[Model Hierarchy]
Keys to Applicability
This model is [generally applicable] (assuming knowledge of the external forces and system constituents), but is especially useful when:
- describing the momentum of systems where external forces are absent (system momentum will be constant).
- estimating the force in a process that occurs in a very short time interval as in collisions (impulse will be easier to determine than force).
"Very short" is ambiguous. In collisions, we generally use the conservation of the system's momentum to yield information about the change in momentum of the individual system constituents. A "very short" collision in that context is one in which the internal forces produce an impulse on the constituent of interest that is so much larger than the impulse from external forces that the external forces can be neglected. Of course, that is a relative criterion as well, and will depend on the desired accuracy of the calculation.
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Assumed Knowledge
Prior Models
Vocabulary
Model Specification
System Structure
Constituents: System is composed of Point particles.
Rigid bodies may be treated as point particles with positions specified by the center of mass positions of the rigid bodies when this model is used.
Interactions: Only external forces need be considered, since internal forces do not change the system's momentum.
Descriptors
[Object Variables]: None.
This model does not require that constituent masses be constant. Thus, mass is a state variable for this model.
[State Variables]: Mass (mj) and velocity (vj) for each object or momentum (pj) for each object inside the system.
[Interaction Variables]: External forces (Fext,k) or, alternately, impulses may be specified (Jext,k).
Model Equations
Relationships Among State Variables
If not directly given, momenta can be obtained using the definition:
\begin
[ \vec
^
= m^
\vec
^
]\end
The relationship implied by the model is most easily expressed in terms of the system momentum, which is the vector sum of the constituent momenta. For a system composed of N point particles:
\begin
[ \vec
^
= \sum_
^
\vec
^
]\end
The number of point particle constituents in the system is not necessarily fixed. A totally inelastic collision, for example, could be viewed as a process where two separate system constituents exist in the initial state, but only one is present in the final state.
Mathematical Statement of the Model
\begin
[ \frac{d\vec
^{\:\rm sys}}
= \:\sum_
^{N_{F}} \vec
_
]\end
or, alternately:
\begin
[ \vec
^
_
= \vec
^
_
+ \sum_
^{N_{F}} \vec
_
= \vec
^
_
+ \int \sum_
^{N_{F}} \vec
_
\:dt ]
\end
Relevant Examples
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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 |
