Dimensional analysis and units

[Biker]

Hello,

I have been doing high school physics for about 3 years now and I am perfectly fine with it. However, Something tricked me off few days ago. It is how we treat units. I was kind of hesitant to post this thread because I have seen multiple threads about it before and some member really posted useful links about it. It is a very common one.

I will define a "Unit" from what I understand. Unit is a measurement of physical quantity. We define how much a Kg weighs and we use it in our mathematics. We say we have 5 Kg which means we have something that weighs 5 times bigger than the unit we chose.

We can add similar units together. It makes intuitive sense.
2 Kg + 3 Kg = 5 Kg
we can subtract, divide such as density.

Now if I think about different units.
I can't simply add:
2 apples + 3 oranges = ... (No physical meaning)

"Multiplication is repeated addition" or we can see it is scaling which consists of addition.
So 2 apples * 3 oranges = shouldn't give you 6 apples x oranges.

However division makes sense. For example speed when I say 5m/s
I am just saying that for every sec passes I move 5 m.

and what about area? I define an area of an object to be made out of a square meter for example which is just a square with length of 1 m

So when I find the area of a rectangle with length 5 and width 2 I am simply stating
that I have 5 square meters for 1 width so you have 10 square meters
We reach to the same result as if we multiply units
5 m x 2 m = 10 m^2

I have read books about this and they just simply state that the derived quantity is just the product of the units without giving explanation

A summary of all this is, Why do we treat units as if they were variables?

Hopefully, Someone can set me straight because it is really irritating that I am not able to figure this out and it is kind of embarrassing :/

 

[Stephen Tashi]

You ask good questions - especially about why there apparently some distinction between addition of unlike units and multiplication of unlike units. Both addition and multiplication introduce ambiguity in recording the results of an experiment. If we allow the unit "apples-plus-oranges" and the value in a experiment is recorded as 6 apples-plus-oranges, we don't know if there were 3 apples and 3 oranges or zero apples and 6 oranges etc. Likewise if the unit recorded in an experiment is given as 6 apples-times-oranges, we don't know if the experiment involved 6 apples and 1 orange or 3 apples and 2 oranges etc.

To accept that 6 apples-time-oranges is a useful unit in recording an experiment, we must accept that whatever we are doing doesn't depend on the individual numbers of apples and oranges, but only on their product. Likewise if we accept apples+oranges as useful unit then what we are doing must depend only on the sum and not the individual summands.

To accept that an equation has significance as a "physical law", its natural to insist that if it is confirmed by experiment A then it can be confirmed by an experiment B where the only change is to use different units of measurement. For example, if a physical law is true when the experiment measures things in ft, lbs,minutes then
it should be confirmed by repeating the experiment using measurements of meters, Newtons, seconds. The confirmation is with the understanding that the numbers used in equation of the physical law are converted in some manner. The equality of the two sides of the equation should be preserved after the conversion.

The usual manner of defining units implies converting units by multiplication ( eg. 1 minute = 60 seconds , 1 apple = 2 half-apples ). It is easy to think of situations where the product or ratio of units gives a complete description of the situation. For example, the same effect of torque on a bolt can be produced by a small force acting on a long wrench handle or a large force acting on a short wrench handle. It is difficult to think to situations where the different ways of realizing a sum of physical units produce the same result - especially since a value like 6 apples-plus-oranges admits the possibility that the experiment might involve zero apples.

There might be a way to begin with simple assumptions and prove that the only sensible system of units involves defining and converting units by multiplications. I've read that James Clerk Maxwell wrote about this topic, but I haven't read his original paper. Perhaps some forum member can give us a link to it or summarize his arguments.

The treatments that I have seen of dimensional analysis simply take for granted that "apples-plus-oranges" makes no sense as a physical unit, but that apples-times-oranges might make sense.

A good text about dimensional analysis is available on the web: The Physical Basis of Dimensional Analysis by Ain A. Sonin http://web.mit.edu/2.25/www/pdf/DA_unified.pdf However, Sonin simply assumes the sum of unlike units has no meaning. He doesn't prove it from more basic assumptions.

However division makes sense. For example speed when I say 5m/s
I am just saying that for every sec passes I move 5 m.

