Quantities, units and dimensions

Contents

Physical Quantities

A physical quantity is any feature of matter that can be measured, for example mass and length. Every physical quantity has a numerical value and a unit, for example we can say your height is 1.87 metres. This means that the numerical value of your height is 1.87 and the units are metres.

Large and small values of quantities can be expressed in scientific notation, i.e. a multiple of a power of 10. For example if the distance between my school and your school is 1 000 m, we can express it in scientific notation as 1×103 m.

Prefixes

A much more easier way of representing large and small units is by using predefined prefixes. For example, 1 000 m can be represented as 1km whereby k is short for kilo which represents 103.

FactorPrefixSymbolExample
1012teraTterabyte(TB)
109gigaGgigahertz(GHz)
106megaMmegawatt(MW)
103kilokkilogramme(kg)
102hectohhectometer(hm)
10–1deciddecibel(dB)
10–2centiccentimetre(cm)
10–3millimmilliamperes(mA)
10–6microµmicrocoulombs(μC)

S.I units

Just like we have different languages around the world, we also have different units worldwide for each quantity. The problem comes when a device calibrated, say in U.S.A, in yards is exported to Zimbabwe where they only know metres and never heard of yards. However just like we can translate one language to another, we can also convert one unit to another. That is if we know the relationship between the two units.

The most ideal method would be to use the same units worldwide. The scientific community came up with an international system of units (S.I units) specifically for that.

S.I units can be divided into two classes :

1. Base units
2. Derived units

Base Units

Base units are the 7 fundamental units upon which S.I units are based on. The table below shows the base units.

QuantityNameSymbol
lengthmetrem
masskilogramkg
timeseconds
electric currentampereA
thermodynamic temperaturekelvinK
luminositycandelacd
amount of substancemolemol

Derived units

Derived units are units which are formed from product or quotient combinations of base units. The table below shows some derived units as products or quotients of base units.

QuantityNameSymbol
areasquare metrem2
volumecubic metrem3
speed or velocitymetre per secondm/s
accelerationmetre per second squaredm/s2
densitykilogram per cubic metrekg/m3
concentrationmole per cubic metremol/m3

Derived Units with Special Names

Some derived units have special symbols which do not show the base units from which they were derived.

These special units are named in honor of scientist who made important contributions to the respective areas in which the units are used. For example, in honor of the Sir Isaac Newton’s contributions to the field of mechanics, the unit of force is named the newton (N). Special units are written starting with a capital letter when written in short form as shown in the table below.

QuantityUnitSymbolDerivation from other unitsDerivation from base units
forcenewtonNkgm/s2
pressurepascalPaN/m2kgm-1s–2
energyjouleJNmkgm2/s2
powerwattWJ/skgm2/s3

Dimensions of quantities

Dimensions of a quantity are the base quantities from which the quantity is derived. We use square brackets to indicate "dimensions of …". For example, [Force] means the dimensions of Force.

Velocity

Since we know that velocity = distance ÷ time and that distance is length, the dimensions of velocity are expressed as:

[Velocity] = LT-1

The expression above shows us that the dimensions of velocity (a derived quantity) are length (a base quantity) and time (a base quantity). Symbols M, L and T are used to denote the base quantities mass, length and time respectively.

Acceleration

Acceleration is the rate of change of velocity.

Acceleration = velocity ÷ time

[Acceleration] = LT-1/T
= LT-2

Force

Force = mass × acceleration

F = ma

[Force] = MLT-2

Area

Area = length × length

[Area] = L2

Pressure

Pressure = force ÷ area

[P] = MLT-2 ÷ L2
= ML-1T-2

Work

Work = force × distance

[Work] = MLT-2 × L
= ML2T-2

Moment

Moment = force × perpendicular distance

[Moment] = MLT-2 × L
= ML2T-2

Dimensionless quantities

Not all quantities have dimensions, some are dimensionless, which means their dimensions are zero. Usually these units are ratios of the quantities with the same dimensions. One common example of a dimensionless quantity is angle (θ).

To dimension angle we use the its definition in radians.

Angle = length of arc ÷ radius of arc

[θ] = L ÷ L (since radius is also length)
= 1

If the dimensions of a quantity are equal to a number or a numeric ratio, the quantity is dimensionless

In the example above we got 1 after dimensioning angle, this means that angle is dimensionless (has no dimensions or its dimensions are zero). We do not say its dimensions are 1.

Dimensioning units

The dimensions of a quantity are the same as the dimensions of its unit.

As an example, lets look at the quantity ‘force’.

[Force] = MLT-2

This means that the dimensions of the unit of Force can be expressed as the dimensions of the units of M,L and T as follows:

[newton] = [kilogram][meter][second]-2
[N] = [kg][m][s]-2

And this therefore means that both N and kgms-2 are suitable units for force.