Goals of this WikiTextBook

Given the many textbooks on Introductory Newtonian Mechanics using Calculus, we would have little intellectual incentive to write another, except that we have important goals that are different from existing texts of either the "standard" or "reform" variety. 

Our three top goals are:
 

  • Give our students the strategic knowledge to attack new and unfamiliar problems in Mechanics using a pedagogy that we call Modeling Applied to Problem Solving
  • Help our students obtain an overview of the subject, and - by using a Glossary - help them understand the specialized vocabulary of Mechanics
  • Explore the advantages of an electronic format to reach these goals, and generally to construct an environment for learning that is superior to a printed textbook.

Modeling Applied to Problem Solving (MAPS) - a Strategic Pedagogical Approach

Problems vs exercises


A problem exists when you understand what would constitute a satisfactory solution but you are unable to see how the knowledge and information you have can be used to obtain a solution.

An exercise exists, by contrast, when you know how to obtain a satisfactory solution as soon as you comprehend the given information.

Here are examples of an exercise and a problem (by the standards of the typical student of this text).

Consider this question: 

An ice skater of mass m=55kg experiences a force of 20N when she is at rest due to the wind.  Find her acceleration neglecting friction with the ice.

For most of you, this question is (or soon will be) an exercise.  An exercise is defined as "Once you understand the question, the plan of solution is obvious."  Once you understand that this question involves mass, force, and acceleration you will think of F=ma.  The plan of attack is then obvious -  rearrange this equation to give a=F/m and substitute the given quantities to obtain a = 0.36 m/s.

The following question should be a problem for you (if it's not, you may not get much from reading this WIKItextBook):

An ice skater of mass m=55kg experiences a force of 30N when she is at rest due to the 10 m/s wind.  Find her position vs. time neglecting friction with the ice and assuming that the drag force of the wind is proportional to the square of its relative velocity.

The plan for solving this question involves 

  1. Find the acceleration of the skater - unfortunately it involves her velocity which is unknown and is hence a differential equation.
  2. This is the only kind of differential equation you will need to be able to solve in this WIKItextBook - it can be solved by separation of variables
  3. The resulting v(t) can be integrated to find x(t)

An Approach Geared Toward Problems

Our overarching goal is to help you learn to solve mechanics problems.

Imagine that you have a problem.  You understand the material in your textbook, and you now understand what's asked in the problem.  But you realize that you don't know how to solve it.  You need to figure out what to do next.  Most textbooks suggest some variant of Polya's (1945) four step process:

  1. Understand (visualizing the motion is very helpful for mechanics)
  2. Plan your solution
  3. Solve the Problem
  4. Check Over for Reasonableness and Lessons learned.

Unfortunately, conventional textbooks don't bridge the gap between 1 and 2 - i.e. how do you plan a solution if you don't understand the problem well enough to solve it?

The Modeling Approach to Problem Solving builds a bridge for connecting your physics understanding and the problem by starting from both ends. It organizes your declarative and procedural knowledge into a Hierarchy of Models that facilitates both learning it and accessing it on the basis of fundamental interactions, and it gives a strategy for understanding the problem in a way that facilitates accessing your hierarchy of factual and procedural knowledge so that it can be applied to the problem at hand. Practiced together, these combine to give you strategic knowledge.

Organization of this WIKItextBook

Conceptual Organization

This WikiTextBook arranges the core concepts and their associated conditions of validity and equations of change into about a dozen models. The models are then arranged into categories that depend on the type of system and its interaction. With this organization, determining the system and interactions present in a problem hints at the category of models that can solve it. Within each category, the models are arranged in a hierarchy to make it clear that some are simply special cases of the one(s) above suitable for a restricted set of circumstances (e.g. that the force does not change with time).

The SIM Approach

If you are a novice skier, your teacher will say something like “Bend your knees and put equal weight on both skis.” as you are about to start down the slope. The teacher would not need to tell this to an experienced skier, who would naturally do this. In the same way, this WIKI will ask you to begin each problem with a prescribed, conceptual rubric. This rubric has been carefully designed to teach steps that every expert problem solver goes through when approaching a mechanics problem, even if they don't always mention the steps explicitly. This rubric is at the heart of your transition from a formula hunter to expert who starts problem solving from fundamental principles.

Our S.I.M. Problem Solving Approach asks you to think in terms of the system and its Interactions (forces) in your problem, rather than to key on the superficial features like “inclined plane” or “pulley” or “collision problem”. The system and interaction serve as a guide to your selecting the best Model from the hierarchy. This will then lead to a graphical approach or an equation of change that will provide the best route to the solution.

Here is an expanded description of the components of the S.I.M. Approach:

System: Specify the object or objects whose motion you are describing and state whether they will be considered as point particles or rigid bodies.

Interactions: Specify the forces that act on the system and classify these forces as internal vs. external, conservative vs. non-conservative, and/or torque-producing vs. non torque-producing.

Model: A selection from the Hierarchy of Models provided to you. The model that you choose to deploy is an idealized mathematical representation of the problem.

When you review example problems or work problems on your own, it is important to keep SIM in mind. What and why was that particular system employed? Review the forces and ask which one is the least obvious. Finally, why was the successful model selected? Did the problem contain several pieces that required different models – either in time or in space? (You should also look for tricky constraints, e.g. from pulleys, geometry, etc.)

This approach is used in all the worked examples presented in this WIKI.

Exploiting Electronic Format

Multiple Parallel Organizations of the Content

A big part of Strategic Knowledge is organizing your knowledge of Mechanics - gaining a perspective on its major categories, what are the characteristics of each, and which of the facts and procedures you know fit under that category (some facts and procedures may fit under more than one).  Not long ago a group of educators reconsidered the textbook http://serc.carleton.edu/textbook.  Chief among their criticisms was that conventional textbooks do a poor job of imparting strategic knowledge.  They partition the subject into chapters, but don't say how the chapters relate to each other, or how to decide when a problem needs the material taught in a particular chapter.  Often the only overview is provided by the table of contents, and this is not very useful to someone who doesn't understand the many technical words it contains.

The Content Guide page of this WIKItextBook offers you several ways to get an overview of mechanics.  Foremost is the organization of the content into four categories, each with its hierarchy of models hierarchy of Models.  In addition, we provide an expandable table of contents.  This table has a series of Traditional Lessons which present the models that cover the core material in much the usual sequence of topics (rather than the more abstract categories and model hierarchy).  It also has a section on Interactions - very important because they are the key to strategic knowledge in mechanics.

Hyperlinks Facilitate Instant Acquisition of Unknown Terms

Our vocabulary glossary helps you to understand with precision the various technical terms used, a necessity to learn any scientific topic.  This is particularly important in mechanics, because so many of its concepts have become everyday words with broad common usage meanings - like acceleration, momentum and energy. We will try to incorporate the Glossary summary when technical words are introduced, and we will make many of them links to remind you to be sure you understand them.  The Glossary entries expand with a click to help you understand these words more fully and appreciate some of their implications. &nbsp

Comments Facilitate Instant Feedback

This Wiki text is still under development, and we greatly appreciate constructive feedback, which can easily be given using the Comments option at the bottom of every page.

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