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# An object of mass m is attached to the lower end of a

ISBN: 9781111427412 173

## Solution for problem 3.102 Chapter 3

Advanced Engineering Mathematics | 7th Edition

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Advanced Engineering Mathematics | 7th Edition

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Problem 3.102

An object of mass m is attached to the lower end of a spring of modulus k. Assume that there is no damping. Derive and solve an equation of motion for the object, assuming that at time zero it is pushed down from the equilibrium position with an initial velocity v0. With what momentum does the object leave the equilibrium position?

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Diversity of Life II Notes Week 11 4/7/16 Dr. Donald Stewart – Fish Aquatic vertebrates with gills and limbs in the shape of fins Ex. ‘Blobfish’ – YES  There are exceptions Diversity ­25,000 living species (conservative estimate) to perhaps over 30,000 ­several hundred new species identified every year (many in Amazon and Philippines) ­482 families, 57 orders ­85 are jawless fishes ­850 are cartilaginous (sharks, stingrays and allies) ­everything else (over 23,000) are bony fishes, about 96% ­bones are comprised of Calcium Phosphate ­during hard times with little food available the fish can sequester phosphate out of their bones as emergency food source Most of the world (over two­thirds) is covered by ocean, but much of it is empty space (wet deserts) – middle of the Pacific Ocean is a ‘wasteland’ ­58% of modern fishes are marine ­freshwater is about 1% of overall area, only 0.01% volume; 41% of fishes are freshwater ­Only 1% move back and forth – takes a lot of effort for osmoregulation during the transition between salt­ and freshwater Properties of Water that influence Fish Functional Design 1.) Water Density – 800x denser than moving through air ­fish can be neutrally buoyant because water reduces effects of gravity ­water’s density and viscosity resists movement (pressure drag and friction drag, respectively)  development of streamlined bodies (teardrop shape) to reduce up to 95% of drag occurring on your body Ex. Bluefin Tuna – optimal design ­swim 24hrs/day (no swim bladders) 2.) Water is nearly incompressible  creates drag ­facilitates detection of vibration in water (sensory mechanisms) ­Facilitates suction feeding ­facilitates breathing with gills Water is slightly compressible  sounds travels very fast (335m/s in air vs. 1433m/s in water) ­most fish have excellent hearing (no external ears) and communicate by sound 3.) Water is universal solvent ­fish take up water with salts and organics needed to survive by feeding or by gills ­waste products also removed using gills (heat, ammonia, CO2) ­oxygen is often limiting nutrient – high surface area of gills allow them to exploit low O2 levels (8ml oxygen/1L water vs. 210ml/L water) 4.) Low Light Penetration – rarely penetrates over 100m deep Evolution of Fish The Pre­Teleost Fishes ­Fish arise from inverts via neoteny – juvenile stage becomes sexually mature and transitions into adult form ­Evolved from Tunicates (Urochordata) – pharyngeal gill slits ­Cephalo­Chordata (Amphioxus) ­possess primitive nerve chord ­no definite head or vertebrae Fish aka Craniates (includes ancestral form of all other tetrapods) Living Jawless Fish (<100extant species, very primitive) Ex. Hagfish (scavengers found on bottom) This diagram shows the juvenile features Lamprey (blood­suckers) transition into adult features, but are not lost ­no limbs altogether (e.g. proportionately large heads and short legs  Tullimonstrum gregarium (3/18/16 article) in Illinois, relative of lamprey 300MYA fossil, McCoy et al. 2016 *Evolution of Jaws was a key evolutionary innovation *Evolution of Jaw Protrusion – Important evolutionary trend among jawed fishes Gnathostomes (Jawed Craniates) ­Placoderms (extinct, many were predatory and huge) ­Sharks ­Bony fish – Lobe­finned fishes and ray­finned fishes Sharks – most primitive living jawed fishes ­full complement of fins (dorsal, pectoral, pelvic, anal fins and tail)  gives more control in 3D space ­cartilaginous skeleton Respiration in Fishes Lamprey use tidal flow (not as efficient) Sharks – use mouth or spiracles to gills Bony fish – uni­directional flow

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