Reading:

Read B & O Sections 2.4 -- 2.6; Griffiths Chapter 1 -- 1.4; KB 361 lecture notes 11 -- 15.

Problems for Review:

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1. Use B & O problem 2-15 for the definition of cylindrical and spherical coordinates (you may also wish to review Griffiths section 1.4).
   1. Calculate a particle's velocity and acceleration in cylindrical coordinates.
   2. Given the particle velocity

      

      in spherical coordinates, derive the particle acceleration

      

2. A natural use of cylindrical coordinates might be to describe motion on the surface of a cylinder, in the absence of gravity.
   1. What force must be applied to keep a particle's motion entirely on the cylinder (i. e. to keep constant)?
   2. For constant , with no forces applied in the z and directions, what are the solutions for the particle's motion? Find their form in terms of t, and sketch them.
3. Griffiths Example 1.3 (p 15).
4. 1. B & O 2-15.
   2. For a potential dependent only on in cylindrical coordinates, sketch the equipotential surfaces, and the local gradient and force directions. Similarly, for a potential dependent only on r in spherical coordinates, sketch the equipotential surfaces, and the local gradient and force directions.
5. Consider the potential energy . Find
   1. The direction in which the potential energy is increasing most rapidly at and the magnitude of the rate of increase. What then is the force at ?
   2. The rate of change of potential energy with distance at in the direction . What is the component of force in this direction?
   3. The direction and magnitude of the force at .
   4. The magnitude of the force at x = -1, any y.
6. B & O 2-11.
7. 1. Griffiths Example 1.5 (p. 19).
   2. For and in Griffiths Example 1.5, find a closed curve C for which the line integral is nonzero, and evaluate the line integral.
8. 1. Griffiths Example 1.6 (p. 25).
   2. For in Griffiths Example 1.6, calculate . How is this consistent with your answer to (a)?

Problems to Solve:

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1. Recall that in spherical coordinates, the position vector has the following Cartesian components:

   

   where we have used the spherical coordinate unit vectors in Cartesian components

   

   1. By differentiating, show that the particle velocity is

      

      Identify velocity components in the directions respectively.

   2. Show that your expression for (a) is consistent with the infinitesimal displacement vector

      

      derived geometrically in Griffiths equation (1.68).

2. 1. Consider a potential energy , where V is in joules and x,y are in meters. Sketch an equipotential map, showing the contours V = 32, 24, 16, 8, 0 J.
   2. Sketch in the local gradient field on your equipotential map.
   3. If you start at the point (3,2) and proceed in the direction , is the potential increasing or decreasing, and at what rate? What is the component of force in this direction?
3. Griffiths Problem 1.27.
4. Treating the vector functions in Griffiths problem 1.15 as forces for each
   1. Calculate the curl.
   2. If the force is conservative, derive a potential V giving the force; if the force is not conservative, find a closed curve C for which the work integral is nonzero, and evaluate the work integral.
5. For the force ,
   1. Evaluate .
   2. Evaluate , for the curve C shown in Griffiths Figure 1.34 (p. 36).

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