Dimensions / units

[jajrussel]

Dimension / units

 

What is the difference?

 

For instance is mass a dimension where kilograms are it's unit?

 

Is energy a dimension where joules are it's unit?

[studiot]

Hello.

 

 

For instance is mass a dimension where kilograms are it's unit?

Is energy a dimension where joules are it's unit?

 

Yes

 

No

 

In mechanical science we recognise three basic dimensions ( Mass, Length, Time) and can express all other mechanical quantities in terms of these three.

 

When considering heat we need to add another dimension and temperature is usually chosen.

 

When considering electromagnetism we again need one more and current is now chosen. (charge was once but that has changed to current).

 

Certain other less common areas of science (such as light) need some further dimensions.

 

So your first observation is right mass is a fundamental dimension (*symbol M) and can be expressed in kilogramme units or many othere for example pounds, grammes, atomic mass units (AMU) etc.

 

Energy however is not one of the fundamental big three so it is not a dimension, although its units are joules or BTU or whatever.

 

For quantities such as speed and energy which are not dimensions in themselves we talk of the dimensions of that quantity.

 

So speed is distance divided by time and has dimensions of length over time or LT^-1

 

Energy can also be expressed in terms of ML&T.

This expression is more complicated and equal to dimensions of energy = ML²T^-2

 

Please note that the dimensions of speed and energy do not change, whatever units you measure in.

 

Dimensions are useful in seeing that certain quantities have the same dimensions (and are therefore the same)

 

For instance

 

Energy = (ML²T^-2) = Work = force times distance = (MLT) x (L) = Power times time = (ML²T^-3) x (T)

 

Here is a long list of dimensions and units.

 

http://www.ebyte.it/library/educards/sidimensions/SiDimensionsByCategory.html

Edited September 20, 2015 by studiot

[jajrussel]

Thanks studiot, this changes some of what I thought I knew.

[studiot]

If you understood that here's a second installment, combining dimensions.

 

1) Pure numbers have no dimensions so distance and length have the dimension L. Doubling a distance does not change this, the doubled distance still has dimension L.

 

2) If we add quantities we do not change the dimension. So 2 inches plus 3 inches = 5 inches still has dimension L.

A further consequence is that you cannot add things of different dimensions in an equation.

So you cannot add inches and seconds.

If an equation contains a plus (or minus) sign than all the terms must have the same dimensions.

3) Since we cannot make compound units by adding, we make them by multiplying (or dividing) to quantities together as in my previous examples of speed and energy.

Equations containing only multiplications have the same dimensions and units on both sides of the equation.

There may be different if there is a unit conversion involved. This conversion factor will not be a pure number but have units of its own eg the 4.2 joules per calorie when converting to each other.

This means that a number may be dimensionless (have no dimensions) but yet have units!

Of course some (pure) numbers such as [math]\pi[/math] have neither dimensions nor units.

Edited September 20, 2015 by studiot

[jajrussel]