A-level Physics/The SI System of Units

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SI units are used throughout science in many countries of the world. There are seven base units, from which all other units are derived.

Base units[edit]

Every other unit is either a combination of two or more base units, or a reciprocal of a base unit. With the exception of the kilogram, all of the base units are defined as measurable natural phenomena. Also, notice that the kilogram is the only base unit with a prefix. This is because the gram is too small for most practical applications.

Quantity Name Symbol
Length metre m
Mass kilogram kg
Time second s
Electric Current ampere A
Thermodynamic Temperature kelvin K
Amount of Substance mole mol
Luminous Intensity candela cd

Derived units[edit]

Most of the derived units are the base units divided or multiplied together. Some of them have special names. You can see how each unit relates to any other unit, and knowing the base units for a particular derived unit is useful when checking if your working is correct.

Note that "m/s", "m s-1", "m·s-1" and \frac{\mbox{m}}{\mbox{s}} are all equivalent. The negative exponent form is generally preferred, for example "kg·m-1·s-2" is unambiguous. In contrast "kg/m/s2" is ambiguous -

is it \frac{\frac {kg}{m}}{s^2}  or is it  \frac{kg}{\frac{m}{s^2}}?

Quantity Name Symbol In terms of other derived units In terms of base units
plane angle radian rad m m-1 = 1
solid angle steradian sr m2 m-2 = 1
Area square metre m2 m2
Volume cubic metre m3 m3
Speed/Velocity metre per second m s-1 m s-1
Acceleration metre per second squared m s-2 m s-2
Density kilogram per cubic metre kg m-3 m-3 kg
Specific Volume cubic metre per kilogram m3 kg-1 m3 kg-1
Current Density ampere per square metre A m-2 m-2 A
Magnetic Field Strength ampere per metre m-1 A m-1 A
Concentration mole per cubic metre m-3 mol m-3 mol
Frequency hertz Hz s-1 s-1
Force newton N kg m s-2 m kg s-2
Pressure/Stress pascal Pa N m-2 m-1 kg s-2
Energy/Work/Quantity of Heat joule J N m m2 kg s-2
Power/Radiant Flux watt W J s m2 kg s-3
Electric Charge/Quantity of Electricity coulomb C A s s A
Electric Potential/Potential Difference/Electromotive Force volt V W A-1 m2 kg s-3 A-1
Capacitance Farad F C V-1 m-2 kg-1 s4 A2
Electric Resistance Ohm Ω V A-1 m2 kg s-3 A-2
Electric Conductance siemens S A V-1, Ω-1 m-2 kg-1 s3 A2
Magnetic Flux weber Wb V s m2 kg s-2 A-1
Magnetic Flux Density Tesla T Wb m-2 kg s-2 A-1
Inductance henry H Wb A-1 m2 kg s-2 A-2
Celsius Temperature degree Celsius °C K - 273.15
Luminous Flux lumen lm sr cd
Illuminance lux lx lm m-2 sr m-2 cd
Activity of a Radionuclide bequerel Bq s-1 s-1
Absorbed dose gray Gy J kg-1 m2 s-2
Dosae equivalent sievert Sv J kg-1 m2 s-2

Symbols usually start with a lower case letter unless the unit was named after somebody - for example "newtons" (note lower case letter when writing about the units) were named after Sir Isaac Newton, "watts" after James Watt, "farads" after Michael Faraday and so on.

Prefixes[edit]

The SI units can have prefixes to make larger or smaller numbers more manageable. For example, visible light has a wavelength of roughly 0.0000005 m, but it is more commonly written as 500 nm. If you must specify a quantity like this in metres, you should write it in standard form. As given by the table below, 1 nm = 1×10-9 m. In standard form, the first number must be between 1 and 10. So to put 500 nm in standard form, you would divide the 500 by 100 to get 5, then multiply the factor by 100 (so that it's still the same number), getting 5×10-7 m. The power of 10 in this answer, i.e.,. -7, is called the exponent, or the order of magnitude of the quantity.

Prefix Symbol Factor Common Term
peta P 10^{15} quadrillions
tera T 10^{12} trillions
giga G 10^9 billions
mega M 10^6 millions
kilo k 10^3 thousands
hecto h 10^{2} hundreds
deca da 10^{1} tens
deci d 10^{-1} tenths
centi c 10^{-2} hundredths
milli m 10^{-3} thousandths
micro µ 10^{-6} millionths
nano n 10^{-9} billionths
pico p 10^{-12} trillionths
femto f 10^{-15} quadrillionths

Homogenous equations[edit]

Equations must always have the same units on both sides, and if they don't, you have probably made a mistake. Once you have your answer, you can check that the units are correct by doing the equation again with only the units.

Example 1[edit]

For example, to find the velocity of a cyclist who moved 100 metres in 20 seconds, you have to use the formula

velocity = \frac {displacement}{time}

so your answer would be 100÷20 = 5 m·s-1.

This question has the units m ÷ s and should give an answer in m·s-1. Here, the equation was correct, and makes sense. In this case a middle-dot (·) has been inserted between the "m" and the "s" to show that this is metres per second, not milliseconds.

Often, however, it isn't that simple. If a car of mass 500 kg had an acceleration of 0.2 m·s-2, you could use Newton's second law

F=ma

to show that the force provided by the engines is 100 N. At first glance it would seem the equation is not homogeneous, since the equation uses the units (kg) × (m·s-2), which should give an answer in kg·m·s-2. If you look at the derived units table above, you can see that a newton is in fact equal to kg·m·s-2, and therefore the equation is correct.

Example 2[edit]

Using the same example as above, imagine that we are only given the mass of the car and the force exerted by the engines, and have been asked to find the acceleration of the car. Using

F=ma

again, we need to rearrange it for a. We do this by setting

a=\frac {m}{F}.

By inserting the numbers, we get the answer a = 5 m·s-2. You already know that this is wrong from the example above, but by looking at the units, we can see why this is the case

ms^{-2}\ne\frac {kg}{kg\ ms^{-2}}.

The units are m·s2, when we were looking for m·s-2. The problem is the fact that "F=ma" was rearranged incorrectly. The correct formula was

a=\frac {F}{m},

and using it will give the correct answer of 0.2 m·s-2. The units for the correct formula are

ms^{-2}=\frac {kg\ ms^{-2}}{kg}=ms^{-2}.