# Intuitive understanding of dimension (units)

I was flipping through a physics text and some of the units seemed pretty 'crazy'. Just wanted to know if they can always be understood intuitively, like you can visualize what's going on

e.g. force, mass x acceleration, if I look at it as kg*m/s^2 then it doesn't really make sense to me

And whilst we're on the topic of dimensions/unit, if you have something like 20J/m^2/s and you simplify it mathematically so it reads 20J/m^2 s, does that still read 20J per square metre per second, or per metre second? If the latter, is that supposed to mean the same thing as metre per second? Metre second doesn't make sense at all!

Simon Bridge
Homework Helper
There are usually tricks to visualize even the most abstract of processes.

20J/m^2 s, does that still read 20J per square metre per second, or per metre second?
both.
is that supposed to mean the same thing as metre per second?
Not really.

acceleration is m/s/s = (m/s)/s = m/s.s the first two are just sloppy notation. can you see how m/(m/s) does not make much sense for acceleration? (it simplifies to seconds).

Don't worry, they start to make sense with familiarity.

^ sorry, when I said metre per second, I was still referring to the joules per metre square per second

I hope so, it's one of the things that slows me down in physics!

sophiecentaur
Gold Member
There's no need to lose sleep over this. A good reason for using dimensional analysis is to check that the units on each side of an equation are in fact balanced. This is one check that the equation could be, in fact, right. The actual 'meaning' shouldn't bother you because you can always come across many different combinations of MLTQ. Some are familiar - like LT-1 but others may look bizarre - just go with the flow.

you can always come across many different combinations of MLTQ. Some are familiar - like LT-1 but others may look bizarre - just go with the flow.
Does each combination identify one and only one "entity/concept"? What happens if 2 different entities share same combination?

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sophiecentaur
Gold Member
Entities and Concepts are in only your head. They are just what we use to try and get things sorted in our minds. Our brains love to categorise things and I think that's why we use special terms like speed, acceleration, power because they frequently turn up as combinations of the more fundamental quantities (MLTetc.)
Can you give an example of your second query about entities and combinations? I can't see where it's going.

we use to try and get things sorted in our minds.... special terms like speed, acceleration, power.
I asked if these "terms/concepts" that are in our minds must correspond biunivocally to combinations.
father of DA, Fourier states that: "the physics is independent of the units", how does this affect the choice of "units"?

sophiecentaur
Gold Member
I'm still not clear what you mean. (Biunivocal is a term you don't come across every day - but whadthehell).

You could take Electrical Resistance as an example. It could be described in terms of 'Volts per Unit Current' or 'Volts Squared per Watt' or 'Watts per Amp Squared' (or even Resistivity per metre). The first option is the one we use most because that involves the most commonly measured quantities, in practice. The second two could be much more suitable / meaningful in some circumstances.
The same Physics applies but we can choose quantities and units to suit.

I'm still not clear what you mean. (Biunivocal is a term you don't come across every day.
is http://wikipedia.org/wiki/Bijection" [Broken], bijective map[ping]/ function , one-to-one correspondence any better?

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sophiecentaur
Gold Member
This is one check that the equation could be, in fact, right..[PLAIN]http://www.uklv.info/g.php..[/QUOTE] [Broken]
I can't see the img.
Or is that a very subtle point. Last edited by a moderator:
Does each combination identify one and only one "entity/concept"? What happens if 2 different entities share same combination?
I think this is a very interesting question and useful conclusions can be drawn from it. Below are my thoughts. Comments are welcome.

In order to identify a concept with a combination one must demonstrate an algorithm that connects/maps combination to concept. E.g., L/T is a combination, the concept is velocity, and my algorithm to connect the two is that velocity is a measure of how much length per unit time. I am using the term algorithm since a 'thinker' must conceptualize these ideas in a given order to reach the desired conclusion of interpretation, and so I consider it a procedure.

