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Phys 325 Spring 2011 Lecture 8

# Phys 325 Spring 2011 Lecture 8 - Physics 325 Lecture 8...

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Physics 325 Lecture 8 Motion in a Central Field (cont.) Recall from our discussing in Lecture 7 that it is sometimes convenient to study the motion of a particle in an isotropic central field ˆ ( ) F f r r by directly solving the differential equations from Newton’s 2 nd Law. We arrived at a differential equation for 1/ u r given by Equation 7.8: 2 1 2 2 2 ( ) d u m u f u d L u   This is the differential equation of the orbit of a particle moving under the influence of a central field. Conversely, if one is given the orbit 1 ( ) r r u , then one can obtain the force function by differentiating the orbit to get 2 2 d u d and inserting this into the above differential equation. Example: A particle in a central field moves in a spiral orbit given by 2 r c Determine the force function ( ) f r and ( ) t . We have 2 1 u c and 2 3 4 2 2 2 6 6 du d u cu d c d c From Equation 7.8, 2 1 2 2 6 ( ) m cu u f u L u Hence, 2 1 4 3 ( ) (6 ) L f u cu u m and 2 4 3 6 1 ( ) L c f r m r r

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To find ( t ) , we can use the fact that 2 constant r L m : 2 2 4 Lu L m mc or 4 2 L d dt mc By integrating, we find 5 2 5 L t mc where the constant of integration has been taken to be zero so that 0 at t 0 . Finally, we can write 1/5 2 5 ( ) L t t mc Gravitational Force and Potential Now we return specifically to the problem of gravity and planetary motion. First with some general considerations regarding gravitational forces, and then we will turn to the question of planetary orbits and Kepler’s laws.
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