Keys to Applicability
If we ignore 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 just 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. It is especially useful for systems where the non-conservative work is zero, in which case the mechanical energy of the system is constant. These can be recognized explicitly by statements like "frictionless surface" "smooth track" or in situations where only forces that may be represented by potential energy are involved.
[Model Hierarchy]
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Assumed Knowledge
Prior Models
Vocabulary
- system
- internal force
- external force
- conservative force
- [non-conservative]
- kinetic energy
- rotational kinetic energy
- [gravitational potential energy]
- [elastic potential energy]
- mechanical energy
System Specification
Constituents
One or more point particles or rigid bodies, the (internal) interactions between them, whether conservative or not, and any fields applied externally such as a uniform gravitational field.
The admission of fields avoids requiring the system to contain the source objects of the conservative interactions that are represented by the fields. In the example of earth's gravity, this is justified because the earth will have no change of its kinetic energy (its infinite mass implies no change of velocity) or of its potential energy (all potential energy is attributed to objects in the system influenced by the fields).
[State Variables]
Mass (mj) and possibly moment of inertia (Ij) for each object plus linear (vj) and possibly rotational (ωj) speeds for each object, or alternatively, the kinetic energy (Kj) may be specified directly.
If non-conservative forces are present, each object's vector position (xj) must be known throughout the time interval of interest (the path must be specified) unless the work done by each force is specified directly.
When a conservative interaction is present, some sort of specific position or separation is required for each object (height hj for near-earth [gravity], separation rjk for universal gravity, separation xjk for an elastic interaction, etc.) unless the potential energy (Ujk) is specified directly.
Alternately, in place of separate kinetic and potential energies, the mechanical energy of the system (E) can be specified directly.
Interactions
All forces that do [non-conservative] work on the system must be considered, including internal forces that perform such work. Conservative forces that are present should have their interaction represented by the associated potential energy rather than by the work.
Occasionally it is easier to consider the work of conservative forces directly, omitting their potential energy.
[Interaction Variables]
Relevant non-conservative forces (FNC,jk) or the work done by the non-conservative forces (WNC,jk).
Model Relationships
Relationships Among System Variables
In the following formulas, N is the number of system constitutents and NFj is the number of non-conservative forces acting on the jth system constitutent.
\begin
[ E = K^
+ U^
]
[ K^
= \sum_
^
\left(\frac
m^
(v^
)^
+ \frac
I^
(\omega^
)^
\right)]
[ W^
_
= \int_
\vec
^
_
\cdot d\vec
^
]
[ W^
_
= \sum_
^
\sum_
^{N_
{j}} W
_
]\end
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.
Law of Change
\begin
$E_
= E_
+ W^
_
$ \end
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RELATE wiki by David E. Pritchard is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License |
