Units (split from What is Time?)

[michel123456]

What I find utterly troubling is the multiplication of units.

OK meters by meters makes sense (square meters) where the m^2 is something that you can represent easily.

M^3 is volume, no problem.

But in one of the tables posted above you encounter m^4. What is this?

Similarly, square seconds (s^2), what is this? As if time (which is single-dimensional) could be squared.

Also, if one looks at the Universal Law of Gravitation, one sees kg^2 in the formula. Mass multiplied by mass: mass squared, what is this?

And so on, physics if full of these.

[StringJunky]

35 minutes ago, michel123456 said:

What I find utterly troubling is the multiplication of units.

OK meters by meters makes sense (square meters) where the m^2 is something that you can represent easily.

M^3 is volume, no problem.

But in one of the tables posted above you encounter m^4. What is this?

Similarly, square seconds (s^2), what is this? As if time (which is single-dimensional) could be squared.

Also, if one looks at the Universal Law of Gravitation, one sees kg^2 in the formula. Mass multiplied by mass: mass squared, what is this?

And so on, physics if full of these.

I had a thread recently on square seconds. See if it helps.

 

[jajrussel]

27 minutes ago, StringJunky said:

I had a thread recently on square seconds. See if it helps.

 

I'm way out of my league here, which is probably why I don't understand this question, but doesn't a second of time represent a portion of space? As a portion of space isn't it okay to think of squaring it? I haven't actually read this whole thread yet so if the questions are out of line I apologize.

[michel123456]

It doesn't explain what s^2 is for a thing.

But on the other hand I understand your question: seconds by seconds (s/s) gives unity, it doesn't give you s^2.

[Strange]

59 minutes ago, michel123456 said:

But on the other hand I understand your question: seconds by seconds (s/s) gives unity, it doesn't give you s^2.

That is because it isn't seconds by seconds; the notation is confusing (as explained in that thread).

It also gives you a good example of what s² means (OK, s^-2)

[studiot]

3 hours ago, michel123456 said:

But in one of the tables posted above you encounter m^4. What is this?

Metres⁴ is a 'section property', called Moment of Inertia by structural engineers, materials engineers and the like.

The fourth power arises because it is defined as the Integral the product of an area and the square of its distance from the pivot axis.

For a given shape

[math]{I_x} = \int {{y^2}} dA[/math]

Where I_x is the Moment of inertia of the shape about the x axis
dA is a small (differential parcel of area of the shape
y is the distance of that area from the x axis

This section property appears in formulae involving bending in beams etc.

Here is a table for various shapes of cross section. Note that the formulae involve four lengths multiplies together to get the fourth power.

Does this help?

 

Funnily enough Prometheus is currently doing a grand job helping a new member with some statistics. Moments of area are also used in statistics.

 

 

Edited April 17, 2018 by studiot

[jajrussel]

5 hours ago, michel123456 said:

It doesn't explain what s^2 is for a thing.

But on the other hand I understand your question: seconds by seconds (s/s) gives unity, it doesn't give you s^2.

Now that I have read the whole thread and enjoyed the reading. I'm still puzzled why you seem to keep singling out the time unit second when the expression is meters per second, per second. Or meters per second squared. With standard Earth gravity the rate of acceleration is 9.8 m/s or is stated as 9.8 meters per second. 9 seconds later at that rate of acceleration the things velocity will be 88.2 meters per second which is 9.8 times the nine seconds of time. When you square the time and multiply it times the acceleration rate you end up with the distance traveled in 9 seconds by the thing, which is simply stated as 793.8 meters. At this point you can forget m/s/s.   The s^2 confusion is probably brought about by thinking how does squaring time give you distance? By itself it doesn't. You need the meters per second times the seconds squared because it is an acceleration rate. If it was simply a velocity of 9.8 meters per second you could just add 9.8, nine times. Then the distance in nine seconds would be 88.2 meters. Also, I am completely lost by the stament of unity said to be given by (s/s). What exactly does that mean?

I hope I managed to explain this correctly. Oh well, I'm sure I'll be corrected if I'm wrong. 

 

 

 

Edited April 18, 2018 by jajrussel

[swansont]

There doesn't have to be a physical meaning behind combinations of units, especially in constants of proportionality. It's a book-keeping exercise.

