Lecture # 10: Hydrostatic Skeletons
This lecture explores the anatomy and physics of hydrostatic skeletons in organisms, focusing on the structures, such as the coelom and tissues, that allow for movement and support. The discussion includes the implications of LaPlace’s Law on stress distribution in hollow animals like worms, highlighting the relationship between size and wall thickness, and the geometric considerations in their anatomy. Through practical examples, including cylindrical and helical forms, we illustrate how these factors influence the strength and adaptability of hydrostatic skeletons in various environments.
Lecture # 10: Hydrostatic Skeletons
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Presentation Transcript
1 cell cellular sheet cellular bilayer one way gut ecto- derm endo- derm mouth anus bilayered canister Body Plan Evolution cephalization mesoderm
1 cell cellular sheet cellular bilayer one way gut ecto- derm endo- derm mouth anus bilayered canister Body Plan Evolution cephalization mesoderm
ectoderm coelom endoderm gut mesoderm coelom
pseudocoelom ectoderm mesoderm endoderm gut
r P d P= internal pressure r = radius d= thickness slice in half How does stress in a worm depend on geometry? Consider a hollow spherical animal…. P P what is stress in wall? Define tension, T, as force/length then T = s x d = r p / 2 T = ½ r p s = force/area = (p r2 p) / (2 p r d) = (r p) / (2 d) LaPlace’s Law: Tension in wall of sphere is proportional to radius and pressure. disk area ~ p r2 rim area ~ 2 p r d
1) longitudinal slice 2) slice in half Consider a cylindrical animal…. Equivalent to spherical case, Thus longitudinal tension, TLis same as in sphere of equal radius: TL = ½ r p 3) cap with hemisphere
d sc 1) transverse wedge slice area =2rd rim area =2 d d r d Circumferential or ‘hoop stress’ is twice than longitudinal stress. TC = 2 x TL Consider a cylindrical animal…. sc = force/area = (2 r d p) / (2 d d) = r p / d Again, TC = sc x d TC = r p
Implications of LaPlace’s Law: • Small worm withstand greater pressure than large worms. • Large worms should have thicker walls. • Square cross sections should be rare. P P P P Pierre-Simon Laplace 1749-1827 tension is infinite
Solve for dV/dq: V = D3 sin2q cos q Maximum volume at q = 54.73o 4 p L Consider a helical worm: L Volume = p r2 L Solve for volume in terms of q (helical angle): D = L cos q r = D sin q /(2 p )
muscle action Permissible Morpho-space V = d3 sin2q cos q 4 p elliptical profile circular section
Ontogenetic scaling of burrowing forces in the earthworm Lumbricus terrestris • Kim Quillin • J Exp Biol 203, 2757-2770 (2000)
