How does dimensional analysis account for additive variables in equations?

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SUMMARY

Dimensional analysis primarily relates dependent variables to independent variables through multiplicative relationships. However, it does not adequately address additive variables, which can complicate the formulation of equations. The discussion highlights that when multiple independent expressions are involved, such as two velocities or masses, dimensional analysis fails to determine the appropriate variable to use. This limitation necessitates the incorporation of additional operations, such as addition or subtraction, to derive accurate equations.

PREREQUISITES
  • Understanding of dimensional analysis principles
  • Familiarity with dependent and independent variables
  • Knowledge of basic algebraic operations (addition, subtraction)
  • Concept of units and their dimensional representations
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  • Explore the limitations of dimensional analysis in complex equations
  • Study the role of additive variables in physical equations
  • Learn about alternative methods for equation formulation beyond dimensional analysis
  • Investigate case studies where dimensional analysis fails to provide solutions
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Students and professionals in physics, engineering, and mathematics who are looking to deepen their understanding of dimensional analysis and its limitations in handling additive variables in equations.

Mr Davis 97
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Under the standard form of dimensional analysis, I know that we relate a dependent variable to a function of the independent variable(s). However, what if there is some additive variable needed in the equation? How does this method, which expresses all of the independent variables as a product of the variables (with the exponents being any real number) times a constant, account for the needed addition of operations such as subtraction and addition? How can we derive correct equations if there is a missing sum or difference needed in the formula?
 
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If you can construct a specific unit with multiple independent expressions, then dimensional analysis for this unit does not work.
A trivial example is a setup where you have two velocities, or two masses or something similar. There is no way to figure out which velocity/mass/... to use just by dimensional analysis.
 

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