You shouldn't think that a physical unit implies a unique physical situation. For example, there could be an experiment where the battery of a toy car is charged for Y seconds. Then the car is turned on and runs for X meters on the charge. The equation that fits the data might be X = 5 Y or X/Y = 5 meters/sec. So meters/sec does not refer to a velocity in this situation. The idea that meters/sec "is" velocity comes from the fact that there are many many equations in physics where the measurement of meters/sec indeed represents a velocity. However, this doesn't imply that "meters/sec" must always represent a velocity.

 

[Biker]

I have read that link before. I have seen you post it in a previous thread.. Very useful!

I really like your argument about the product of different units gives us what happens in reality. We can apply the same idea to forces
F = m a
1 2
2 1
respectively result in the same feeling of force
However, If we chose F = m +a
We might get the same result by taking m =3 and a = 0 which defies the reasoning of forces and ruins the concept.
Also, the machine example was a pretty good example :D.

Sometimes, the reason of things end up being that this is how our world works. Even though multiplying or summing units don't make physical sense. Maybe that is why most of writers just glance over it by saying in some sense we can multiply and divide units without giving an intuition approach

When I asked my math teacher, He simply replied "It is a philosophical thing to think about, You have moved out of simply applying physics to think about how it works which is good but can often lead to confusion if you don't have someone to clear it out in your level" But he was happy that I asked those questions
I really hope to read Maxwell's paper about this topic.

 

[PeroK]

Why does 2 apples + 3 oranges make no sense? That makes perfect sense to me. That might currently describe the contents of my fruit bowl.

 

[Biker]

Why does 2 apples + 3 oranges make no sense? That makes perfect sense to me. That might currently describe the contents of my fruit bowl.

"Then what you're really doing is defining a more general category, which contains both of the things you're adding. You're not really adding 2 apples and 3 oranges, you're just adding 2 fruits and 3 fruits"

A quote from a website. Just saved time and copied it XD

 

[PeroK]

"Then what you're really doing is defining a more general category, which contains both of the things you're adding. You're not really adding 2 apples and 3 oranges, you're just adding 2 fruits and 3 fruits"

A quote from a website. Just saved time and copied it XD

Well that's just hokum. To get the answer 5 of something you need to generalise to fruit. But, adding 2 apples to a bowl containing 3 oranges is a valid physical process.

 

[Khashishi]

We can treat units like constants because they are constants, for all intents and purposes. We can also use a constant as a unit if we want. For example, measuring speed in units of c.
We write units differently from constants by convention (upright vs italic, and we always put the units after the number) but mathematically they are the same thing.

 

[Svein]

Well that's just hokum. To get the answer 5 of something you need to generalise to fruit. But, adding 2 apples to a bowl containing 3 oranges is a valid physical process.

It can also be a valid mathematical statement:

My space X consists of all thing edible. The subset "Fruit bowl" (abbreviated to B) consists of {apple, apple, orange, orange, orange}.

Unions and intersections are valid operations...

 

[Charles Link]

Units can sometimes be tricky, but I find one thing helpful with conversions. A vector $\vec{Q}$ can be written as quantity $x$ of unit $\hat{ X}$ can also be written as quantity $y$ of unit $\hat{Y}$.i.e. $\vec{Q}=x \hat{X}=y \hat{Y}$. If we know the conversion, e.g. $\hat{X}=b \hat{Y}$ (e.g. $\hat{X}$ can be a "foot" and $\hat{Y}$ can be an "inch" so that 1 foot=12 inches) then $1=b \hat{Y}/\hat{X}$. If we begin with $\vec{Q}=x \hat{X}$, we can convert to $\hat{Y}$ units by multiplying by $1=b \hat{Y}/\hat{X}$ by writing $\vec{Q}= (\vec{Q})(b \hat{Y}/( \hat{X}))=(b x) \hat{Y}$. This gives $y=bx$. With our example, Let $\vec{L}=50 ft=50 ft (12 \,inches/1 ft)=600 \, inches$ . Most conversions of any vector quantity can be done by this process.

 

[Biker]

Well that's just hokum. To get the answer 5 of something you need to generalise to fruit. But, adding 2 apples to a bowl containing 3 oranges is a valid physical process.

Well I guess that is a point.
I think I might have to change the first example to something else
5 kg + 5m/s = No physical meaning
5Kg * 5 m/s = 25 Kg m/s

 

[olivermsun]

You can make the conversion explicit if need be:
2 apples * (1 fruit / apple) + 3 oranges * (1 fruit / orange) = 5 fruit.