of J1
Systems Particles: Observations
We have developed tools for use in analysis of problems that include
models more complex than single particles, that is, a system that is
modeled as a general collection of n particles (no restrictions on model)
For systems with large n, tools may be difficult to use for various
reasons, including the following two major problems:
1
For general system of n particles, cannot get ALL EOM just by
utilizing differential equations involving the center of mass.
2
Relationships we do have for H , W, KE require summations
over all particles
applied one at a time
Impractical for large n
General class of "systems of particles" must be decomposed into
subclasses. Mechanical tools can be modified for each subcategory to be
applied easily and efficiently.
J2
Each subcategory can be defined by additional assumptions (or
constraints) associated with the model. The additional assumptions
(constraints) are useful to reduce the differential equations to forms
better suited for analysis.
(None of this is helpful unless a subset, or subcategory, will represent a
large range of actual problems; analysis; will analysis produce
physically reasonable results; good approximations to a wide range of
'real world' problems)
A system of particles that can be modeled as a rigid body is one such
model (subcategory).
This model works very well in problems for which it is reasonable to
assume that the distance between individual particles remains constant.
The single additional assumption allows differential equations
to reduce to an extremely useful form
(Applies especially well when number of particles that
is, continuous medium)
J3
Rigid Body
collection of particles such that the distance between
each pair of particles remains fixed
Utilizing RB model helps beat the two major problems we noted
previously.
To modify the differential equations, it is necessary to discuss some
additional concepts. The utility of the RB model to overcome our two
main problems is associated with concepts:
1
Too many EOMs required
Reduce the number of EOMs required to some maximum
value (concept of degrees of freedom)
2
New representations for H , W, KE are necessary
Transformations and mass moments of inertia
(new representations to evaluate summations)