit size v circ 2 = 4 " 2 R 2 P 2 = 4 " 2 ( ) R 2 P 2 = 4 " 2 # K $ % & ' ( ) R 2 R 3 = # # K ( ) 1 R * 1 R From Kepler’s Third Law we have: P 2 = K ! (R 3 ) " R 3 Squaring the orbit equation and substituting for P 2, The instructions were not to worry about the constants. Then v circ = " " K R # 1 R This is called “Keplerian” rotation (I wonder why!) If we use Newton’s form of Kepler’s 3rd law we can find the value of K # (we’ll see this later in the semester). Kepler’s Third Law: Velocity variation with orbital radius ! We have just shown that the average speed of a planet in a larger orbit is slower than a planet in a smaller orbit
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“escape velocity” - depending on its initial velocity, the cannonball will either fall to Earth, continually free-fall (stay in orbit) , or escape the force of Earth’s gravity. Fate of a Falling Object Orbital Paths Extending Kepler’s First Law, Newton found that ellipses were not the only possible orbital paths: ellipse (bound)
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