Visualizing Dimensional MLT quality

[seanwperry]

My problem is a general one concerning how to best visualize/get a good mental construct for what's happening when we get simple units by multiplying dimensions other than length^2 or length^3. I don't seem to have any trouble with the intuition about the quotient of values: the notion of 'per' suffices pretty well here. I'm going to use MLT for mass, length, time. L/T (velocity) is a ratio comparing how many units of length per unit(s) of time. So here's the crux of it. Say we're talking force with dimensional units of M*L*T^-2. I don't have a problem with per T per T to get T^-2. But how can I visualize what a M*L, like a kg*m is? I've tried thinking about this as a analogous to area being L*L but one of the dimensions is mass, but my visualization breaks down when the units aren't both L. Is there a more illuminating way to think about it, or is a analogy to area going to be as far as I should try to pursue this?

Thanks! -_-

PS are there any units where we raise units to the power of units? L^T, etc.

 

[UltrafastPED]

Here is an introductory site for dimensional analysis: http://joneslhs.weebly.com/

But I think your question is not "How do I analyze this for a formula?" but rather "How best to think about compounding of multiple units?"

The latter comes from our attempt to define a set of mutually independent "base units" of measure, each one corresponding to one of the "dimensional units". Since the SI system is built on this basis, you can simply accept it. That is, "Force", which appears to be a fundamental idea within physics has a dimensional unit, the Newton, which can be expressed in terms of more "elementary" dimensional units.

Most people, in my experience, simply memorize the relationships that they commonly use. Since I rarely work with magnetic properties, I always have to look these up. But most of the others I simply recall - much like how I know that 4 x 6 is 42 without any thinking.

I suppose I'm suggesting that you just "get used to it", but I never teach this topic, so I'm curious what the teachers will have to say.

PS: AFAIK, you will never see something like L^T; everything more complicated than multiplication/division is always dimensionless, while addition/subtraction is only valid if the units are the same across all of the terms.

 

[sophiecentaur]

PS are there any units where we raise units to the power of units? L^T, etc.

What possible meaning could "Five metres to the power of six seconds" have? An exponent has to be dimensionless.