1 Class #24 of 30 Exam -- Tuesday Additional HW problems posted Friday (also due Tuesday). Bring Index Card #3. Office hours on Monday 3:30-6:00 Topics Central Force  Kepler’s Laws  Gravitational PE and Force  Small Oscillations about equilibrium  Reduced Mass  Momentum conservation  Pseudopotential :02

2 Class #24 of 30 Central Force  Polar form of orbital equations Elliptical orbits Hyperbolic orbits Parabolic / Circular Orbits  Scattering Accelerated Reference Frames  Effective Gravity  Centrifugal force  Coriolis force :02

3 Problem Review KKR9-4 Holy Earth KKR9-6 Eccentric comet KKR9-8 Escape to the moon Taylor 8-9 Small oscillations Taylor 9-1/9-2 Buoyant doughnuts Taylor 9-9 Tilted Plum-line Taylor 9-10 Spin the bucket :02

4 :08 Gravity and Electrostatics Gravity Electrostatics Universal Constant Force Law Gauss’s Law Potential

5 Kepler’s 1 st, 2 nd and 3 rd laws (1610) :37 1 st Law – Planets move in ellipses with sun at one focus Third law demonstrated previously relates period to semi-minor radius 2 nd law is direct consequence of momentum conservation “Equal areas are swept out in equal times” True for ALL central forces

6 E, L and Eccentricity :30 The physics is in E and L. Epsilon is purely a geometrical factor. Epsilon equation applies to ALL conic sections (hyperbolae, ellipses, parabolas).

7 Central Force :30 <- Completely general for inverse square forces … All types of orbits. <- “Gamma” makes it specific for gravity. Key constants are, (E and L), OR (c and L) or (L and epsilon) or (c and epsilon) <- Specific for elliptical orbits.

8 Planetary Scattering Angle :37  Sketch for epsilon=2

9 Reduced two-body problem :15

10 Equivalent 1-D problem :30 Relative Lagrangian Radial equation Total Radial Force

11 Class #24 Windup :60