Dimensions and Units

Having an understand of dimensions and units is important.  This is because any physical quantity can be characterized through the use of dimensions.  While a unit is normally assigned to a dimension, since it represents the magnitude of the dimension.

Primary vs Secondary Dimensions

There are two distinct types of dimensions.  They are primary and secondary dimensions.  First, let’s take a look at primary dimensions.  Some primary dimensions, or basic dimensions, are mass $m$, time $t$, length $L$, and temperature $T$.  On the other hand, some examples of secondary dimensions are velocity $v$, density $ρ$, and energy $E$.  The reason why these are secondary dimensions is because they are made up of a combination of primary dimensions.  For example velocity is mass $m$ over time $t$.

Unit Systems

Through out the ages different unit systems have been developed.  As of today there are two sets of unit systems that are commonly used.  They are the English System and the Standard International (SI) system; also known as the metric system.  There is currently a push to unify the world to the SI system.  However, some countries, mainly the United States, have not completely switched over, and are still using the English System.

The main reason why the SI system is so popular is because it is simple and logical.  What makes it so logical is that it is based off of a decimal relationship between the various units.  Refer to the table below.

Standard Prefixes in SI Units

Multiple

Prefix

$10^12$

tera, $T$

$10^9$

giga, $G$

$10^6$

mega, $M$

$10^3$

kilo, $k$

$10^2$

hecto, $h$

$10^1$

deka, $da$

$10^{-1}$

deci, $d$

$10^{-2}$

centi, $c$

$10^{-3}$

milli, $m$

$10^{-6}$

micro, $μ$

$10^{-9}$

nano, $n$

$10^{-12}$

pico, $p$

 

For example if you measured an object and determined that it was 10 cm long, from this table you can also conclude that it would be .01 m long as well as 100 mm long.

The English system, on the other hand, does not follow a systematic numerical base like the SI system.  Due to this it is not as easy to convert from one type of unit to another.  Instead you have to memorize the number that is needed to make the conversion.  For example, 12 inches = 1 foot, while 5280 feet = 1 mile.

Finally, below is a table showing several fundamental dimensions with their SI unit.

Dimensions

Unit

Length

meter (m)

Mass

kilogram (kg)

Time

second (s)

Temperature

kelvin (K)

Electric Current

ampere (A)

Amount of Light

candela (cd)

Amount of Matter

mole (mol)

SI and English Systems

Due to the fact that both the SI and English systems are both used, there will be times that you will need to convert from one system to the other system.

First, let’s take a look at the SI system.  For the SI system the units of mass, length, and time are kilogram (kg), meter (m), and second (s).   On the other hand, for the English system mass, and length are pound-mass (lbm), and feet (ft).  The units for time is the same between the two systems.

(Eq 1) $1~lbm~=~0.45359~kg$

(Eq 2) $1~ft~=~0.3048~m$

Force

You may have noticed that for the English system mass is pound-mass instead of just pounds. The reason why it is written like this is to avoid confusion when pounds are used to describe the force.  Which in that case, for force the English system would be pound-force (lbf).  On the other hand, the unit for force for the SI system is newtons (N).  Finally, to determine the force acting on an object you use Newton’s second law of motion which states that a force is derived from a mass times the acceleration acting on that mass.

(Eq 3) $F=ma$

Hence,

(Eq 4) $1~N~=~1~kg·m/s^2$

(Eq 5) $1lb~f~=32.174~lbm·ft/s^2$

The reason why equations 4 and 5 are written the way they are is because of the following statements. In the SI system a force of 1 newton (N) is defined by the force required to accelerate a 1 kg mass at the rate of 1 $m/s^2$.  On the other hand, for the English system the force of 1 lbf is defined as the force required to accelerated a 32.174 lbm (1 slug) at a rate of 1 $ft/s^2$.  The constant 32.174 $ft/s^2$ also happens to be the gravitational constant on earth for the English system. For the SI system the equivalent gravitational constant is 9.81 $m/s^2$.

Finally, weight (W), which is often incorrectly expressed as a mass, is the force caused by the gravitational constant $g$.  To calculate weight the following equation is used.

(Eq6) $W=mg$

Work

Work is a form of energy that is the product of a force times a distance.  For the SI system this would be a newton-meter (N·m).  This is often refereed to as a joule (J).

(Eq 7) $1~J~=~1~N·m$

On the other hand the energy unit for the English system is the Btu (British thermal unit).  One Btu represents the amount of energy needed to raise the temperature of 1 lbm of water at $68 ^oF$, by $1^oF$.

(Eq 8) $1~Btu~=~1.0551~kJ$

In the SI system one calorie (cal) is the energy required to raise 1 g of water that is at $14.5 ^oC$ by $1^oC$.

 

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