I need to learn how to solve metric conversions through dimensional analysis. Can someone please help any info would be greatly appreciated.
I think I know what you are asking but I might be wrong. I think the best way to explain is with an example.
We want a formula to describe how long it takes a ripple moving through a liquid to die down.
We assume that the time taken is a function of the fluid's density $\displaystyle \rho$, viscosity $\displaystyle \eta$ and the speed of sound in the fluid $\displaystyle v$
Assume that the formula is in the form
$\displaystyle t=\rho^a\cdot \eta^b\cdot v^c$
Examining the units of each quantity
$\displaystyle t: s$
$\displaystyle \rho: kg m^{-3}$
$\displaystyle \eta: Ns^{-1}= kgms^{-3}$
$\displaystyle v: ms^{-1}$
The formula must be such that the units are balanced
$\displaystyle t=\rho^a\cdot \eta^b\cdot v^c$
$\displaystyle s=(kg m^{-3})^a\cdot (kgms^{-3})^b\cdot (ms^{-1})^c$
$\displaystyle s^1=kg^am^{-3a}kg^bm^bs^{-3b}m^cs^{-c}$
The indexes on the units must match up
seconds, $\displaystyle 1=-3b-c$
meters, $\displaystyle -3a+b+c=0$
kilograms, $\displaystyle a+b=0$
If you solve this set of equations you find that b=1, c=-4, a=-1.
So we know the equation must be
$\displaystyle t=\rho^{-1}\cdot \eta^1\cdot v^{-4}$