Models in Physics

A model is a "simplified description of a complex entity or process".  It often highlights some feature of the modeled entity or process by blatantly ignoring others.  For example scale models of some particular airplane (e.g. SR-71) are faithful in appearance, but can't fly' other models of that same plane can fly but don't look much like an SR-71.  Other models are not tangible, e.g. computer models of the supersonic air flow into the air inlets for the engines of the SR-71. 

 Models intermediate between laws of nature, which are relationships among abstract quantities, and experimental/experiential reality.  While the laws of nature apply to any situation in the real world, the model simplifies consideration of which laws and how they apply, and allow the mind to consider the world in much larger and richer "chunks" that does starting from fundamental laws for each new problem.  However, one must be careful that the assumptions and simplifications made in discussing the model accurately apply to the physical situation at hand. 

According to David Hestenes, one of the founding fathers of modeling instruction,

"A physical model (in physics) is a representation of structure in a physical system and/or its properties."  A physical model will describe the system, the state of its constituents (including perhaps geometric and temporal structure), their internal interactions, external interactions, and the changes of state (i.e. behavior).  Physical models combine the definitions, concepts, procedures, interactions, laws of nature and other relationships that model some aspect of the physical world. A physical model is a mentally linked collection of physical laws, concepts, equations, and associated descriptions that relate to a particular common patterns found in nature. 

Examples of physical models relevant to Newtonian mechanics are motion with constant acceleration, harmonic motion, energy conservation, and applying ΣF = ma to a point particle - see Hierarchy of Models.  A model consists of the following pieces:

1. the physical systems/situations where the model applies and vocabulary of involved objects, state variables, and agents(interactions)  involved.
2. specification of the independent and dependent (measurable) state variables that characterize the system and which the model interrelates
3. what idealizations and physical theories underlie the model and the resulting equations, representations
4. descriptions of the model and interpretation of its predictions as expressed in all various useful representations
5. the behavior/change in state, geometric, temporal, and interaction structure
   

A physical 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.

System: is the portion of the physical or mental universe described by the model, thereby creating internal and external regions.  The system is the object or the group of objects (sometimes including electromagnetic fields) that are being modeled. It is the part of the universe under consideration.

The environment is the rest of the universe which contains the object interacting with the system under consideration.  The external interactions completely determine the effects of the rest of the universe on the state variables of the system, and allow the system to be considered in isolation from the rest of the universe.

state variables will completely describe the system (in a particular physical model).  In the examples above these might be the volume of the space in the engine cylinder above the piston and the temperature and pressure of the gas in this volume (heat engine), or by the torque and revolutions per minute of the shaft exiting the engine (powerplant).  Whatever the state variables, they change due to Laws of Change that generally are due to Interactions.

Interaction is the action of one physical object upon another. In general interactions are between two objects, although in electromegnetism it is frequent that one of the "objects" is a magnetic or electric field. If an interaction is between two objects both in the system it is "internal".  If one of the objects is outside the system it is considered external.  If both interacting objects are outside of the system then this interaction does not affect the system.  Generally the interactions may be represented as mathematical expressions, for example the force laws that interrelate the masses, their separation, and the gravitational constant to the gravitational force.  

Law of change: A great deal of mechanics is about change.  Central to change is the physical law describing how the state variables of a system or some object in it change, often because of its interaction with some other object in the system or with the environment. For example, F = ma , F is the cause of making velocity  change. (Similarly heat and work cause changes in the internal energy of a system). Generally laws of change are expressed in

differential form, e.g.F=dp/dt,

but they may also be expressed in corresponding

integral form that results from integrating the differential form and relates the final condition to the initial condition and an integral of the interaction, e.g,

Physical Model

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Characteristics of Models: 

Name:  Each model must have a name 

Verbal Description:  A sentence or two

Motivation/Examples: Common physical situations where model is a good approximation to reality

Assumptions and Limitations:
Preconcepts - concepts involved that should be known
PriorModels - models assumed known
New Vocabulary 

Separating Model from Environment - Structure of Models:
Internal constituents
External Agents
Interactions Considered
Assumptions and Approximations 

Key Descriptors Necessary
Internal objects
Description of State
Interaction(s), Agent(s) of Change 

Representations
Different ways to represent model 

Laws of Interaction
How does agent of change behave? 

Laws of Change
Often this is differential/integral Eq. (F=dp/dt)
And may be a special case (e.g. F=ma) 

A useful model fits many real situations to a good approximation.  Some useful models used by physicists to think about the physical include: motion with constant acceleration, the harmonic oscillator, the two level quantum system, Feynman diagrams and the Schwartzschield metric (which applies the law of General Relatively to find the warping of space-time due to a central sphere of matter).  A models Hierarchy for mechanics is available.

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, 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, and 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. 

Models are idealizations of physical reality involve a particular structure or pattern.  Models can be mathematical, logical, pictoral, or a physical object (but not usually in physics).  Models only approximate reality; they represent an idealization of reality (e.g. with no friction, ignore bending of rigid bodies, etc.), but generally they are applicable to many situations (else the model is not useful).  Models generally involve a cluster of several concepts and theories (e.g. harmonic motion involves kinematics, F=ma, and a linear restoring force).  Models almost always may be expressed in several representations, and the cross connection of these representations provides a richness for the models.  For example, motion with constant acceleration may be represented with standard equations, strobe pictures of the object, graphs or tables of kinematic variables vs. time, or a concise verbal description.  The two level system may be represented using the equations following from time-dependent perturbation theory, the Bloch vector, and the density matrix.   

Physicists and educational psychologists agree that understanding a model implies fluency with, and ability to transfer between, all of its commonly used representations.  A physicist familiar with any model can recognize/describe/understand/quantitatively predict situations that fit within the model's assumptions with little effort (as an exercise), but is typically confronted with a "problem" if even a small discrepancy exists between the situation at hand and the relevant model. 
Being able to understand and use a model involves:

1. understanding the various representations and their interrelationships
2. developing an ability to recognize (even novel) physical situations where the model applies
3. being able to map the reality onto the model (i.e. to ignore the unimportant)
4. being able to carry through the solution in any of the model's representation(s)
5. 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 in which a group of several students follows a more or less guided discovery procedure (but never a completely cookbook prescription) followed by discussion conducted skillfully 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.   

However it is done, successful learning involves the student understanding the pieces of the model and being able to use it in context-rich problems (i.e. extract the relevant variables from a real world story as well as the minimalist presentation typically found in textbook problems).  Hestenes would contend that a key to modeling is that students become skillful at constructing models for new situations, so he and many true modelers would disprove of our approach.  Making a model and learning to apply it typically takes two weeks, limiting the number of models to ~6 for the course.  Our hope is that we can teach ~ one model/week by starting with only 4 general models and indicating how several other models are subcases of the general models.