Keys to Applicability
Can be applied to any system for which the change in mechanical energy can be attributed to work done by [non conservative forces] (as opposed to processes like heat transfer, radiative losses, etc.). 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 specially useful for systems where the non-conservative work is zero, in which case the mechanical energy of the system is constant.
Description
[Model Hierarchy]
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Assumed Knowledge
Prior Models
Vocabulary
- system
- internal force
- external force
- conservative force
- [non-conservative]
- kinetic energy
- [gravitational potential energy]
- [elastic potential energy]
- mechanical energy
Model Specification
System Structure
Constituents: One or more point particles or rigid bodies. Technically, the system must be defined in such a way as to contain all objects that participate in any non-negligible conservative interactions that are present.
For example, for systems subject to earth's gravity, the earth should technically be included in the system, though it is usually sufficient to treat it as a rigid body that is at rest and has infinite mass. If this approximation is made, the earth will have zero kinetic energy (it will not change its velocity, since it has infinite mass).
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 a potential energy rather than by work.
Descriptors
[Object Variables]: None.
Object masses and moment of inertia can technically change in this model, so they are state variables.
[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.
[Interaction Variables]: Relevant non-conservative forces (FNC,jk) or the work done by the non-conservative forces (WNC,jk).
Model Equations
Relationships Among State Variables
\begin
[ E = K^
+ U^
]
[ K^
= \sum_
^
\frac
m^
(v^
)^
+ \frac
I^
(\omega^
)^
]
[ W^
_
= \int_
\vec
^
_
\cdot d\vec
^
]
[ W^{NC,{\rm sys}} = \sum_
^
\sum_
{N_{F,j}} W
_
]\end
where N is the number of system constitutents and NF,j is the number of non-conservative forces acting on the jth system constitutent.
The system potential energy is the sum of all the potential energies produced by interactions between system constituents.
There are always two system constituents involved in any interaction. It is technically important to count each interaction only once, even though two constituents are involved.
Some common potential energy relationships are:
Near-Earth Gravity
\begin
[ U_
^
= m^
gh^
]\end
Universal Gravity
\begin
[ U_
^
= -G\frac{m^
m^{k}}{|\vec
_
}} ]\end
Elastic
\begin
[ U_
^
= \frac
k^
(\vec
^
-\vec
^
-x_
^
)^
]\end
where xjk0 is the natural length of the spring.
Mathematical Statement of the Model
\begin
$E_
= E_
+ W^{NC,{\rm sys}} $ \end
where WNCi,f is the work done by the all the non-conservative forces on the system between the initial state defined by Ei and the final state defined by Ef and is given by
\begin
$ W_
^
= \int_
^
\sum \vec
^
. d\vec
$ \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 |
