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Determine the moment of inertia Ixx of the composite plate

Engineering Mechanics: Statics & Dynamics | 14th Edition | ISBN: 9780133951929 | Authors: Russell C. Hibbeler ISBN: 9780133951929 123

Solution for problem 21-12 Chapter 21

Engineering Mechanics: Statics & Dynamics | 14th Edition

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Engineering Mechanics: Statics & Dynamics | 14th Edition | ISBN: 9780133951929 | Authors: Russell C. Hibbeler

Engineering Mechanics: Statics & Dynamics | 14th Edition

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Problem 21-12

Determine the moment of inertia Ixx of the composite plate assembly. The plates have a specific weight of 6 lb>ft2 .

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Life 103 Notes *Adapted from the lecture notes of Dr. Tanya Dewey* Animal Taxa: Vertebrates II (Mammals and Metabolic Trade-offs) ctn. • Mammalia -Sound conducted through middle ear by 3 bones (malleus, incus, stapes) -Each side of lower jaw made up of a single bone, the dentary -Remember, our perceptions about perception are influenced by our evolutionary history. Primates are very visual. Most use olfaction and hearing more. • Mammal evolution -Pelycosaurs-evolution of homoiothermy (or homeothermy) ~Homeothermy: stable internal body temperature in spite of external influence; can be regulated behaviorally or metabolically • Thermoregulation -Homeotherm: relatively constant temperature -Poikilotherm: variable temperature -Ectotherm: uses environmental heat -Endotherm: generates metabolic heat • Trends -Larger brains -Larger and more muscles -Upright limb posture -Nocturnal habits ~All of these trends are very metabolically expensive • Metabolic expense -How to address this problem ~Heterodonty (specialized teeth): efficiency in processing food; pre- process before it enters the stomach, most other animals don’t chew ~Secondary palate: can breathe while eating; uninterrupted oxygen supply ~Endothermy: blood vessels in fossilized bones; evidence of whiskers (maybe fur) Body Structure & Function • Form and Function -Animals must obtain nutrients and oxygen, excrete wastes, and move -Animals live in nearly every conceivable kind of environment (temperature, pressure, salinity, oxygen concentrations, light levels, selective pressures, etc.) -Natural selection acts on heritable variation to favor the most successful solutions to these challenges -So if a trait has an adaptive function (influences the successful survival and reproduction of individuals) then its form reflects its function ~Ex) Darwin’s finches -Natural selection acts on variation in bill shape and size in populations to favor particular bill morphologies well suited to particular foods -Bill shape (form) tells you what the diet is (function) -Exchange with the environment is ultimately at the cellular level- substances in solution travel across the plasma membrane of cells -How is this accomplished in complex multicellular organisms -Reflected at all scales ~Molecular-cellular-tissues-organs-organ systems-organism -Molecular-enzyme structure and substrate specificity; phospholipid characteristics -Cell structure- cell with extensive endoplasmic reticulum and Golgi apparatus is likely to have what function Protein synthesis ~Cell shapes -Neuron cells-function in communication -Epithelial cells-function in lining surfaces -Muscle cells-function in contraction ~Tissues- groups of cells that together serve a particular function -Embryonic tissue layers: endoderm, mesoderm ectoderm ~Give rise to 4 kinds of tissues 1. Connective-support, connect, or separate 2. Nervous-regulation, control, communication 3. Muscle-support, movement (contraction) 4. Epithelial-lining/covering -Tissues reflect function (see textbook figure 40.5) ~Example: Hydrodynamics is a physical constraint of the environment -Water is 1000x denser than air -Problem-how to travel quickly and efficiently through water -Solution-fusiforms (streamlined) body shape -Diverse organisms face similar challenges and arrive at similar solutions=convergent evolution ~Example: Increasing stride length to run fast ~Example: Evolution of flight in vertebrates -Similar physical constraints (generate lift) -Solutions reflect evolutionary constraint as well -Benefits of flight-powered flight has evolved four times (three times in vertebrates)-insects, pterosaurs, birds, bats -Problem-How to generate sufficient lift for sustained flight -Solution ~Flight membranes ~Generally small body size ~Lighten bony elements -Specific solutions vary ~Pterodactyl-elongate single digit ~Bird-fuse distal bony elements and feathers ~Bat-elongate four digits • Locomotion -Adaptations for different forms of locomotion are based on the same basic tetrapod body plan • Form and Function: two general principles that influence form and function in organ systems -Body size and surface area to volume relationships -Homeostasis and Regulation • Body size and surface are to volume relationships -Large ~Need more food ~Take longer to mature ~Lost heat slowly ~Lose water slowly ~Reproduce slowly -Small ~Need less food ~Shorter time to maturity ~Lose heat quickly ~Lose water quickly ~Reproduce quickly -There are many consequences of body size ~While the large aren’t very subject to the environment, they have much slower cycles ~While the small have fast cycles, they are very subject to environmental extremes -Example: body mass (size) and metabolic rate (see textbook figure 40.20) ~Metabolic rate-oxygen consumption per unit of time=a measure of metabolic activity ~Metabolic rates vary with activity, so basal metabolic rate is measured (at rest) ~What is the relationship between body mass and basal