Buckingham Pi theorem: Why do we need dimentionless parameters? I've been trying to understand Buckinghams Pi theorem for my lab work on rolling cylinders. I don't exactly get why you need dimensionless parameters? I do understand that having a (not sure of the English word here) quota between two parameters will in some cases simplify the problem, in my case the inner and outer radius of the cylinders. But why is it interesting that they are dimensionless?
Tankfull for answers!
 A: Consider you are working with a law $a=f(a_1, ..., a_n, b_1, ..., b_m)$ where the $a_i$ are parameters with indipendent dimensions while $b_j$  are the parameters whose dimensions can be built up by the ones of the $a_i$'s. For example in such a way, $$[b_i]=[a_1]^{p_i}...[a_n]^{r_i}$$ ($[a_i]$ means dimension of $a_i$). $a$ instead is the quantitiy you are interested in.
You can surely define the following dimensionless parameters $\Pi_i=\frac{b_i}{[a_1]^{p_i}...[a_n]^{r_i}}$ with $i=1,2 ...,m$ and $\Pi=\frac{a}{[a_1]^{p}...[a_n]^{r}}$.
So your initial expression
$a=f(a_1, ..., a_n, b_1, ..., b_m)$ becomes $$\Pi=\frac{f(a_1, ..., a_n, \Pi_1a_1^{p_1}...a_n^{r}, ..., \Pi_m a_1^{p_m}...a_n^{r_m})}{a_1^{p}...a_n^{r}}=g(a_1, ..., a_n, \Pi_1, ..., \Pi_m)$$
now take a look at the final expression $$\Pi=g(a_1, ..., a_n, \Pi_1, ..., \Pi_m)$$
on the lhs you have a dimensionless quantity, on the rhs indipendent variables and dimensionless qunatities. The $a_i$, being indipendent, can change its value with the others $a_j$ remaining constant. But nothing varies with $a_i$, so by a covariance principle the function g is constant on rescaling of the $a_i$'s and so indipendent by them. Hence you get $$\Pi=g(\Pi_1, ..., \Pi_m)$$ and so we arrive at $$a=f(a_1, ..., a_n, b_1, ..., b_m)=a_1^p...a_n^rg(\Pi_1, ..., \Pi_m)$$
By simple dimensional analysis you have reduced your problem of finding a law for n+m variables in one for m variables. In some cases this higly simplify your obejctive of finding a law for $a$. As a trivial example you can check that is possible to determine the law for the period of the pendulum $T\alpha\sqrt\frac{l}{g}$, given an initial guess $T=T(g, m, l)$. Others more interesting examples can be made, such as the law for the flow of a fluid thorugh a cylindrical pipe. The dependencies by the non indepdent parameters must still be determined by experiments. But such work is higly reduced, since now there are only m variables.
This is why you need to identify the dimensionless paramaters for this procedure. The point is that Buckingham theorem stops to the dependencies of these parameters, by dimensional analysis alone you can't continue.
