1. 1 Class #17 of 30 Central Force Motion Gravitational law Properties of Inverse-square forces Center of Mass motion Lagrangian for Central forces Reduced Mass and CM reference frame Central Forces DVD BRIEF Exam Review :02

2. 2 Tides :05 Why are there bulges on BOTH sides of the earth?

3. 3 :08 Gravity and Electrostatics Gravity Electrostatics Universal Constant Force Law Gauss’s Law Potential

4. 4 Potential of a spherical shell :72 Unique property of inverse- square forces. For r>r shell, treat the entire shell mass as if concentrated at a point at the center of the shell

5. 5 Potential of a sphere :15 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

6. 6 Potential and Force for solid spheres :18

7. 7 Vectors and Central forces Vectors Many forces are of form Remove dependence of result on choice of origin Origin 1 Origin 2 :20

8. 8 Two particles with central forces :30 Origin

9. 9 Two particles with central forces :35

10. 10 Earth and Moon :30 1. Write down the relative Lagrangian for the earth-moon system 2. Use it to get the relative radial equation for the earth-moon system 3. What is the reduced mass to use (in kg)? 4. How many percent different is it than the lunar mass 5. What is theta-dot? 6. What is the radius of a circular orbit? 7. How would this change if the earth were fixed in space by the hand of God or a Borg tractor beam?

11. 11 1. Write down the relative Lagrangian for the earth-moon system 2. Use it to get the relative radial equation for the earth-moon system 3. What is the reduced mass to use (in kg)? 4. How many percent different is it than the lunar mass 5. What is theta-dot? 6. What is the radius of a circular orbit? 7. How would this change if the earth were fixed in space by the hand of God or a Borg tractor beam? Earth and Moon :50

12. 12 Tides :55 Why are there bulges on BOTH sides of the earth?

13. 13 Class #17 Windup Office - Tues 3-5, Wed 4-5:30 Midterm grades will be posted on web immed. after lunch :60 <- Gravitational Lagrangian <- General Central Force “1-D eqn”

14. 14 Exam 2-3 :72 In Figure (a), a spring with spring constant “k” has been added to an Atwood machine that is glued to the top of the pulley. [If the pulley turned a large fraction of a revolution, the spring would no longer be horizontal, which makes it pretty messy. Ignore this complication.] Assume the wheel only turns a few degrees from the position in which it is shown so that we may assume that the string compresses or stretches but remains horizontal. a) Define an appropriate generalized coordinate. Write down constraints. Write down T and U and finally the Lagrangian for this case. The pulley is massless and has a radius “R” (if you feel you need to know). (10) b) Write down Lagrange’s equation(s) to find the differential equation for the acceleration of m1 (8) c) Initially, at time t=0, we are told that the spring is unstretched (it has length L), and that the masses are at the same height (just as drawn). What is the acceleration of mass m1 at t=0? (5) d) The system has an equilibrium point at which the spring length is not L. How far are the masses displaced from their initial positions shown in the figure at this equilibrium point? (You don’t need to solve the differential equation to get this answer!) (5)

15. 15 Exam 2-2, 2-4 [A hoop and a sphere both start from rest at height h, and race down two identical ramps as shown below. They have identical radius “R” and identical mass “m” (and the mass is distributed uniformly as you should expect in a physics problem!)] a) Define the generalized coordinate you will use. Define which direction is positive. Write the equation of constraint (6). b) Write down the Lagrangian for the hoop AND the Lagrangian for the sphere in terms of the generalized coordinate. You may use "I-hoop" and “I-sphere” as the moments of inertia (8 pts) c) Write down Lagrange's equations, still using I as the moment of inertia. (6) d) Use Lagrange's equation to get the acceleration of the hoop and the sphere along the ramp using.. [You are not required to solve for position as a function of time. Just get acceleration.] 5) e) What is the outcome of the race? (3) f) Why? (One sentence) (2) A bead of mass “m” is constrained to slide on a rod. The rod is rotating about the z-axis such that its angle “phi” at any time is omega-t. The rod is not horizontal or vertical, it is inclinded at an angle “theta” (which doesn’t change). a) The correct generalized coordinates for this problem are where is the angle the rod makes with respect to the z-axis (the axis of rotation), as shown and s is the position “s” of the bead defined as its distance on the rod from the origin. Write x, y and z in terms of. (hint …this is very similar to using spherical coordinates). (5) b) Write down the Lagrangian for the system. (Gravity matters) (5) c) Write down Lagrange’s equation for “s” (only).(5) Extra Credit – There is an equilibrium position for the bead at any. What is it? (5)