定标率的实际应用（量纲分析/半定性的物理学）

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1 定标率的实际应用（量纲分析/半定性的物理学）
Square-cube law Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface-area-to-volume ratio decreases with increasing volume such that A = V 2/3.

2 考虑一个物体，密度不变的情况下有它的体积V 质量M 均和尺寸L 成立方关系，表面积A 为 L的 平方关系。
F=Ma; Pressure=F/A=M (a/A) M’=x3M; A’=x2A; F’= x3Ma; P’=F’/A’=x P scaling up the size of an object, keeping the same material of construction (density), and same acceleration, would increase the thrust by the same scaling factor. This would indicate that the object would have less ability to resist stress and would be more prone to collapse while accelerating. This is why large vehicles perform poorly in crash tests and why there are limits to how high buildings can be built. Similarly, the larger an object is, the less other objects would resist its motion, causing its deceleration. As the size of a balloon is increased, the cost in surface area increases quadratically but the lift generated from volume increases cubically.

3 假定骨截面由动物重量决定A~M，而骨长度L~M1/3，那么骨骼重量应该正比于M1.33
Small and large animals therefore behave differently. Mice and squirrels jump and run; Elephants can gallop, but usually walk, and do not jump.

4 Resting metabolic rates of mammals
The straight line that fits the data, however has a slope of 0.75 instead of the expected 0.67 This a puzzle yet to be explained.