It's similar to being befuddled that a negative entry can be present in a ledger, since negative money can't physically exist.

[jajrussel]

we've changed rooms? I kinda left my note in the other room. It has a little to do with this subject on it, but I understand the bookkeeping analogy, so I should be able to follow along...

[Endy0816]

8 hours ago, jajrussel said:

Also, I am completely lost by the stament of unity said to be given by (s/s). What exactly does that mean?

He's talking about the old chessnut,

a/a = 1

1 being Unity

though for m/s^2, we're not dividing seconds by seconds, rather (1/seconds) by seconds, so it can't be used here.

(1/a)/a = 1/(a^2)

 

[jajrussel]

Thank you. I tried to up vote your post in thanks, but I either don't understand how it works or it isn't working on my end. Had a similar problem yesterday. I'm almost afraid that mentioning it will result in another room change, so I will eventually search it out through the proper channels when I get tired if trying to up vote with apparent null results.

Once again I thank you.

[Strange]

15 minutes ago, jajrussel said:

I tried to up vote your post in thanks, but I either don't understand how it works or it isn't working on my end.

*Whispering so the mods won't notice* 

There is a bug in the software so that the votes don't appear immediately. If you come back to the thread, you should see it.

 

1 hour ago, swansont said:

There doesn't have to be a physical meaning behind combinations of units, especially in constants of proportionality.

And sometimes, there might appear to be a physical interpretation but it doesn't really mean anything. For example if you work out the fuel use of your car in, say, gallons per mile (for the Americans) the result has the same units as area. But I'm not sure what that area would mean.

[Sensei]

17 hours ago, michel123456 said:

What I find utterly troubling is the multiplication of units.

There is needed context in which they are used.

e.g.

If you have energy E₀ at the beginning of experiment at time t₀=0s

and after 1s delay at time t₁=1s there is energy E₁,

you can write equation

(E₁-E₀)/(t₁-t₀) = dE/dt

and then you use multiplication/division of units to find out what are new units of newly created power P.

Energy was in kg*m^2*s^-2, subtraction E₁-E₀=dE and t₁-t₀=dt don't change units, but division by time does.. (kg*m^2*s^-2) *s^-1 gives you kg*m^2*s^-3..

Second to third power looks weird, but you have to remember how to get to this result knowing equations P=I*U=dE/dt .. It will help.

 

17 hours ago, michel123456 said:

But in one of the tables posted above you encounter m^4. What is this?

Equation where energy has been multiplied by energy (J*J).. or where volume has been multiplied by velocity (V * v)... or where area has been multiplied by other area (A*A)... etc. ?

[swansont]

56 minutes ago, Strange said:

 And sometimes, there might appear to be a physical interpretation but it doesn't really mean anything. For example if you work out the fuel use of your car in, say, gallons per mile (for the Americans) the result has the same units as area. But I'm not sure what that area would mean.

Good point. Unit cancellation can cause such issues —  the original portrayal of information does have a physical significance, but after units are canceled that information is no longer there.  

(Americans tend to use miles per gallon, even though it has a murkier physical significance — it's nonlinear — than gallons per mile)

[Sensei]

 

17 hours ago, jajrussel said:

I'm way out of my league here, which is probably why I don't understand this question, but doesn't a second of time represent a portion of space? As a portion of space isn't it okay to think of squaring it? I haven't actually read this whole thread yet so if the questions are out of line I apologize.

Imagine following experiment. We start with object at altitude/height h=20m, then we release object, and while it's flying we record position of object on timeline (easily done with modern high speed cameras).

We received graph h(t). It can be put to e.g. Excel or OpenOffice Calc Spreadsheet. They have simple unit meter m. One entry of table per single exact time. Altitude measured from the ground.

If you subtract entries in spreadsheet, A2-A1, fill it down, you will receive distance traveled each second, you can (in memory) divide this distance by e.g. 1s time delay between them. Put it in column B in spreadsheet. First row will be empty because of lack of data (B2=A2-A1... B3=A3-A2... B4=A4-A3... and so on..)

