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Author Topic: Units, formulae, Equations. READ BEFORE YOU QUOTE UNITS INCORRECTLY!  (Read 14474 times)
Ivan
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« on: April 03, 2009, 12:58:15 AM »

This Sticky is to help people with their units (maybe formulae and equations later). Conversion between units or SI multipliers is the biggest source of error in calculations...and often results in an error of a factor of 1000 or more (eg confusing kilo-units with units). Sometimes people use the wrong units and we know what you mean; other times, the wrong units are used and we haven't a clue.

This is a work in progress. I've tried to make it readable, but I'll try to highlight the important stuff. As my worthy physics teacher would say, as he hit me across the head with a ruler 'You won't get anywhere, unless you learn to use your units correctly'


Numbers
Units use numbers. We need to have a universal way of writing numbers. The Romans could have been good mathematicians, but weren't because their way of writing numbers was illogical in a mathematical context. Separators cause problems. For example, in the UK, we use the symbol '.' to represent the decimal point. In the decimal number '2.5' we mean two units plus five tenths of a unit. However, elsewhere in Europe, countries often use the symbol ',' to represent the decimal point - so they would write '2,5'.

Numbers higher than one thousand, expressed nummerically are somewhat difficult to read, as the human brain has trouble visualising more than three similar objects in close proximity (anything above three is 'counted' rather than visualised). So it's easier to read numbers if they are separeated into groups of three digits. To avoid confusion, the groups are started from the unit rather than from the highest number. For example, we might write:

1
10
100
1 000
10 000
100 000
1 000 000
and so on. Note that I've used spaces for separators. This is very deliberate, and I'd encourage everyone to do this. The reason is that whereas in the UK, often people use a comma as a separator eg 1,000 , on the continent, it might be written like this 1.000 which can be confused with the number one (to an accuracy of 3 decimal places).

Units

There are lots of units in use. Some units even have different meanings depending on where you're discussing them (very confusing). In the UK we primarily deal with two sets of units 'imperial' and 'metric'. Although imperial units often have some advantages (eg if 12 smaller units equal one bigger unit, then you have the advantage of dividing the big unit by 12, 6, 4, 3, 2 or 1 and you still have a whole number of smaller units), some are equally rooted in something vague and imprecise  eg 'horsepower'

A long time ago, someone came up with a good idea for counting - metric. Metric meant units were counted in tens or tenths. Which makes good sense to humans as we have ten digits on our hands, which we've used for counting in decimal probably ever since we lost our hooves. Computers, on the other hand like a counting system based on two digits - On and Off (0 and 1) - known as 'base two' or 'binary'. Seems like hard work when you think that the number one million is expressed like this in binary '11110100001001000000'...but computers love it. The french also had a good idea (it's true!). They came up with a system of standard international units (SI units) which were deemed to be universal (aren't everybody's?) and critically, used metric in their units system. This made it a lot easier to use, and they also cleverly related different types of units to each other. For example 1 LITRE of pure water weighs 1 KILOGRAM (ok, don't ask me why it wasn't 1 GRAM) and measures 0.1m x 0.1m x 0.1m. The standard international base units are:

metre        m    length
kilogram     kg    mass
second       s    time
ampere       A    electric current
kelvin         K    thermodynamic temperature
candela     cd    luminous intensity
mole        mol    amount of substance

and other SI units are derived from these base units. Note that international units have abbreviations, and the abbreviations are case-sensitive.
****Simple Rule: Symbols of units named after people are upper case (W, K, J), symbols of simply named units are lower case (m, kg, s, h).****

The units can use prefixes to indicate metric multiples or fractions of units. Many people tend to work mainly with multiples of 1 000 ie g, mg, g, kg, Mg....although other prefixes exist eg

SI prefixes
Factor    Name     Symbol
1024      yotta       Y
1021      zetta       Z
1018      exa         E
1015     peta        P
1012      tera         T
109       giga         G
106       mega       M
103       kilo          k
102       hecto       h
101       deka        da
        
