Dimensional Analysis, How to determine the right order for the power relation? I came across this question in  Solved Problems in Classical Mechanics by O.L. de Lange and J. Pierrus. Question 2.12 is as follows:

Use dimensional analysis to determine the dependence of the period $T$ of a simple
pendulum on its mass $m$, weight $w$, length $\ell$ and arc-length of swing, $s$.

Solution:
$$
T=k m^\alpha w^\beta \ell^\gamma s^\delta .
$$
Hence
$$
M^0 L^0 T^1=M^\alpha\left(M L T^{-2}\right)^\beta L^\gamma L^\delta,
$$
and so
$$
\alpha+\beta=0, \quad \beta+\gamma+\delta=0, \quad-2 \beta=1 .
$$
These yield $\alpha=-\beta=\frac{1}{2}$ and $\gamma=\frac{1}{2}-\delta$. Consequently, becomes
$$
T=k \sqrt{\frac{m \ell}{w}}\left(\frac{s}{\ell}\right)^\delta,
$$
where $\delta$ is an undetermined number.
Question
My question is how do I know that  $\delta$ is the "correct" undetermined? I could have rearranged so $\delta=\frac{1}{2}-\gamma$ and obtained:
$T=k \sqrt{\frac{m s}{w}}\left(\frac{\ell}{s}\right)^\gamma$,
This seems wrong to me as the equation for a pendulum is: $T=k \sqrt{\frac{m \ell}{w}}$.
what am I misunderstanding?
 A: There's no way to know, since you're starting with four parameters and only 3 equations to solve them. Thus, both $\gamma$ and $\delta$ can be thought as undetermined numbers.
Nevertheless, the results you'll obtain will be identical with any of those two choices, since from a dimensional point of view you have powers of the quantity $\dfrac{l}{s}$ or $\dfrac{s}{l}$, which are scalar quantities.
In general, just from the dimensional analysis it's possible to show the following relations:
$$
T=k\sqrt{\dfrac{ml}{w}} \chi\left(\dfrac{s}{l}\right)\\
T=k\sqrt{\dfrac{ms}{w}} \chi'\left(\dfrac{l}{s}\right)
$$
Where $\chi$ and $\chi'$ are undetermined functions. The two expressions are nevertheless the exact same expression, maybe one function will be easier to compute than the other one but at the end the result for T is uniquely defined.

This seems wrong to me as the equation for a pendulum is: $T=k\sqrt{\dfrac{mℓ}{w}}$. what am I misunderstanding?

It seems wrong because you know a priori the answer, but just looking at the dimensional analysis you cannot state which one of the two expression is correct. They both are, the real challenge is to find an analytic expression for the two functions $\chi$ and $\chi'$ so that at the end they explain the same physics.
