. The Biological bulletin. Biology; Zoology; Biology; Marine Biology. Figure 4. (a) Location of the peristomial membrane in an urchin. The star in both figures marks the point of attachment to the edge of the penstome. (b) Geometric model of the penstomial membrane. The angle 6, and the radius of curvature of the membrane are not independent. Zero vertical displacement occurs when the membrane is horizontal. by the test. The vertical force, ft,, exerts a force, f,,,, in the membrane. (6) cos (6) ' where 0 is the angle between the vertical and a tangent at the central margin of the membrane (at
. The Biological bulletin. Biology; Zoology; Biology; Marine Biology. Figure 4. (a) Location of the peristomial membrane in an urchin. The star in both figures marks the point of attachment to the edge of the penstome. (b) Geometric model of the penstomial membrane. The angle 6, and the radius of curvature of the membrane are not independent. Zero vertical displacement occurs when the membrane is horizontal. by the test. The vertical force, ft,, exerts a force, f,,,, in the membrane. (6) cos (6) ' where 0 is the angle between the vertical and a tangent at the central margin of the membrane (at the point of at- tachment of the peristomial membrane to the teeth) (Fig. 4b). The force, f,,,, on the membrane corresponds to a tension, T, (force per length) in the membrane of T = (7) where r, is the radius of the central margin of the peristo- mial membrane. From Laplace's equation (4) Ap = — , (8) where r,,,,, is the radius of curvature of the membrane. In using equation (4) rather than (5) we make two simplifying assumptions: that a second horizontal radius of curvature can be ignored, and that the curve formed by a vertical cross section of the peristomial membrane has a single radius of curvature at every point. In reality this curve may have variable radii of curvature. A more realistic model would add an unjustifiable degree of complexity 2 cos (arctan (v/h)) cos (6 + arctan (v/h)) Substituting through equations 6, 7 and 8, Ap = ft, cos (arctan (v/h)) cos (6 + arctan (v/h)) Trhr, cos (6) (9) (10) which is shown in Figure 5. This graph shows that many possible combinations of pressure, protraction, and 9 are possible when only the force balance on the membrane is considered. Initially, this may seem counterintuitive. Intuition suggests that as the lantern protracts, the internal pressure should get more and more negative relative to outside as the membrane pulls more and more on the constant volume of water inside the urchin. That this pressure pattern is not implie
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Keywords: ., bookauthorlilliefrankrat, booksubjectbiology, booksubjectzoology
