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A geometrical theorem for the static equilibrium of a
common-point-force system has been proven by means of virtual-work principle:
The equilibrium point of a common-point force system has a minimal weighted
distance summation to every fixed point arbitrarily given on each force line
with a weighing factor proportional to corresponding force value. Especially
the mechanical simulating technique for its inverse problem has been realized
by means of pulley block. The conclusions for the inverse problem derived from
mechanic method are in accordance with that given by the pure mathematical
method, and the self-consistence of the theorem and its inverse problem has
been demonstrated. Some application examples in engineering, economy and
mathematics have been discussed, especially the possible application in the
research of molecular structure, has also been predicted.
The virial theorem is written by using the canonical equations of motion in classical mechanics. A moving particle with an initial speed in an n-particle system is considered. The distance of the moving particle from the origin of the system to the final position is derived as a function of the kinetic energy of the particle. It is thought that the considered particle would not collide with other particles in the system. The relation between the final and initial distance of the particle from the origin of the system is given by a single equation.
In this paper, an approach to
Pythagoras’ Theorem is presented within the historical context in which it was
developed and from the underlying intellectual outline of the Pythagorean
School. This was analyzed from a rationalism standpoint. An experiment is
presented to the reader so that they, through direct observation, can analyze
Pythagoras’ Theorem and its relation to the creation of knowledge. The theory
of knowledge conceptualization is used.