If a single 'unit combination' is mathematically reordered, then the concept it maps to must be unchanged. Thus the difference must manifest in the interpretation, i.e., in the mental algorithm, that maps combination to concept. Note however that the end result of the algorithm here must be the same since one identifies this combination with a single concept beforehand. An example of this would be considering force as the product of mass and acceleration and later interpreting it as the time derivative for momentum--the conclusion hasn't changed, only the path/algorithm (although one could argue they're isomorphic, I will leave that alone for now).

The only way that two or more distinct entities/concepts could be mapped to by the same unit combination would be if there existed two distinct algorithms connecting the same combination to distinct entities. Below is an example of how this could occur.

Consider energy: Newton*Meter. Define this concept as a force of one Newton applied over a meter distance.

Consider torque: Newton*Meter. Define this concept as a force that causes rotational motion.

It is worth noting that multiple algorithms can be developed to map a unit combination to an entity, yet they may represent different concepts to the thinker. I think of force quite differently when considering mass times acceleration and when considering the derivative of momentum. Although I alluded to above that I believe these thoughts are in some sense isomorphic, their effect on the thinker's conceptualization, and hence their use in problem solving, is not.

One might also wonder if this points to some of our definitions as being non-fundamental.

Fundamentally torque is an approximation. If a beam of say Silver and one of Aluminum, both with identical macroscopic parameters, were subjected to the same perpendicular force relative to some axis, and we could measure torque with arbitrary precision, at some point there'd be a measurable difference due to differences in their particular constituents.

cepheid
Staff Emeritus
Gold Member
Does each combination identify one and only one "entity/concept"? What happens if 2 different entities share same combination?

Pressure and energy density have the same physical dimensions.

EDIT: the pressure and the energy density of an ideal gas are not exactly the same, but they are related to each other by a dimensionless factor.

I think this is a very interesting question and useful conclusions can be drawn from it. Below are my thoughts. Comments are welcome.
if you think so, you can discuss theoretical aspects here [post]3536678[/post]

1) Consider energy: Newton*Meter. Define this concept as a force of one Newton applied over a meter distance.
2) Consider torque: Newton*Meter. Define this concept as a force that causes rotational motion.
... I think of force quite differently when considering mass times acceleration and when considering the derivative of momentum..
This is a very interesting example:
I have shown here [post]3582794[/post], that there is no difference between 1 and 2, if you consider lever/torque for what it really is: when you realize that m/r in N*m / F*r1,2 , is not the r radius/arm of lever but the r rad[ian], the distance each weight travels. Nobody refuted that.

If you are able to make a drawing or, even better, an animation, you'll see with your own eyes that there is no difference when you lift a weigth on r2 with your hands or by means of [weight on r1] a lever. only that path [r] is slighly curved.

That suggests a reflection on vector L and τ

... pressure and the energy density of an ideal gas are not exactly the same, but they are related to each other by a dimensionless factor.

1) what is "http://wikipedia.org/wiki/Dimension" [Broken] in dimensional analysis [=DA], how is it related to a "quantity", say: "time"

2) "term/concept" speed has really "dimensions" LT-1 or is http://wikipedia.org/wiki/Fine_structure_constant#Physical_interpretations" like α?. Is α 0.0073 really dimensionless or has "dimension" speed : 0.0073 V [=C]

3) is DA useful only to check balance of units in equations [post#4]?

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Ken G
Gold Member
I think a lot of the confusion about units could be resolved by adopting two fairly simple but uncommon conventions:
1) all mathematical expressions should express truths about pure numbers (i.e., dimensionless quantities), and
2) all constants should be replaced by conventional values of the observables and dimensionless numbers required to make the conventions self-consistent.
When we do this, we would replace, for example, Newton's force of gravity, normally written F = GMm/d2, with F/F* = (M/M*)(m/m*)/(d/d*)2. All the subscripts * mean "the conventional value" for that observable, and note the conventions must be self-consistent in the sense that if all the quantities take on their conventional values, the equation must hold.

