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Basic MathematicsChanging UnitsR Horan & M LavelleThe aim of this package is to provide a short self assessment programme for students who wish to learn how to convert between different

units.Copyrightc?2003rhoran@plymouth.ac.uk,mlavelle@plymouth.ac.ukLast Revision Date: February 26, 2004 Version 1.0

Table of Contents1.Introduction2.Powers of Units3.Equations and Units4.Final QuizSolutions to ExercisesSolutions to Quizzes

Section 1: Introduction 31. IntroductionThe use of different units in science and everyday life makes it impor-

tant to be able toconvert between different units.Example 1One kilometre is roughly58of a mile. What is a mile in

kilometres? 1km =

58mile (multiply by8)

8km = 5miles (divide by5)

85km = 1mile

Therefore one mile is roughly

85of a kilometre.Exercise 1.Express the following quantities in the units requested

(click on thegreenletters for the solutions).(a)One year inseconds.(b)0.016 miles inkilometres.(c)A speed of 10miles per hour

inkilometres per hour.(d)One million pounds inpennies.

Section 2: Powers of Units 42. Powers of UnitsIt is important to take care with powers of units.Example 2Consider theareaof the square drawn below. From the

package onDimensional Analysiswe know that this has dimensions of length squared (L

2). Its numerical value depends on the units used.1mm?

1cmTwo possible ways of expressing the area areeither:(1cm)×(1cm) = 1cm2or(10mm)×(10mm) = 100mm2.

Section 2: Powers of Units 5In general, different units are linked by aconversion factorand an area given in one unit may be expressed in another unit by multiplying the area by the square of the conversion factor. Since 1cm = 10mm, theconversion factor is 10.So an area of one square centimetre is 10

2= 100 square millimetres.

Now try these two short quizzes:QuizWhich of the following is theareaof the square whose sides are

1cm long in theSI units of square metres?(a)0.01m2(b)100m2(c)0.0001m2(d)10,000m2QuizAnareaof a square mile is a mile times a mile. Recalling that

8km≈5miles, what is this in square kilometres?(a)2564km2(b)6425km2(c)1610km2(d)1016km2

Notethat we multiply the area by the square of the conversion factor.

See also the following example.

Section 2: Powers of Units 6Example 3Consider theareaof the rectangle drawn below.1mm?

2cmIts area is given by the product of the sides. This iseither:area = (2cm)×(1cm) = 2cm2orarea = (20mm)×(10mm) = 2×102mm2= 200mm2Note the square of the conversion factor in the last line.

Section 2: Powers of Units 7In general an area ofAcm2can be expressed in square millimetres as

Acm2=A×(10mm)2

=A×102mm2 = 100Amm2 Similarly avolume(whosedimensions are L3) can be converted from one set of units to another by multiplying by thecube of the appro-

priate conversion factor.QuizWhat is avolumeof 3cm3expressed in cubic millimetres?(a)300mm3(b)27,000mm3(c)3×104mm3(d)3,000mm3QuizAlitre (l)is a cubic decimetre, where 1dm = 0.1m. Express

0.5l in cubic metres.(a)5×10-4m3(b)1.25×10-2m3(c)0.005m3(d)5,000m3

Similarly, withnegative powersof a unit wedivideby the appropriate power of the conversion factor. An example of this follows. Section 2: Powers of Units 8Example 4To convert a density (dimension ML-3) of 1kgm-3into kgdm -3we use 1kgm -3= 1kgm3 = 1 kg(10dm)3 = 1 kg1000dm3 = 0.001kgdm-3

This may now be practised in the next quiz:QuizExpress adensityof 3×102kgm-3in terms of kilograms per

cubic centimetre.(a)3×10-4kgcm-3(b)2.7×1011kgcm-3(c)3×108kgcm-3(d)3×105kgcm-3 energy in half a microsecond. (Power is the rate of conversion of

energy with time. See the package onUnits.)(d)A kilowatt hour is the energy used by a device with a power output

of one kilowatt in one hour. Express this inJoules.

Section 3: Equations and Units 103. Equations and UnitsGenerally it is best in equations to express all quantities in SI units

before performing calculations.Example 4The electron Volt (eV) is a widespread unit of energy in atomic and sub-atomic physics. It is related to the SI unit of energy, the Joule, by:1eV = 1.6×10-19J.The energy of a photon is related

to its frequencyνbyE=hν, whereh= 6.6×10-34Js. What is thefrequency of a photon with energy 2.2eV?The energy of the photon in SI units isE= 2.2×1.6×10-19=

3.52×10-19J. It follows that the photon"s frequency is

ν=Eh

3.52×10-19J6.6×10-34Js

3.526.6×10-19+34s-1

?ν=5.3×1014s-1 Section 3: Equations and Units 11Exercise 3.Put the quantities below into SI units to perform the

requested calculation (click on thegreenletters for the solutions).(a)Ohm"s lawV=IRlinks the voltageVacross a resistanceRto

quotesdbs_dbs7.pdfusesText_5

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