(h(1)-h(0))/(t(1)-t(0) = dH/dt = local velocity at 1st second

(h(2)-h(1))/(t(2)-t(1) = dH/dt = local velocity at 2nd second (depends on initial altitude whether there will be 2nd second of flight)

(h(3)-h(2))/(t(3)-t(2) = dH/dt = local velocity at 3rd second (depends on initial altitude whether there will be 3rd second of flight)

dH/dt has unit m/s.

Now imagine that you're interested in how this velocity changes over time. It's acceleration.

You have to take B3 and subtract from it B2 (In memory you divide by 1s).

You have dV/dt. With units (m/s) / s = m/s^2 = m * s^-2

And this is how such example spreadsheet could looks like:

 

I used 0.1 time delay between entries to show how it changes.

 

Initial A1 = 20 - 1/2*9.81*T1^2 and then Edit > Fill > Down

 

 

Edited April 18, 2018 by Sensei

[michel123456]

8 hours ago, swansont said:

There doesn't have to be a physical meaning behind combinations of units, especially in constants of proportionality. It's a book-keeping exercise.

It's similar to being befuddled that a negative entry can be present in a ledger, since negative money can't physically exist.

Nicely said.

However it is still bizarre that you can use quantities that do not exist in order to get a correct result. It looks more like a prestidigitator trick than a representation of the way the laws of nature are acting.

[swansont]

2 hours ago, michel123456 said:

Nicely said.

However it is still bizarre that you can use quantities that do not exist in order to get a correct result. It looks more like a prestidigitator trick than a representation of the way the laws of nature are acting.

Are you familiar with imaginary numbers, and their use in physics?

Remember that this is about how nature acts (as you note) and not the way nature is. These are parts of models.

[michel123456]

On 4/18/2018 at 1:16 PM, swansont said:

There doesn't have to be a physical meaning behind combinations of units, especially in constants of proportionality. It's a book-keeping exercise.

It's similar to being befuddled that a negative entry can be present in a ledger, since negative money can't physically exist.

Do accountants use dollars squared?

21 hours ago, swansont said:

Are you familiar with imaginary numbers, and their use in physics?

 

 

Imaginary numbers have no units.

On 4/18/2018 at 12:43 AM, studiot said:

For a given shape

Ix=∫y2dA

Where I_x is the Moment of inertia of the shape about the x axis
dA is a small (differential parcel of area of the shape
y is the distance of that area from the x axis

 

y^2 that is the square of the distance again. Which is an area, and not a very long distance. So you have an area multiplied by an area (A). Correct?

And what about kg^2? mass multiplied by mass? The Earth multiplied by the Sun. A banana multiplied by an umbrella (remebering Pr. E.R. Laithwaite)

 

 

Edited April 19, 2018 by michel123456

[studiot]

Indeed, my apologies, you have the correct integegral.

it is really difficult to get this stupid editor to do what I want.

Nevertheless was it sufficient answer to your question?

19 minutes ago, michel123456 said:

Do accountants use dollars squared?

Don't they use square dollars in the Lone Square State?

[swansont]

1 hour ago, michel123456 said:

Do accountants use dollars squared?

I have no idea. I'm not an accountant.

Quote

Imaginary numbers have no units.

Are you sure about that?

The impedances of capacitors and inductors are complex. For an inductor the imaginary part of the impedance Z  is iwL (w is frequency, L is inductance) That's got units. There are also instances where complex velocities and times appear in problems. Units. (and probably more examples)

 

[michel123456]

16 hours ago, swansont said:

 

Are you sure about that?

 

 

I am less sure now.

I was supposing that a number has no units. Being positive or negative.

[StringJunky]

17 hours ago, studiot said:

Don't they use square dollars in the Lone Square State?

Jeff Bezos is worth $168291.41²   

[Strange]

38 minutes ago, StringJunky said:

Jeff Bezos is worth $168291.41²   

Unimaginable.

[StringJunky]

3 minutes ago, Strange said:

Unimaginable.

It's not real money though. It's just numbers, most of it. Value is not money in your hand....or truck in his case.

Edited April 20, 2018 by StringJunky

[studiot]

1 hour ago, StringJunky said:

Jeff Bezos is worth $168291.41²   

 

Is that because he has two ffs to his name?

 

Perhaps I should start a thread entitled Who is Jeff Bezos? (because I heave never heard of the bozo)