System: Any system that does not undergo significant changes in internal energy. — Interactions: Any interactions that can be parameterized as mechanical work. Notable exceptions include heat transfer or radiation. |
Introduction to the Model
Description and Assumptions
If we ignore non-mechanical processes like heat transfer, radiative losses, etc., then we arrive at a model involving only mechanical energy which changes due to the application (or extraction) of the work done by non-conservative forces The non-conservative forces can be external forces exerted on the system or internal forces resulting from the interactions between the elements inside the system.
Learning Objectives
Students will be assumed to understand this model who can:
Relevant Definitions
Mechanical Energy
\begin{large}\[E = K + U\]\end{large} |
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Kinetic Energy
\begin{large}\[ K = \frac{1}{2}mv^{2} + \frac{1}{2}I\omega^{2}\]\end{large} |
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Work
\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}(\vec{s}) \cdot d\vec{s} = \int_{t_{i}}^{t_{f}} \vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large} |
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The system potential energy is the sum of all the potential energies produced by interactions between system constituents. Even when there are two system constituents involved (for example in a double star) each interaction produces only one potential energy. |
S.I.M. Structure of the Model
Compatible Systems
One or more point particles or rigid bodies, plus any conservative interactitons that can be accounted for as potential energies of the system.
In mechanics, the only commonly encountered conservative interactions are gravity and springs. |
Relevant Interactions
Any external force that performs that perform work on the system must be considered, and also any internal non-conservative forces that perform work. Any internal conservative forces that are present should have their interaction represented by the associated potential energy rather than by the work.
Law of Change
Mathematical Representation
Differential Form
\begin{large}\[ \frac{dE}{dt} = \sum \left(\vec{F}^{\rm ext} + \vec{F}^{\rm NC}\right)\cdot \vec{v} \]\end{large} |
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Integral Form
\begin{large}\[ E_{f} = E_{i} + \sum W^{\rm ext}_{fi} + \sum W^{\rm NC}_{fi} \] \end{large} |
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Diagrammatic Representations
Relevant Examples
 Examples Involving Constant Mechanical Energy
 Examples Involving Non-Conservative Work
 Examples Involving Gravitational Potential Energy
 Examples Involving Elastic (Spring) Potential Energy
 Examples Involving Rotational Kinetic Energy
 All Examples Using this Model
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