Final answer:
The shape assumed by a freely deforming cable in response to concentrated loads is known as a Catenary Curve. It is important in the design of suspension bridges and similar structures where it appears naturally due to gravity.
Step-by-step explanation:
The term used to describe the shape assumed by a freely deforming cable in direct response to a set of concentrated loads is a Catenary curve. This shape is often observed in cables and chains hanging under their own weight and supported at the ends, forming a natural curve under the influence of gravity. The Catenary curve is significant in the design of structures like suspension bridges, where the cables often follow this form due to evenly distributed weight along their length. In contrast, a funicular polygon results from a set of discrete loads applied at specific points along a cable, which shapes the cable into a series of straight-line segments. Additionally, other shapes such as Archimedean Spirals and Parabolic Arcs have specific mathematical definitions and applications, but the Catenary Curve is thus distinctive due to its appearance under uniform gravity.
An actively contracting cable network can take on a Circular Arc morphology, especially within cellular structures. This has been studied in various models, including tensegrity and finite element models (FEM), which consider the unique mechanical properties of biological materials such as elasticity and the generation of tension. Such models consider the asymmetrical mechanical response of filaments and tension generation by molecular motors, aiding in understanding biological processes and engineering applications.
It's noteworthy to mention that while both catenary curves and circular arcs are curves created under different types of forces or constraints, the underlying physics and resulting curves can be quite distinct.