It is obvious from simple grouping of the terms what the value of G is, in terms of the conventional quantities, and that's all G ever was-- the value you get from a collection of self-consistent conventional choices. So although the form I suggest looks more complicated (and that's why it isn't used), it has conceptual advantages-- we pay a conceptual price for using "G". The form I suggest expresses two independent types of information-- it shows the functional dependences that characterize the law, and it also explicitly indicates a self-consistent convention has been adopted. The usual form focuses on the former goal, but compromises the latter, and obscures the role of convention in the whole concept of what a unit is.

Note also that we can recover the simple form, indeed a simpler form, by simply adopting the implicit convention that what we mean by any of the variables is actually their ratio to the conventional choices, so F is actually F/F* where the value F* is assumed to be implicit, and then the force of gravity becomes simply F=Mm/d2. This is the "business end" of the expression, the use of "G" is just a confuser and only adds tedium to doing physics problems. This form of the equation is I believe the reason that Fourier said that the physics is independent of the units-- all we need to do is do observations to tell us what a consistent convention is, and then we never need units in the equations of physics.

So what happened to the "G" in this form of the equation? Apparently, we don't need G if we reference all quantities to a convention that is consistent with the equation. So the entire reason for the presence of "G" is that we don't usually do that-- we choose our conventions for force, mass, and distance in an arbitrary way that is not consistent with the force of gravity. There's a reason for that-- the kinds of masses and distances we generally deal with yield negligible gravitational forces, so our self-consistent force convention would correspond with a very tiny force, and our actual forces would seem huge by comparison. But these are contexts where we don't calculate the force of gravity in the first place, we just use mg, saving us from having to measure the mass of interest and the mass of the Earth in the same units. We can still do that-- just use m/m* and a/g for masses and accelerations, and F=ma becomes F/F* = m/m* a/g, where F* = m* g is the self-consistent convention. When that convention is implicit, we again have F = ma, but now the quantities are dimensionless-- they are ratios to the self-consistent convention that generalize from a conventional observation to any other observation.

We don't usually do this because our everyday values are generally not self-consistent with the equation we want to use them in. Then we need constants that have dimensions in our equations-- but it is a high price to pay, because there is an actual lesson in the smallness of the gravitational force, and we completely miss that lesson when we select inconsistent unit conventions and have to include constants of conversion in our equations. I think the conceptual price we paid to get everyday kinds of numbers is too high-- I think we made the wrong choices for our unit conventions, and we pay the price of obscuring some of the more important lessons of physics by doing that.

Now, it should be mentioned that there will not be one single set of conventional values that will be consistent with all the equations of physics-- we still have to choose which equations we want to use to set the consistency of the conventions, and then other equations will have to include dimensionless constants (like the fine structure constant) to allow that consistency to continue to hold. But there is a lesson in these dimensionless constants-- they are pure numbers, so in a sense are "numbers that nature knows", and their values are meaningful independent of our conventions. Again, by choosing inconsistent conventions, we miss this lesson, the lesson of the dimensionless constants that nature actually exhibits-- they get lost in all the G, and k, and epsilon and so on.

An alternative is using "rational" units, which many theoretical physicists, who don't want to miss these lessons, do all the time. But they are not viewed as practical for everyday usage, as they don't translate well to people who want numbers they can picture from experience like square meters and kilograms and seconds. It's a compromise made to the engineers, in effect, but it obscures the meaning of the physics, and I think it was a mistake. It's basically the mentality that you "take the theory to the observations", meaning it is the theorists job to package everything in the language of the observer so the observer can test it without understanding what it is really saying. I think that's wrong-- I think the purpose of the theory is to understand the observations, so the observations must be converted into the language of the theory as a key step in understanding them. The observations are the reality, yet we must process them to understand their lessons, they are not just means of testing theories that need to be dumbed down into everyday numbers.

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gmax137