What is Non-Dimensionalising and How Can It Help Solve General Linear Equations?

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Discussion Overview

The discussion revolves around the concept of non-dimensionalising, particularly in the context of general linear equations. Participants explore its meaning, applications, and methods for implementation, focusing on both theoretical understanding and practical approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses uncertainty about the meaning of 'non-dimensionalising' and seeks guidance on how to start applying it to general linear equations.
  • Another participant questions whether radians can be considered a non-dimensional unit, prompting a clarification about radians being dimensionless due to the cancellation of units in its definition.
  • A participant explains that dimensionless expressions can be formed by dividing by a fixed quantity with the same units, providing an example related to radians and angles.
  • A link to the Buckingham π theorem is shared as a systematic approach to obtaining results using non-dimensional variables.

Areas of Agreement / Disagreement

Participants generally agree on the definition of radians as a non-dimensional quantity, but the overall understanding and application of non-dimensionalising in the context of linear equations remain unresolved, with multiple viewpoints and questions present.

Contextual Notes

Some limitations include potential missing assumptions about the specific types of linear equations being discussed and the dependence on definitions of non-dimensional quantities. The discussion does not resolve how to systematically apply non-dimensionalising to various equations.

newstudent
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Hi All,

I have often heard the term 'non-dimensionalising', and am unsure as to what it really means. I gather that it literally means non dimensionalising the units such that it may be applied to a wider range of situations. My question is, if i have a general linear equation and wish to non dimensionalise it, where should be my starting point? I would appreciate if someone could point me in the right direction. Thank you.

=)
cheers,
newstudent
 
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newstudent said:
Hi All,

I have often heard the term 'non-dimensionalising', and am unsure as to what it really means. I gather that it literally means non dimensionalising the units such that it may be applied to a wider range of situations. My question is, if i have a general linear equation and wish to non dimensionalise it, where should be my starting point? I would appreciate if someone could point me in the right direction. Thank you.

=)
cheers,
newstudent

I'm curious too. Would radians be a undimentionalising unit?
 
Last edited:
"Radians", not "radiants". Yes, radians are an example of a non-dimensional quantity. Given a circle of any radius, the radian measure of an angle is the length of the arc it cuts from the circle, divided by the radius of the circle. Since those are both lengths, any units of length will cancel out. For example, a 60 degree angle, in a 40 inch in radius circle, would cut an arc length of (60/360)(2\pi 40)= 41.9 inches long. The angle, in radians, is 251.3/40= 6.28. It is because the "radian" is really "dimensionless" that we can use it in purely algebraic equations with no mention of angles:
f(x)= cos(x) assumes x is "in radians".

In general, one forms dimensionless expressions by dividing by a fixed quantity having the same units as the original expression.

You might want to look at this:
http://astro.nmsu.edu/~aklypin/PM/pmcode/node2.html
 
Last edited by a moderator:

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