hierarchy of models

The following hierarchical list has been developed and organized with several goals in mind:

• Each model must apply (approximately) to many situations in the world
• The models should cover mechanics as completely as possible
• The models should be ranked hierarchically with most general on top
• Each model should have a descriptive name and be accompanied by its most frequently used formula 

Even these requirements create some difficulties.  Firstly, we have to add a model for general energy conservation including thermal energy, even though this is usually considered part of Thermodynamics; Mechanics uses only the special case of Mechanical Energy, treating heat as "Lost Mechanical Energy".  Arranging the many models into a hierarchy with only four principle models (Kinematics, Energy, Momentum, and Angular Momentum) properly stresses that there are only a few basic models in Mechanics and that many of the most used ones are simply special cases of these few; however it obscures the logical chain of proof and derivation of the laws of mechanics from only F=ma and the definitions of kinematics.  (This usually starts with F=ma for point particles, then builds up and out to rigid bodies, systems of particles, momentum, angular momentum and energy.)  A further critique concerns the equations we associate with each model.  It is a simple operation of calculus to express the laws of physics in either differential (v = dx/dt, ΣF = m d²x/dt² , ΣT = I a), or integral form (E^final = E^initial + W^{nonConservative} ).  By presenting only the most frequently used form, we obscure this simplification for the benefit of helping students link titles and verbal concepts to equations. 

Hierarchy of Mechanics Models

*h4. 3D Motion

• • 2D Motion

• • • Circular Motion
      • Circular Motion with Constant Speed
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        $ (\alpha = 0)$

    • 1D Motion
      • 1D Motion with Constant Acceleration
        • 1D Motion with Constant Velocity
    • Simple Harmonic Motion

• Energy, Work and Heat

• • Work-Energy Theorem
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    $(Q = 0, \Delta U_

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    = 0)$

  • Mechanical Energy and Non-Conservative Work
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    $(Q = 0, \Delta U_

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    = 0)$

• • • Constant Mechanical Energy
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      $ (W_

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      = 0) $

• Momentum and Force

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  $ \vec p(t_f) - \vec p(t_i) = \int_

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  ^

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  \sum\vec{F_{ext}} $

• • Point Particle Dynamics
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    $ (\sum\vec F = m \vec a ) $

  • Constant Momentum

• Angular Momentum and Torque

• • Fixed-Axis Rotation
    • Statics
  • Angular Momentum Constant