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Quantities! are they all dimensional uniques??

July 23, 2009

Small Talk
Is it really important to have a unit associated to value.
I really don’t know the answer

I assumed it is important to know the dimension of the quantity.
* To be able to sum, and subtract different units but same dimension quantities.
* To be able to predict that the outcome of the division and multiplication process of quantities.
* To deduce how can I reach specific quantity from another quantity.

Article Begins
So let us focus on the important objective of the library and its calculator.
The main objective was to replace the Dimensionless Variable Concept into Quantity Concept

Reviewing the SI model there are 7 basic base units that form all of our Quantities experience in Life

🙂 do I really have to list them ?? I think I should 😦

1- kg (kilogram) [Mass]
2- m (metre) [Length]
3- s (second) [Time]
4- A (Ampere) [Electric Current]
5- K (Kelvin) [Temperature]
6- mol (mole) [Amount of Substance]
7- cd (candela) [Luminous Intensity]

these are 7 base quantities with 7 base units
(I feel they made it 7 so we can tie our self to the 7 number like 7 days and more legendary 7s)

if you encounter a fluid course in your study you will encounter the dimensional analysis course that teach you MLT system

the lecture will teach you about the dimensionality of the quantity
so Length is equal to M=0 L=1 T=0 or for simple writing M0L1T0

Speed or Velocity: M0L1T-1
Acceleration : M0L1T-2

Based on the dimensional Analysis course any quantity in universe have a unique dimensions

and we have special quantities that have M0L0T0 which means Dimensionless quantity

Types of Dimensionless quantities are many
Mach number.
Reynolds number.
Affinity Laws.

also based on professors philosophy Dimensionless numbers are a high class numbers because they always give a lot of information.

for example if I say I am traveling to Alexandria from Cairo
and you asked me about the distance I covered from the beginning of the trip!!
If I told you I passed 100 km from the distance you will wonder and may feel that information is not enough.
But if I told you that I passed half of the distance to Alexandria
you can deduce many things.
because you know roughly when I begin traveling.
you know the maximum speed between towns.
you can deduce when I should get there and in what hour.

practically you can sense and understand whats going on and what will happen.
or simply REALIZE

so dimensionless numbers are important for our life 🙂

Different Quantities and The same dimensions

BACK to quantities again

Force quantity are taken from the classical newton equations and it is Mass multiplied by Acceleration

F = kg * m/s^2

this is awesome so Force as a derived quantity = M1L1T-2

(NOTE: in dimensional analysis there is another approach to define dimensions by FLT system due to the fact that we can’t get mass without knowing the gravity of the location we are on now 🙂 however let us stick with MLT)

BUT can we REALLY say in honest words that all quantities have a different unique dimensions ??

{NO}

and let me list them as I can remember for now:

1- (Work) vs (Torque):
———————————————–

The two concepts are referring to the same units which are Joule, calorie, or BTU

Work = Force * Length

The same as Torque

Torque = Force * Length

yes you can say that they are different quantities but look at them in units they are the same units N.m 😦

2- (Dimensionless) vs (Angle)
——————————
Angle is measured by Length of circumference to the length of radius.
it will be dimensionless

3- (Angle) vs (Solid Angle)
—————————-
as angle units, steradian for solid angle
Solid angle is Area to Area

4-(Frequency) vs (Angular Velocity)
————————————
Frequency is equal to M0L0T-1 or 1/s i.e. (1/Time)
Angular Velocity = rad/s

and rad = 1 because it is dimensionless.

so summing frequency with angular velocity should be logically correct (because they resemble the same dimensions) WHICH is WRONG in literature.

do you remember the equation of Omega = 2 * Pi * f
where
Omega: Angular Velocity
f: Frequency.

5- (Luminous Intensity) vs (Luminous Flux)
——————————————–
The first quantity is lm/sr and the second one is (cd.sr)=lm

Units can differentiate between the two quantities

but the unit is dimensionless and in SI standards you can replace either rad and sterad with which the dimensionless unit.

Conclusion:

  1. These quantities will prevent any implementation to differentiate between quantities by dimensions and in return will prevent the intelligence of knowing the outcome quantity of an operation
  2. You can’t take the dimensional analysis approach in differentiating between quantities.

So in return most of the implementation of dimensionality of variables took the shorter approach of supporting only units side.

but where is the quantity ???

can the unit approach differentiate between Torque as N.m and Work as N.m

if you implemented the J (Joule) unit you will have to invent a unit for Torque also, so you can make a distinction between units.

But how about the rest of quantities.

In my next post I’ll introduce to you the hypothetical solution of this dilemma and how I solved this problem in my Quantity System implementation.

Thank you
Ahmed Sadek

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