page_title: Flotation of flexible objects
section_title: Two rigid plates connected by a torsional spring
hierarchical_section_title: Flotation of flexible objects / Mathematical model: flotation of hinged plates / Two rigid plates connected by a torsional spring
caption_attribution_description: English: Diagram detailing the submerging of two plates hinged by a torsional spring.
context_page_description: Flotation of flexible objects is a phenomenon in which the bending of a flexible material allows an object to displace a greater amount of fluid than if it were completely rigid. This ability to displace more fluid translates directly into an ability to support greater loads, giving the flexible structure an advantage over a similarly rigid one. Inspiration to study the effects of elasticity are taken from nature, where plants, such as black pepper, and animals living at the water surface have evolved to take advantage of the load-bearing benefits elasticity imparts.
context_section_description: The following analysis is taken largely from the work of Burton and Bush, and offers some mathematical insight into the role that flexibility plays in improving load-bearing characteristics of floating objects.
  Consider two plates of infinite width, thickness , and length  that are connected by a torsional spring with spring constant per unit width . Furthermore, let  be the angle between a plate and the horizontal, and  the from where the meniscus meets the plate to the horizontal. The distance from the undisturbed water line to the plate's outer edge is . The density of water is , the density of air is considered negligible, and the plate density, , shall be varied. All systems naturally assume a configuration that minimizes total energy. Thus, the goal of this analysis is to identify the configurations (i.e., values of  and ) that result in a stable equilibrium for a given value of .
  For a total system energy of , it is natural to distinguish sub-components:
   
  : Work done on the system
  : System potential energy
  In defining , there are several associated components:
    
   is the work done on the interface by hydrostatic pressure
   is the work done on the plates by hydrostatic pressure
   is the work done on the plates by gravitational force
   is the work done on the plates by surface tension forces
  Similarly, the system potential energy, , is taken to be composed of two terms:
   
   is the surface energy of the water/air interface
   is the energy stored in the torsional spring and is equal to  
  There are two ways in which the system energy can change by an incremental amount. The first is a translation of the center of mass of the plates by some distance . The second is an incremental change,  in the hinge angle. Such a change will induce a new moment.
  As mentioned, the system will seek the orientation that minimizes  in order to find point of stable equilibrium. Writing out these terms more explicitly:
   
  Here,  is the equation air/water interface,  is the incremental displacement of the interface, and  is the surface tension of water.
  For a given value of , stable equilibrium configurations are identified as being those values of  and  that satisfy
   
  Taken in a different light, these conditions can be seen as identifying  and  that result in zero net force and zero net torque for a given .
caption: Two rigid plates connected by a torsional spring and submerged between the water line. Since the material is taken to be hydrophobic, the water deforms, forming the meniscus at the plate edge.
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