Dimensional Analysis: Combining 3 Variables

[Saladsamurai]

Homework Statement

Hi. I have a function that contains 4 variables: Q, R, $\mu$, ^dp/_dx

I wish to choose 3 of them, such that they cannot be combined into a dimensionless product.


I have chosen (correctly) R, $\mu$, ^dp/_dx and I would like to know if my method sounds correct:

If we know the dimensions: [R]=[L] [$\mu$]=[ML^-2T^-2] and [^dp/_dx]=[ML^-1T^-1]


and I know that in order for them to be dimensionless, their power product must equal zero:

[L]^a[ML^-1T^-1]^b[ML^-2T^-2]^c=[MLT]⁰

or the system

a-b-2c=0
b+c=0
-b-2c=0

By inspection, this system can only be satisfied if a=0 but that does not make any sense since R is a physical quantity.

Hence I have reasoned that these 3 variable cannot form a non-dimensional parameter by themselves.


Does this work? Thanks!

 

[Pengwuino]

Ignore length real quick. $$[MT^{ - 1} ]^a [MT^{ - 2} ]^b$$ can NEVER be dimensionless (except for a=b=0)

 

[Saladsamurai]

Ignore length real quick. $$[MT^{ - 1} ]^a [MT^{ - 2} ]^b$$ can NEVER be dimensionless (except for a=b=0)

Okay cool!

Also, though longer and more tiresome, my method above works right? For the same reason.

I just want a general method just in case inspection is not that obvious.

 

[Pengwuino]

Yes, that method works.