Tutorial 1 - Symbols and Units

 Contents

Using Symbols

An equation is a mathematical model that sums up how a system behaves.  For example, we know that, if we have a current flowing through a wire and double the voltage, the current will double as well.  We know that the quantities of current and voltage are related by the simple rule:

V = IR

In physics problems we are given certain quantities and we have to use them to find an unknown quantity with an equation.  In all problems in AS level, you will only ever have ONE unknown.  You will never be expected to tackle a problem with two or more unknowns.  That said, you may need to look up some quantities from the data sheet.

In GCSE you were often given equations in words:

Distance (m) = speed (m s-1) × time (s)

You will notice from the data sheet at the end of these notes that the equations are given in symbols, which in my notes I refer to as Physics Code.  The symbols all mean something; they are abbreviations.  The symbols used in exams and most textbooks are those agreed by the Association of Science Education.

Some symbols are easy; V stands for voltage.  Some are not so easy.  I for current comes from the French intensité du courant, since it was a French physicist who first worked on it.  In print you will always find the codes written in Times New Roman Italics.  In my notes, I do try to, but sometimes I miss it.  As you can’t do italics in normal handwriting, then don’t worry.  Here are some examples:

 a Acceleration A Area F Force m Mass I Current p Pressure Q Charge

 This question is about Physics codes.  What do they refer to? F = ma E = 1/2 mv2 c = f.

You will come across codes written in Greek letters.  The normal (Latin) alphabet has 26 characters.  No codes are used which are like ä (a – umlaut) or ê (e – circumflex). The Greek alphabet adds another 24.  The Greek Alphabet is this:

 Greek Name Letter Greek Name Letter a alpha a n nu n b beta b x xi x g gamma g o omicron Short o (ŏ) d (D) delta d (D) p pi p e epsilon Short e (ĕ) r rho r z zeta z s (S) sigma s (S) h eta Long e (ē) t tau t q theta th u upsilon u i iota i f (F) phi ph [or f (F)] k kappa k c chi ch l (L) lambda l (L) y (Y) psi ps m mu m w (W) omega Long o [ō (Ō)]

Some quantities share the same physics codes, e.g. Q for charge, and Q for energy.  You will need to be aware of this when you do the exam, and knowing what each code stands for is part of your examination preparation.

Units

Physics formulae use SI (Système International) units based on seven base units:

• Distance – metre (m);

• Mass – kilogram (kg);

• Time – second (s);

• Temperature – Kelvin (K);

• Current – ampere (A);

• Amount of substance – mole (mol);

• Intensity of light – candela (cd) [which you will not come across at A-level.]

 This little character is about to walk into a bear trap (something students often get wrong through carelessness).  I have fallen into bear-traps often...   Note that you write 7.5 metres as 7.5 m, NOT 7.5 ms.  If you see 7.5 ms, that would mean 7.5 milliseconds, which gives a completely different meaning.    And please do not abuse the apostrophe by writing m’s.  I have seen this often.  It will cost you marks as a unit error.

Notice that the physics codes are typed in italics, while the symbols for the units are typed in plain text.  In most books, physics codes are typed in Times New Roman font.  (The font in which this document is written is called Tahoma.).

Notice that the unit for mass has the kilo- prefix.  This says that the base unit for mass is 1000 grams.  Originally the gram was the base unit, but was displaced by the kilogram in 1960, after twelve years work and discussion by many leading academics.  If you want to find out the fundamental definitions, the National Physics Laboratory has a link, http://www.npl.co.uk/reference/measurement-units/ .

Notice also that the SI unit for temperature is kelvin, not Celsius.  Note that you do not write “degrees kelvin”.

Traditional Imperial units (for example, pound, foot, yards) were once used in mechanics many years ago.  They have not been used for at least fifty years.  There are no imperial measurements for areas such as electricity.   Imperial units are never used in any modern science, technology, or engineering.  They certainly have no place whatever in a modern technological society.   Imperial units may have a place in an historical novel or poem, or use by MPs of the Conservative Party and certain right-wing newspapers that are one hundred years behind the times.  (The Daily Telegraph house style demands that journalists carefully convert every metric unit into an imperial unit, thinking that their readership are incapable of using metric units.  It's rather patronising.)  The sooner they are permanently consigned to the museum of historical culture, the better.

Combining Units

Clearly these base units on their own are rather limited, but we can combine them.  We know that speed is measure in metres per second, which we write as:

m s-1

Make sure that you:

• Separate the symbols with a space;

• Do not use capitals unless the unit symbol is a capital.

In many texts, you will see it written as with the terms separated by a solidus (/), i.e. m/s, which is quite acceptable, although not as correct as m s-1.  Note also the need for a space between the 'm' and the 's'.  If we write ms-1 it actually means "milliseconds to the minus one".

There is a more compelling reason to use the “power” notation over the separation with a solidus.  Consider the units for specific heat capacity:

Joules per kilogram per kelvin

J/kg/K looks clumsy and could be confusing.  The correct way to write it is:

J kg-1 K-1

Often a unit is derived from a definition, where the definition is quantity per unit something.  For example voltage is defined as:

Energy per unit charge.

The units for this are:

Joules per coulomb

J C-1

Note that if a unit is divided by itself, the units cancel out.

