Dimensional analysis seems wrong for this equation: z = -1/(x^2+y^2)

[dyn]

Hi
I've just done a question regarding a marble moving on a surface given by z = -1/(x²+y²)
In this case what happens with dimensional analysis ? x and y have dimensions of length while z has dimensions of 1/(length)².
Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?
Thanks

 

[scottdave]

So if z is a distance, and x and y are distances, then there needs to be some implied dimension in the 1 in the numerator, to make the dimension of z be correct.

It's kind of like you can have y = x^2 mathematically, but if it represents something physical, then you might have to multiply by a coefficient (could be 1) with some dimension (or units).

 

[vanhees71]

HI
I've just done a question regarding a marble moving on a surface given by z = -1/(x²+y²)
In this case what happens with dimensional analysis ? x and y have dimensions of length while z has dimensions of 1/(length)².
Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?
Thanks

The question is badly formulated. There's some constant with the appropriate dimension missing. If it's in a math textbook it's simply that mathematicians nowadays don't care for correct physics. If it's a physics textbook it's simply a bad book ;-)).

 

[Ibix]

Is this question badly written or do i just accept that i won't be able to perform dimensional analysis on any later terms that are produced ?

It's a badly written question. But you can argue that the real surface is $z'=Az$, where $A$ has dimensions of length cubed and equals one in whatever units you are using. At any point in your working you can simply substitute $z'/A$ for $z$ and check dimensions.

 

[ergospherical]

just an opinion but I think $z=-1/(x^2+y^2)$ is better written than $z = -(1\ \mathrm{m^3})/(x^2 + y^2)$ or $(z/\mathrm{m}) = -1/((x/\mathrm{m})^2 + (y/\mathrm{m})^2)$ or whatever because the emphasis is clearly on studying the properties of the surface and not confusing the student with weird dimensional factors

 

[vanhees71]

I'd simply write $z=-A/(x^2+y^2)$, where $A=\text{const}$. Of course, $A$ has the dimension $\text{length}^3$. A dimensionally wrong equation is a nogo in any physics book!

 

[dyn]

The question is badly formulated. There's some constant with the appropriate dimension missing. If it's in a math textbook it's simply that mathematicians nowadays don't care for correct physics. If it's a physics textbook it's simply a bad book ;-)).

It was on a university maths exam paper !

 

[vanhees71]

Well, that explains it. Mathematicians don't care much for physics nowadays anymore :-(.