Derivação de unidades baseada nas equações

Sometimes, we can gain understanding about how some quantities affect others by looking at their units.

The best way to demonstrate this is to show you an example. I will show perhaps the most famous example of all: determining how long it will take an object to fall.

Suppose that the time it takes to fall is $t$. It is reasonable that the time may depend on these three quantities: the mass of the object $m$, the gravitational acceleration $g$, and the height $h$. It is intuitive that all other factors have an incredibly tiny effect on the time, so let us look at these three.

We have the following: $$t=m^\alpha g^\beta h^\gamma$$ for real numbers $\alpha$, $\beta$, and $\gamma$.

Obviously, this equation must be dimensionally correct. Thus, if we let the dimension for length be $[L]$, the dimension of mass be $[M]$, and the dimension of time be $[T]$, then we have the following: $$[T]=[M]^\alpha \left(\frac{[L]}{[T]^2}\right)^\beta [L]^\gamma$$ $$[T]=[M]^\alpha [L]^{\beta+\gamma} [T]^{-2\beta}$$ Now, we may separately balance the equations for each of the dimensions: mass, length, and time, respectively: $$0=\alpha$$ $$0=\beta+\gamma$$ $$1=-2\beta$$ Simple algebra yields the solution: $$(\alpha,\beta,\gamma)=\left(0,-\frac{1}{2},\frac{1}{2}\right)$$ Thus, for some dimensionless constant $C$, we have: $$t=C\sqrt{\frac{h}{g}}$$ As you will learn later, or perhaps have already learned, the constant $C$ equals $\sqrt{2}$.

After seeing this, people sometimes wonder how the ancients missed a solution that was so simple.

Is this completely legitimate? Perhaps not.

It certainly doesn't feel right; it seems like some cheap gimmick, a bit like mass points (I'm kidding, of course).

This is not helped by the fact that dimensional analysis can go wrong if you don't use the correct factors.

For example, we know that the universal gravitational constant is not dimensionless, but actually has some strange units to make Newton's Law of Universal Gravitation dimensionally correct. Thus, a dimensional analysis would have spit out solutions that are inconsistent with our observations.

Thus, you must proceed with caution. But regardless, dimensional analysis is still a powerful and useful tool.