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I think what symanzik138 wrote is the right way to think about this. The laws of physics that we know about and use, should be considered to be effective laws that one could in principle derive from fundamental laws by integrating out the degrees of freedom that the effective laws do not describe. Whether the topic is quantum field theory, fluid dynamics or ...

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Good question, and I don't have a good general answer. However in the specific example you quote it's because the minimal dimension of space you can inscribe the circle is a surface of some kind ie a plane, or the surface of a sphere. Similarly, with the formula for the volume of a sphere, the exponent is three - which refers to the fact a volume can only ...

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What you say is generally true if you consider the linear nature of resonant materials. But no materials are perfectly linear. Furthermore the frequency is not the frequency at which individual atoms resonate, but rather the system of atoms that form a resonant structure. The shape, size, etc also has to do with what frequency you get. In real structures ...

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As docscience has already commented, it's actually not true in general. Usually the linear models in which the frequency is amplitude independent are accurate enough for many purposes but not for all of them. I would recommend you to study some weak nonlinear systems where the hybrid solution of slightly generalised linear solution are good enough. ...

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A priori, they could have been different things. The Equivalence Principle - the hypothesis that they are actually the same - is a core input the General Relativity. To the extent that General Relativity is empirically validated, we have evidence that these really are the same. There's no complete theory of quantum gravity, so I think we'd have to say ...

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The amount of work performed during "a period of time" $\Rightarrow$ Average Power $= \frac{\Delta W}{\Delta t}$ for example, the work is $W_1=3 \, \mathrm{ J}$ at time $t_1= 2\, \mathrm{ sec.}$ and $W_2=7 \, \mathrm{ J}$ at time $t_2= 13\, \mathrm{ sec.}$ $\Rightarrow$ the duration is $\Delta t = t_2-t_1=13-2\, (\mathrm{ sec.})$ $\Rightarrow$ ...

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