Copasetic Flow

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Pages: 176-178 (Chapter 5, Example 5.2) Reading physics books, it often occurs to me that the authors must be aware of some patterns or 'rules of thumb' that the reader may not be privy to. Today's post expands a very truncated example from Marion and Thornton and hopefully clarifies it. This post also poses several questions in search of those patterns and rules mentioned above. After explaining the calculus of variations and the importance of Euler's equation Marion and Thornton follow up with a concrete example: the brachistochrone. The problem of the brachistochrone is to determine the path for a particle to move from point A to B under the influence of a constant force, (gravity for example), in the least amount of time. The 'least amount of time' phra...

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 68 This post looks at Example 2.10 that investigates the motion of a charged particle in a magnetic field. The example is fairly straightforward with one exception. When determining the equations of motion, the authors propose a solution for the system of differential equations discussed below and reference example C.2 of Appendix C. It’s not immediately apparent how to use Example C.2 to arrive at the authors’ solution, so the steps are outlined in detail here. If you have questions, or suggestions, all comments are always welcome! The original system of coupled differential equations is: First, the authors’ differentiate both equations and then substitute the results into the other: at this point, the book suggests using the technique of ex...

This installment of “It’s Obvious. Not!” looks at: Book: “Classical Dynamics of Particles and Systems” Edition: third Authors: Jerry B. Marion and Stephen T. Thornton Publisher: Harcourt Brace Jovanovich Page: 52 This installation of the series provides a few clarifications into the example presented in the textbook and asks even more questions. I have a feeling that readers steeped in differential equations will immediately follow the reasoning of the example as it is written in the textbook. Please, if you have answers to the remaining questions below, or even ideas, please comment. Thanks! The first example of the chapter titled “Newtonian Mechanics” asks the reader to find the velocity of an object sliding down a ramp. The solution for the acceleration, (second derivative of the position x), has already been derived as: The process for deriving the velocity as a function of position illustrated by the author starts with the above equation for accelera...