I What are the applications of dimensional analysis and why does it work?

[Leo Liu]

I know that dimensional analysis is a rough way to check the correctness of the solution to a homework problem. However, what exactly is it and what are the other applications of it? Why does it work?

Thanks.

 

[berkeman]

I know that dimensional analysis is a good way to check the correctness of the solution to a homework problem. However, what exactly is it and what are the other applications of it? Why does it work?

Thanks.

https://en.wikipedia.org/wiki/Dimensional_analysis

 

[Lnewqban]

I recommend the the factor-label method, which you can find in the linked article above.

 

[kuruman]

Dimensional analysis does not check whether a solution is correct because a dimensionally correct equation is just that and not necessarily a correct equation. For example, one may claim that the surface area of a sphere of radius $R$ is $A=3\pi R^2.$ That makes it dimensionally correct because it has dimensions of area, namely length squared, but it is factually incorrect because it is the total surface area of a hemisphere, not a sphere. If you don't know what the area of a sphere and a circle are in terms of the radius, you would be able to assess its correctness purely on dimensional grounds. However if one claimed that the area of a sphere is $A=4\pi R^3$, then you would know purely on dimensional grounds that it has imensions of volume, not area and is, therefore, incorrect.

 

[Master1022]

Hi,

Dimensional analysis is a method in engineering used to analyze a system or machine by using 'dimensionless groups'.

What is a dimensionless group?
An expression which has no units (e.g. Reynold's number, Prandtl Number, etc.)

Why would we use this?
1. Use it to describe the physics of a system in a transferable way
We can analyze smaller scale prototypes or compare different machines (hydraulic machine analysis comes to mind) by looking at non-dimensional groups instead of actual lengths and sizes. That is, we can forget about the individual sizes from machine to machine and instead consider dimensionless groups of variables (e.g. power number) and just make comparisons based on that.

2. Cost-effective
This is important in engineering and as mentioned above, if we can simulate a scenario using a smaller scale prototype with the dimensionless groups equated in such a way as to make the physics the same, then it is much cheaper than building a real life model.

3. Powerful
Lets us discern which variables are most important

These are some of the benefits of dimensional analysis.

Hope that was of some use.

[EDIT]: As mentioned above, there are other uses as well. There are also applications of dimensional analysis to PDEs. For example, you can non-dimensionalise the Navier-Stokes equation.

 

[etotheipi]

Some other really useful references are these online notes:
https://web.mit.edu/2.25/www/pdf/DA_unified.pdf

and also check out @Orodruin's excellent insights article on dimensional analysis, which includes a nice discussion of the very important Buckingham pi theorem!