Conversions

Many physics formulae will give you the right answer ONLY if you put the quantities in SI units.  This means that you have to convert.  You will often find units that are prefixed, for example kilometre.  The table below shows you the commonest prefixes and what they mean:

 Prefix Symbol Meaning Example pico p ´ 10-12 1 pF nano n ´ 10-9 1 nF micro m ´ 10-6 1 mg milli m ´ 10-3 1 mm centi c ´ 10-2 1 cm kilo k ´ 103 1 km Mega M ´ 106 1 MW Giga G ´ 109 1 GWh

 Failure to do conversions is a very common bear trap.

Converting areas and volumes causes a lot of problems.

• 1 m2 100 cm2.

• 1 m2 = 100 cm × 100 cm = 10 000 cm2 = 104 cm2

When you write out your answer, you must always put the correct unit at the end.  The number 2500 on its own is meaningless; 2500 J gives it a meaning.

In the AQA exam, at least one question will ask you for the correct unit. This will be made clear in the question:

 Convert the following quantities to SI units: 15 cm 500 g 3 km 35 mV 220 nF

Many units are made up from base units.  They are sometimes called derived units.  We all know that force is measured in Newtons.  We also know the familiar equation for force:

Force (N) = mass (kg) × acceleration (m s-2)

We multiply the units for mass (kg) and acceleration to give kg m s-2.  Therefore 1 N = 1 kg m s-2.

Note:

While it is perfectly OK to write the units for speed as m/s (metres per second), we should get into the habit of writing m s-1

 Convert the following quantities to SI units or non-SI units as appropriate:  1.0 m2 to mm2  0.45 mm2 to m2  1 cm3 to m3  22.4 dm3 to m3

Conversions from other Units

SI units and derived units may not be appropriate to use in some contexts.  For example, joules may be far too big for particle physicists.  Or they can be far too small if you are buying electricity.

When you buy electricity, you use kilowatt hours.  The kilowatt hour is the amount of energy used by an appliance of 1 kilowatt used for one hour.

You will soon find out that particle physicists use electron-volts (eV) in preference to joules when they talk about energy:

1 eV = 1.6 × 10-19 J

So when you see for example, 3.6 MeV, you will know that it will need to be converted into joules before it can be used in a formula.

And it’s important to convert from joules to eV, by dividing by 1.6 × 10-19.

Unit Analysis

Unit analysis are now part of the syllabus, and it is a useful technique for understanding the relationship between units.  This is done by expressing quantities in base units.  We have seen these above, but here they are again:

• Distance – metre (m);

• Mass – kilogram (kg);

• Time – second (s);

• Temperature – Kelvin (K);

• Current – ampere (A);

• Amount of substance – mole (mol);

• Intensity of light – candela (cd) [which you will not come across at A-level.]

In GCSE, you will have come across the idea that:

Work done (J) = Force (N) × distance moved in the direction of the force (m)

So we can say that:

1 Joule (J) = 1 Newton (N) × 1 metre (m)

1 J = 1 N m

You will have also come across Newton's Second Law which is summed up by:

Force (N) = mass (kg) × acceleration (m s-2)

So we can say that:

1 Newton (N) = 1 kilogram (kg) × 1 metre per second squared (m s-2)

1 N = 1 kg m s-2

So we can combine the two to say:

1 J = 1 kg m s-2 × 1 m

which gives:

1 J = 1 kg m2 s-2

In some old textbooks you may well come across energy as being described as M L2 T-2 (mass × length2 × time-2).  I  will always use the SI units.

 (a) What is the electrical quantity Coulomb (C) in base units? (b) 1 Volt (V) = 1 Joule per Coulomb (J C-1).  What is this in base units?

Your answer to Question 4 will show you that quantities expressed as base units are rather clumsy.  What is a cubic second (s3)?  Use the accepted units.

Checking Homogeneity of Equations

A use of base units is to check for whether an equation is correct.  If we break everything down to base units, the units on the left hand side are the units on the right hand side.  Homogenous means made up of identical parts.

The table shows a number of important derived units:

 Quantity Definition Base unit Derived Unit Density Mass (kg) ÷ volume kg m-3 Momentum Mass (kg) × velocity (m s‑1) kg m s-1 Force Mass (kg) × acceleration (m s-2) kg m s-2 Newton (N) Pressure Force (N) ÷ area (m2) kg m-1 s-2 Pascal (Pa) Work Force (N) × distance moved (m) kg m2 s-2 joule (J) Power Energy (J) ÷ time (s) kg m2 s-3 watt (W) Charge Current (A) × time (s) A s coulomb (C) Voltage Energy (J) ÷ charge (C) kg m2 A-1 s-3 Volt (V) Resistance Voltage (V) ÷ current (A) kg m2 A-2 s-3 Ohm (W)

Consider the equation:

Energy is kg m2 s-2 and is on the left hand side.  ½ is a number and has no units.  Mass, m, is in kg, and speed, v, is in m s-1.  Squaring speed gives us:

m2 s-2

So using the equation:

kg m2 s-2 = no units × kg × m2 s-2

It doesn’t take a genius to see that:

kg m2 s-2 = kg m2 s-2

 In an examination, a student is not sure whether the equation linking power and electrical resistance is:     or    Use base units to check the homogeneity of each of these equations and decide which is the correct one.