On a Class of Planar Geometrical Curves with Constant Reaction Forces Acting on Particles Moving Along Them

In this paper, we find the dependence of the reaction force N ( y ) of a curved trough of arbitrary shape described by a function y ( x ). Based on the extremum condition dN/dx valid for any point of the abscissa axis, we examine the equation N ( y, y', y'' ) = const whose solutions determine the desired class of curves y ( x ). We obtain an analytic solution of this equations and perform numerical simulations for various values of parameters. Examples of functions y ( x ) for which N = const are presented.