1 Class #19 Central Force Motion Gravitational law Properties of Inverse-square forces Center of Mass motion Lagrangian for Central forces Reduced Mass and CM reference frame

2 Reading Read Ch. 6 – Answer 4 Questions 1. Why is it called the “Principle of Least Action”? 2. What does Chapter 6 have to do with the Lagrangian method? 3. What part of the chapter did you find most interesting or novel? 4. What question(s) did the reading leave you with?

3 Test #2 Lagrangian Method Setup Generating equations of motion Looking for equilibria or solving equations for limiting conditions Energy Work and Work-KE Theorem  Line integrals and gradients  Conditions for conservative forces Energy methods and dissipation Rolling and KE of rotation 1-D potentials and small oscillations about stable points Oscillations Types of solutions Damped oscillators Driven oscillators

4 Gravity and Electrostatics Gravity Electrostatics Universal Constant Force Law Potential

5 Gauss’s Law – I - Flux Flux (Phi) may be thought of as the number of liters of fluid per second flowing through an area. The field-strength (g) may be thought of as # liters/second-m 2

6 Gauss’s Law - II Total Flux through a surface depends ONLY on the amount of mass contained inside that surface.

7 Gauss’s Law - II Unique property of inverse-square forces. For spherical shell and r>r shell, treat the entire shell mass as if concentrated at a point at the center of the shell

8 Potential of a sphere Using above formulae – 1) What is potential at radius r outside a solid sphere of radius R and mass M? 2) What is potential inside a solid sphere of radius R (at radius r<R)? 3) What is force? Give answers in terms of r, R, and M

9 Vectors and Central forces Vectors Many forces are of form Remove dependence of result on choice of origin Origin 1 Origin 2

10 Tides Why are there bulges on BOTH sides of the earth?

11 Two particles with central forces Origin

12 Two particles with central forces

13 Reduced two-body problem

14 The power of

15 The power of - part 2

16 Equivalent 1-D problem Relative Lagrangian Radial equation Total Radial Force

17 Pseudopotential and Energy

18 Pseudopotential and Energy

19 Tides Why are there bulges on BOTH sides of the earth?

20 Earth and Moon 1. What is the reduced mass for the earth-moon system (in kg)? 2. How many percent different is it than the lunar mass 3. What is theta-dot? 4. What is the radius of a circular orbit? 5. How would this change if the earth were fixed in space by the hand of God or a Borg tractor beam?

21 Earth and Moon 1. What is the reduced mass for the earth-moon system (in kg)? 2. How many percent different is it than the lunar mass 3. What is theta-dot? 4. What is the radius of a circular orbit? 5. How would this change if the earth were fixed in space by the hand of God or a Borg tractor beam?

22 Potential and Force for solid spheres

23 Class #19 Windup <- Gravitational Lagrangian <- General Central Force “1-D eqn”