Factor    Name     Symbol
10-1       deci          d
10-2       centi         c
10-3       milli          m
10-6       micro        
10-9       nano         n
10-12     pico          p
10-15     femto        f
10-18     atto          a
10-21     zepto        z
10-24     yocto        y

****simple rule: kilo and below are lower case, above kilo are upper case*****


Energy

Energy is conserved absolutely. This is the first law of thermodynamics ('Energy can neither be created nor destroyed. It can only change forms'). When we talk about energy production, consumption or use, we actually mean the conversion of energy from one form to another. Energy usually ends up as heat, sooner or later, and it cools down by heating a larger mass of material to a lower temperature. Eventually everything that is or will ever be, will be reduced to a temperature only very slightly above  Absolute Zero and nothing more will happen. This is known as the heat-death of the universe, and is a consequence of the second law of thermodynamics, namely concerning entropy (The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium) which basically means that everything will become less and less ordered as energy is converted from higher forms to lower forms until all energy reaches it's lowest form (low grade heat).

Units of Energy
Joule (J) - a very small amount of energy - defined as the amount of electrical energy equal to the work done when a current of one ampere is passed through a resistance of one ohm for one second ALSO The work done by a force of one newton acts through a distance of one meter
kJ - a more useful unit of energy. Equals 1000 Joules. Even a kJ isn't really very much
MJ - Equals a million Joules. This is more like it.
kWh - This one is confusing, because it sounds a bit like a unit of power (kW). Actually, it's the amount of energy produced by 1kW of power over a period of an hour. In my opinnion this is one of the most useful units of energy you can relate to. 1kWh= 1000Watts x 3600seconds=3 600 000Joules (3.6MJ)
Wh - a very small unit of energy (ie 1/1000 of a kWh). Most frequently used in terms of battery storage or energy used by battery-powered products.

Units of Power (power = rate of conversion of energy from one form to another).
watt (W) - a very small amount of power. Equivalent to a rate of energy conversion of 1Joule per second.
kW - Equals 1000 Watts
MW - Equals 1 000 000 Watts
horsepower (hp) = 746Watts
BTU/hr [NB the abbreviation 'BTU' is used a lot by heating engineers, plumbers etc incorrectly as a unit of power. Are you surprised?]  1kW =3412BTU/hour

MISCONCEPTIONS
Do not use kW/h - this is not a unit of energy or power. It's actually a unit that is very infrequently used (at least, not correctly). It means 'rate of change of power' - and could be used, for example to describe how quickly a power source comes on line or perhaps how quickly the radiant heat of an object changes as it cools


Speed/Velocity

The words speed and velocity are often used interchangeably, and yet there is a very important difference. Velocity isn't simply a posh word for speed. It's a vector - meaning that you have to state which direction it is moving in. For example a car might be travelling with a speed of 100km/h. If it travelling North, then you could say it has a velocity of 100km/h in the direction of North. Or, you could say it has a velocity of 0km/h in the east direction. If on the other hand, the same car was travelling at 100km/h in the North East direction, you could say it has a speed of 100km/h or a velocity of 71km/h in the east direction AND a velocity of 71km/h in the North direction.




Common Units of Speed:
km/h (kilometres/hour), mph (miles/hour), m/s (metres per second)

Conversion between units
1m/s = 3.6km/h = 2.237mph

Volume
This one is easy, but also important when you're looking at American sites, as they use the same unit name 'gallon' but it means something a little different.

1 UK gallon = 4.546 litres
1 USA gallon = 3.785 litres (and bizarrely, the Americans have a 'dry' gallon of 4.40litres)
1m3 = 1000 litres

also of interest:

1 litre of water weighs 1kg (not a coincidence - this is how the units were defined).




Credits:
Thanks to Eccentric Anomoly for included comments
« Last Edit: October 09, 2009, 04:56:27 PM by Ivan » Logged
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