Learn the Basics of Dimensional Analysis

[Orodruin]

As a university teacher and as a PF member, I have often noted that students are largely unaware of or not using dimensional analysis to help them in their pursuit of knowledge or to check their results. A number of recent threads on PF have also highlighted this issue. The intent of this Insight is therefore to provide a basic introduction to the subject with a number of examples with which the reader may be familiar. The discussion on the Buckingham pi theorem is a bit more involved and can be skipped without missing the basic concepts.
What is physical dimension?
A common misconception when dimensional analysis is invoked is that students mix up the subject with the number of spatial dimensions, which is not what we want to discuss here. Instead, physical dimensions refer to the type of quantity we are dealing with. It is related to, but not exactly the same as, what units we can use to describe a physical quantity. For example, a length can be measured in centimeters or inches...

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[jedishrfu]

Great article!

Small nit, latex not rendering in second line of buckingham pi theorem. It says \emph explicitly.

 

[Orodruin]

Great article!

Small nit, latex not rendering in second line of buckingham pi theorem. It says \emph explicitly.

I typically write in LaTeX first before converting to Wordpress. Fixed now, thanks.

Edit: I also picked a few other nits that hopefully nobody noticed ...

 

[Chestermiller]

To provide a more complete and usable treatment of this subject, it might have been worthwhile to include a section on how to reduce the partial differential equations for a system (or ordinary differential equations and algebraic equations) to dimensionless for by applying the method of Hellums and Churchill: "Mathematical Simplification of Boundary Value Problems," AIChE. J., 10 (1964) 1121. This method us very useful for a great variety of problems.

 

[Orodruin]

To provide a more complete and usable treatment of this subject, it might have been worthwhile to include a section on how to reduce the partial differential equations for a system (or ordinary differential equations and algebraic equations) to dimensionless for by applying the method of Hellums and Churchill: "Mathematical Simplification of Boundary Value Problems," AIChE. J., 10 (1964) 1121. This method us very useful for a great variety of problems.

I did not want to involve anything but basic algebra based physics. I think it is already quite heavy for the beginner as it is without discussing differential equations. I do discuss this in my book though and it would make a good subject for a separate Insight.

An example I thought about bringing up is the time it would take to heat the core of a metal sphere to a certain temperature given that you know the time taken for a sphere of a different size. That can be done without actually solving the differential equation.

 

[kent davidge]

An example I thought about bringing up is the time it would take to heat the core of a metal sphere to a certain temperature given that you know the time taken for a sphere of a different size. That can be done without actually solving the differential equation.

Uniqueness theorems?

 

[Orodruin]

Uniqueness theorems?

No. Dimensional analysis. :)

 

[Stephen Tashi]

Perhaps students of dimensional analysis are actually helped by making the conceptual mistake that physical dimensions and units have a definite physical interpretation. For example, thinking that L/T "is" velocity will be statistically correct in terms of how often that situation arises in physics texts. However, in an introduction to the subject, I think its worth mentioning that dimensional analysis does rely on assigning a definite physical interpretation to quotients and products of physical dimensions.

What's a good example to illustrate this? Perhaps work and torque? A contrived example would be a toy machine where a person holds down a button for t seconds and after the button is released this input causes the machine to move across the floor for a distance of d feet. There is a physical relation between the input and output of the toy that can be described in dimensions of L/T but this doesn't refer to a velocity.