Note that many Barger & Olsson problems have final answers in the back of the book, so you can check your work!

Reading:

Read B & O Chapter 2 -- 2.3; KB 361 lecture notes 4 -- 6.

Problems for Review:

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1. 1. A particle is under the influence of a force , where k and are positive constants. Determine V(x) and discuss the motion, for all allowed values of E. Note whether the motion is bounded or unbounded, and any turning points.
   2. For the problem of part (a), what minimum speed must a particle at the origin (x=0) have, in order to escape to infinite radius? A particle at ?

2. 1. B & O 2-7. Note that this force and potential are defined for all x (positive and negative).
   2. For the potential in part (a), consider a particle with energy . What are its turning points, for positive and negative x? Describe the allowed motions, noting whether they are bound or unbound. Similarly, for , find the turning points, and describe the allowed motions, noting whether they are bound or unbound.

3. Describe how to determine whether an equilibrium is stable or unstable when at the equilibrium point.

Problems to Solve:

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1. B & O 2-6.
2. Recall the hanging cable of problem set 1, problem 3, which experiences force F = mgx/l when 0 ;SPMlt; x;SPMlt;l.
   1. Find the potential energy V(x) of the hanging cable. Sketch the potential over its range of validity, 0 ;SPMlt;x ;SPMlt; l.
   2. Solve for the velocity v(x) using potential methods (B & O Equation 2.19). Check that this agrees with your result from Homework 1, problem 3b.
   3. If the cable starts its fall from rest, what fraction of the cable initially hanging down leads to the greatest final velocity at x = l, when the cable has entirely slipped off the table? What is this greatest attainable final velocity for the cable?
3. Consider the potential

   

   where W and d are positive constants.

   1. What are the equilibrium values of this potential? Are they stable or unstable?
   2. Sketch the potential and discuss the motion, for all allowed values of E. Note whether the motion is bounded or unbounded, and any turning points.
   3. Specifically, for the cases E = -W/16 and E = -W/4, find the turning points and describe the allowed motion.
   4. What is the escape velocity for a particle at the origin?

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