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Study on the Problem of “Catenary Pendulum”

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Author: Changrun Zhao

Abstract: The study of the “catenary pendulum” problem reveals intrinsic connections between the catenary curve and the brachistochrone (cycloid), leading to three key conclusions: (i) Among pendulums of equal length—the catenary pendulum, simple pendulum, and compound pendulum—the catenary pendulum exhibits the shortest oscillation period. (ii) The motion of the system's center of mass forms a brachistochrone curve, which constitutes the fundamental dynamic characteristic of the catenary pendulum. Notably, the midpoint of the rope has a significantly smaller amplitude than the system's center of mass. (iii) The free end of the catenary pendulum (its terminal point) represents the most dynamically active particle. Its trajectory forms an arc-shaped elliptical orbit symmetric with respect to the y-axis.

Keywords: Catenary Pendulum; Catenary; Optimal Sag; Cycloid; Rolling Circle's Initial Contact Angle

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