TITLE:
The Study on the Cycloids of Moving Loops
AUTHORS:
Gennady Tarabrin
KEYWORDS:
Curves, Loops, Cycloids
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.4,
April
24,
2018
ABSTRACT: The infinite set of cycloids is created. Each
cycloid of this set is defined as a movement trajectory of a point when this
point circulates on the convex closed contour of arbitrary form when this
contour moves rectilinearly without rotation on the plane with a velocity equal
to the tangential velocity of a point on circulation contour. The classical
cycloid is elements of this set. The differential equation of a cycloid set is
derived and its solution in quadratures is received. The inverse problem when
for the given cycloid it is necessary to fine the form of a circulation contour
is solved. The problem of differential equation of the second order with boundary
conditions about a bend of big curvature of an elastic rod of infinite length
is solved in quadratures. Geometry of the loop which is formed at such bend is
investigated. It is discovered that at movement of an elastic loop on a rod when
the form and the size of a loop don’t change, each point of a loop moves on a
trajectory which named by us the cycloid and which represents a circumference
arch.