metabolic rate -Bigger animals eat less proportionally because they generally have a slower metabolism -Example: Relationship between body mass and lifespan ~What is the relationship between body mass and lifespan -Larger animals tend to live longer -As volume increases, surface area increases less -Surface area is critical for exchange -Example: salmon hatchlings ~Oxygen exchange across skin and gills ~At hatching, most oxygen exchange is across the skin ~As they grow, the gills take over -Every cell must exchange with its environment (see textbook figure 40.3) -Complex animals: vastly increased surface areas for exchange of nutrients, oxygen, and wastes (see textbook figure 40.4) -Adaptations to increase surface area ~Flattening ~Folding ~Branching Regulating Internal Environments: Homeostasis and Thermoregulation • Homeostasis and Regulation -Conformers vs. Regulators (see textbook figure 40.7) ~Conformers-conform to the environment; as the environmental temperature changes, so does the body temperature ~Regulators-don’t conform to the environment; as the environmental temperature changes, body temperature stays relatively the same -Homeostasis: maintaining internal chemical and physical environments within a tolerable range, independent of external environmental states -Why is homeostasis important ~Enzyme function is impacted by temperature, pH, other reactants, etc. ~Other chemical reactions in the body are sensitive to conditions ~Extremes also destroy proteins (high heat, pH) and cells (freezing, high heat) -Metabolic rate of a thermoconformer ~Do thermoconformers operate equally well at all temperatures -No, thermoconformers have an optimal range of metabolic activity that is influenced by environmental temperatures ~Acclimatization-organisms adjust to gradual environmental changes to maintain optimal (or near optimal) performance (see textbook figure 40.10) -Homeostasis: How is it maintained ~Set point-optimal temperature ~Stimulus-a variation from optimal temperature ~Sensor/control center-senses the stimulus and regulates response ~Response-brings temperature back to optimal ~Negative feedback-response is to minimize the effect- this is the feedback mechanism used to maintain homeostasis ~Positive feedback- response is to amplify the effect -Thermoregulation (thermoregulators): temperature maintained within a normal range ~Example: circadian rhythms-normal daily variation in body temperature (see textbook figure 40.9) • Energetic trade offs of ectothermy and endothermy -Able to maintain activity in different conditions ~Endotherms are able to do this much more efficiently than ectoderms -Require more or less energy ~Ectotherms require substantially less energy than endotherms • Mechanisms of Heat Exchange (see textbook figure 40.12) -Conduction: direct contact between solids -Convection: direct contact between solid and gas or liquid -Radiation: no direct contact -Evaporation: high heat of vaporization; a special property of water -In which direction does heat flow ~Heat always flows from areas of high heat to areas of low heat; from high to low; from hotter to colder temperatures • Generating and retaining heat -Insulation helps retain heat in mammals and birds -Behavioral heat absorption and regulation -Circulatory adaptations ~Vasodilation and vasoconstriction-delivering different amounts of heated blood to skin surface to heat or cool ~Countercurrent heat exchange (see textbook figure 40.13) -Evaporative heat loss -Metabolic heat production ~Shivering and non-shivering thermogenesis ~Some ectotherms use shivering thermogenesis to generate heat • Thermoregulation and energy conservation -Torpor: a state of decrease physiological activity ~Small endotherms (birds and mammals) that are active, with high metabolic rates ~Daily pattern -Hibernation: seasonal state of reduced physiological activity ~Mostly mammals ~Aestivation-“summer hibernation”- often facultative ~”Set point” is greatly reduced, sometimes to nearly 0 degrees Celsius ~Often accompanied by periodic arousals -Adaptive heterothermy ~The ability of an endothermic animal to allow its body temperature to fluctuate in response to some form of environmental stress, saving significant amount of energy and water ~Heterothermy: endothermic animals allow body temperature to fluctuate in response to an environmental stress (hibernation, aestivation, torpor, daily changes) Dietary Strategies and Digestion: Tongues, Teeth, and Venom • Dietary Strategies and Digestion -Remember that all animals are ingestive heterotrophs -Why eat ~Obtain chemical energy (sugars; respiration) ~Obtain organic molecules (building blocks for tissues) ~Obtain essential nutrients-cannot synthesize, must obtain from food -Essential amino acids-for protein synthesis -Essential fatty acids-for fatty acid synthesis -Vitamins-organic compounds with a variety of functions; “a substance that makes you ill if you don’t eat it” -Minerals-inorganic compounds, diverse functions, includes electrolytes -This is all done to support ~Survival ~Growth ~Reproduction -3 classes of molecules are sources of chemical energy, building blocks, and essential nutrients ~Carbohydrates-sugars and polymers of sugars, including glycogen and cellulose ~Proteins-polymers of amino acids ~Lipids-complex carbon molecules that are hydrophobic (fats-made up of fatty acids, phospholipids, steroids)

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Chapter 21, Problem 21-12 is Solved
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Textbook: Engineering Mechanics: Statics & Dynamics
Edition: 14
Author: Russell C. Hibbeler
ISBN: 9780133951929

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Determine the moment of inertia Ixx of the composite plate