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A more detailed introduction to Models and how they are used in introductory physics instruction. |
Models and Physical Models
A model is a simplified description of a complex entity or process. Models often highlight some particular feature of the modeled entity or process while blatantly ignoring others. For example some scale models of a particular airplane (e.g. an SR-71) are faithful in appearance, but can't fly. Other models of that same plane can fly but don't look very much like an SR-71. Still other models are not tangible, e.g. computer models of the supersonic air flow over the wings and into the air inlets for the engines of the SR-71. As George Box (an industrial statistician) once said, "All models are wrong. Some are useful."
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This WIKI aims to enable you to apply these models to the physical world. It therefore adds another category: what are the key restrictions and requirements about the system and its interactions, and the typical physical cues, that trigger the mind to recognize that a particular model applies.
Models in Newtonian Mechanics
Newtonian mechanics is a restricted domain that is concerned only with describing certain effects of interactions between objects. The power of Newtonian mechanics is that the small number of idealized frameworks (mechanical energy conservation, constant acceleration, momentum conservation, etc.) presented in the course are sufficient to describe many seemingly disparate situations.
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- One-Dimensional Motion with Constant Acceleration
- Simple Harmonic Motion
- Mechanical Energy, External Work, and Internal Non-Conservative Work
- Point Particle Dynamics (applying Σ F = m a to a point particle)
For a formal organization of these models, see Hierarchy of Models.
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A model will generally model only some of the structure in a physical system. For example, the engine of a car can be regarded as a "heat engine" to turn heat into mechanical energy (work), or as the "powerplant" - a source of a certain amount of power that can accelerate the car. The particularization of the model therefore relies critically on the selection of which state variables to include or exclude.
Specification of Basic Models for Mechanics
Name: Each model must have a name
Description: What separates this model from the others.Problem Cues: Common physical situations where model is a good approximation to reality.
Compatible Systems: The restrictions needed to ensure a given system can be adequately described by the model.
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Laws of Change: The mathematical rules that govern the evolution of a system that is described by the model. Often this is differential equation or an integral equation ( for example, F = dp/dt ).
Hierarchical Organization of the Models
A Models Hierarchy for Mechanics is used to help the learner understand that there are only four basic conceptual domains in Newtonian mechanics:
- Motion
- Momentum
- Mechanical Energy
- Angular Momentum
and that each domain has an associated class of relevant interactions (the class of interactions that cause evolution of the principle quantity)
- Motion is altered by acceleration, or equivalently, net force.
- Momentum is altered by external forces.
- Mechanical Energy is altered by non-conservative work.
- Angular Momentum is altered by external torque.
Pedagogical Usefulness
The key pedagogical reason for using models is to provide a framework within which students can organize the many facts and procedures they learn in introductory physics into a small number of useful models. They can then relate these models to the few overall theories that underlie the material, and think about the real world by recognizing situations or simplifications where these models apply. This leads to an understanding of the world through the ability to simplify and model physical situations that are new. From an expert/novice perspective models organize the many formulae and graphs on the novice's formulae sheet into a much smaller number of "chunks" of related things that are of reflective of nature's organization.
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- understanding the various representations and their interrelationships
- developing an ability to recognize physical situations where the model applies (even novel ones)
- being able to map the reality onto the model (i.e. to ignore unimportant things)
- being able to carry through the solution in any of the model's representations
- at a high level, being able to generalize the model
Understanding/Learning a Model
Understanding a model requires each student to reconstruct and interrelate its components in their own minds, and to understand the relationships among its representations. This is usually achieved by a laboratory course in which a group of several students follows a guided discovery procedure (but never a completely cookbook prescription) followed by a discussion skillfully conducted by a trained person. DEP feels that interactive lecture demonstrations together with problems involving transfer of representations should be able to perform much of this function.
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