Models in Physics

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. 

Physical Models in science model the response of a system (generally composed of one or a few objects) to physical interactions.  Physical Models embody concepts and mediate between experiential/experimental reality and the laws of nature, which are fundamental relationships among abstract quantities (for example, F = ma or the conservation of energy).  While the laws of nature apply to any situation in the real world, a model is generally much more specialized and concentrates on a concept, a common pattern or a situation (for example, uniform circular motion), highlighting the relevant physical laws and how they apply.  Models generally include several representations, each pf which is a different way to conceptualize the model's applicability and implications.  The modeler's mind typically recognizes key concepts in a particular physical situation that are similar to those of one of these representations, suggesting which model is applicable.  The model's different representations are then all available.  Typically, one or two will give intuitive insight, and another will lead to a solution (often analytic or numerical).  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 (that is to say, the system's 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 representations and descriptions that relate to a particular common pattern found in nature. 

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.

Examples of physical models relevant to Newtonian mechanics are:

• motion with constant acceleration
• harmonic motion
• energy conservation
• linear dynamics (applying Σ F = m a to a point particle)
  For a formal organization of these models, see [*Hierarchy of Models*]. 

A model consists of the following pieces:

1. the situations, conditions, and idealizations under which the model applies
2. vocabulary of involved objects, state variables, and agents (interactions)  involved
3. specification of the independent and dependent measurable state variables that characterize the system and which the model interrelates
4. what physical theories underlie the model and the resulting equations and representations
5. descriptions of the model and interpretation of its predictions as expressed in all various useful representations
6. 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.

Model and Conditions of Applicability: short statement of the idealized situation described by the model, including conditions under which it applies to reality

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 (if the engine is being modeled as a heat engine). Or they might be the torque and the number of revolutions per minute of the shaft exiting the engine (if it is being modeled as a powerplant).  Whatever the state variables, they change according to the Laws of Change, generally due to Interactions.

Interaction is the action of one physical object upon another. In general interactions are between two objects, although in electromegnetism one of the "objects" may be a magnetic or electric field. If an interaction is between two objects both in the system it is an "internal" interaction.  If one of the objects is outside the system it is considered an external interaction.  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 law that relates the masses of two objects, their separation, and the gravitational constant to the gravitational force between them can be expressed as a simple formula.  

Law of change: A great deal of mechanics is about change.  The physical laws describe how the state variables of a system change, often because of interaction with some other object in the system or with the environment. For example,  in the expression F = ma , the force F is the cause of the acceleration, a, which is the rate of change of an object's velocity. Similarly, heat and work cause changes in the internal energy of a system. This is the First Law of Thermodynamics. Generally laws of change are expressed in two ways: instantaneous change involves the

differential form, for example:

F=dp/dt,

whereas the change occurring from one point in time and space to another (between an initial and a final state) involves the

integral form that results from integrating the differential form, for example:

Physical Model

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
Interactions, Agents of Change 

Representations
Different ways to represent model 

Laws of Interaction
How does agent of change behave? 

Laws of Change
Often this is differential equation or an integral equation ( for example, F = dp/dt )
This may be a special case (for example, F = ma ) 

A useful model is one that fits many real-life situations to a good approximation.  Some models used by physicists to think about the physical world 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 used to help the learner understand that there are only four basic concepts

• Motion,
• Momentum,
• Energy, and
• Angular Momentum,

but that each has several models that apply under special circumstances (for example, when the object moves precisely in a circle).

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 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. 

Models are idealizations of physical reality that 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 (where we can, for example, exclude friction, or ignore the bending of rigid bodies, and so on), but generally they are applicable to many situations (if they weren't, the model would not be useful).  Models generally involve a cluster of several concepts and theories (As an example, 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 quantum mechainical 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 among, 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, 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 physical situations where the model applies (even novel ones)
3. being able to map the reality onto the model (i.e. to ignore unimportant things)
4. being able to carry through the solution in any of the model's representations
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 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.   

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, and so he would disapprove of our approach.  Making a model and learning to apply it typically takes two weeks. This limits the number of models we can study to about 6 for the course.  Our hope is that we can teach roughly one model/week by starting with only 4 general models and indicating how several other models are subcases of the general models.