Potency product approach

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The power product approach is often used to approximate functional relationships in physical systems.

For the function f a power product of the form

${\displaystyle f(x_{1},\dots ,x_{n})=c\prod _{k=1}^{n}x_{k}^{\alpha _{k}}}$

elected. The characterize the variables that can be changed in the experiment and a natural constant as a proportionality factor. The free parameters can be determined by varying these quantities in the experiment . ${\displaystyle x_{k}}$ ${\displaystyle c}$ ${\displaystyle \alpha _{k}}$

In practice, the measured values ​​can e.g. B. be plotted on logarithmic paper. The respective exponent can easily be determined from the slope of the regression line .

example

With drop tests, the relationship between the time of fall and the height and weight of the falling body should be clarified. The function from the power product approach corresponds to this fall time . The height and weight correspond to the variable sizes and . ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle t}$ ${\displaystyle h}$ ${\displaystyle M}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$

${\displaystyle t(h,M)=c\cdot h^{\alpha _{1}}\cdot M^{\alpha _{2}}}$

The equation is obtained by adapting the exponents to the result ${\displaystyle {\alpha _{k}}}$

${\displaystyle t(h,M)=c\cdot h^{\frac {1}{2}}\cdot M^{0}=c\cdot h^{\frac {1}{2}}}$

Limits

The parameters must be rounded sensibly in order to form a physical model of the experiment. A value of e.g. B. 0.498 for would be of little help. ${\displaystyle {\alpha _{k}}}$ ${\displaystyle {\alpha _{1}}}$

However, the power product approach fails in many places because the correct physical model cannot be mapped with it. Various contradicting radiation laws , e.g. Partly based on potency product approaches ( Stefan-Boltzmann law ), z. B. only merged by Planck's law of radiation , which does not satisfy any power product approach.