dimensionalreasoning：空間推理-資料下載頁
2025-01-12 07:46本頁面
【正文】 the obvious fact that the total area is the sum of the two smaller ones, by applying the previous equation we have: c2 ? f (α, π/2) = a2 ? f (α, π/2) + b2 ? f (α, π/2). And, eliminating f: c2 = a2 + b2, . Proof of the Pythagorean Theorem Dimensions Scaling, modeling, similarity ? Types of “similarity” between two objects/processes. – Geometric similarity – linear dimensions are proportional。 angles are the same. – Kinematic similarity – includes proportional time scales, ., velocity, which are similar. – Dynamic similarity – includes force scale similarity, ., equality of Reynolds number (inertial/viscous), Froud number (inertial/buoyancy), Rossby number (inertial/Coriolis), Euler number (inertial/surface tension). Dimensions ? Distorted models – Sometimes it’s necessary to violate geometric similarity: A 1/1000 scale model of the Chesapeake Bay is ten times as deep as it should be, because the real Bay is so shallow that, with proportional depths, the average model depth would be 6mm, too shallow to exhibit stratified flow. Scaling, modeling, similarity Dimensions Dimensions Scaling, modeling, similarity ? Scaling – What’s the biggest elephant? If one tries to keep similar geometric proportions, weight ? L3, where L is a characteristic length, say height. – However, an elephant’s ability to support his weight is proportional to the crosssectional area of his bones, say R2. – Therefore, if his height doubles, his bones would have to increase in radius as 2?2 R, not 2R. – [Note: A crosssection of 8 R2 = (2?2 R)2]. So, with increasing size, an elephant will eventually have legs whose crosssectional area will extend beyond its body Dimensions Biological scaling
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