Contents. TOPIC 1 Working as a Physicist 8 1 Units 8 2 Estimation 10

Transcription

1 7 Behaviour can Conens be learned1 Conens How o use his book 6 TOPIC 1 Working as a Physicis 1 Unis Esimaion 1 TOPIC Mechanics.1 Moion 1 1 Velociy and acceleraion 1 Moion graphs 14 3 Adding forces 17 4 Momens 19 5 Newon s law of moion 6 Kinemaics equaions 5 7 Resolving vecors Projeciles 3 Thinking Bigger 3 Exam-syle quesions 34. Energy 36 1 Graviaional poenial and kineic energies 3 Work and power 41 Thinking Bigger 44 TOPIC 3 Elecric circuis 3.1 Elecrical quaniies 6 1 Elecric curren 6 Elecrical energy ransfer 65 3 Curren and volage relaionships 6 4 Resisiviy 71 5 Conducion and resisance 73 6 Semiconducors 76 Thinking Bigger Exam-syle quesions 3. Complee elecrical circuis 4 1 Series and parallel circuis 6 Elecrical circui rules 9 3 Poenial dividers 9 4 Emf and inernal resisance 96 5 Power in elecric circuis 9 Thinking Bigger 1 Exam-syle quesions 14 TOPIC 4 Maerials 4.1 Fluids 16 1 Fluids, densiy and uphrus 1 Fluid movemen Drag ac Terminal velociy 115 Thinking Bigger 11 Exam-syle quesions 1 4. Solid maerial properies 1 1 Hooke s law 14 Sress, srain and he Young modulus 17 3 Sress-srain graphs 19 Thinking Bigger 13 Exam-syle quesions 134 TOPIC 5 Waves and he paricle naure of ligh 5.1 Basic waves Wave basics 13 Wave ypes 141 Thinking Bigger 144 Exam-syle quesions The behaviour of waves 14 1 Wave phase and superposiion 15 Saionary waves Diffracion Wave inerference 159 Thinking Bigger 16 Exam-syle quesions Opics Refracion 16 Toal inernal reflecion 17 3 Lenses 17 4 Image formaion Polarisaion 177 Thinking Bigger 1 Exam-syle quesions 1 Exam-syle quesions Quanum physics 14.3 Momenum 4 1 Momenum 5 Conservaion of linear momenum 5 Thinking Bigger 56 Exam-syle quesions 5 1 Wave-paricle dualiy 16 The phooelecric effec 19 3 Elecron diffracion and inerference 19 4 Aomic elecron energies 19 Thinking Bigger 19 Exam-syle quesions Mahs skills Exam preparaion 6 Glossary 1 Index 16 3

2 7 How Behaviour o use his can book be learned1 Thinking Bigger A he end of each chaper here is an opporuniy o read and work wih real-life research and wriing abou science. These spreads will help you o: Chaper openers TOPIC 5 CHAPTER Waves and paricle naure of ligh 5.3 Opics Inroducion Humans have long wondered a he appearance of rainbows and why fish appear o be larger underwaer. The developmen of undersanding of he physics behind such naural phenomena has led us o be able o develop more and more complex imaging echniques. A mobile elephone, lying on a desk, will soon be projecing a cinema screen ono your bedroom wall. In some insances, undersanding he physics has led o immensely useful low-ech soluions. Specacle lenses ha can have heir srengh alered by filling a plasic pouch wih differen amouns of waer mean ha glass lens grinding faciliies are no needed, which is very useful for remoe or undeveloped socieies. Wha have I sudied before? Wha will I sudy laer? Definiions of he properies of waves, such as frequency, wavelengh and speed The focusing of a paricle beam in a paricle acceleraor (A level) Models of waves and wave processes The use of elescopes in asronomy (A level) Calculaions of wave speed The ransmission of waves Some of he opical ideas behind an elecron microscope (A level) Some consideraions abou elescopes used in asronomy Ineracions of ligh wih is medium a an aomic level (A level) This chaper will explain he deails of refracion in differen maerials and how his leads o he properies of differen ypes of lenses. Lenses and combinaions of lenses are used in many opical insrumens, such as elescopes and microscopes, and you will also see how hese can produce a clear image of an illuminaed objec. Wha will I sudy in his chaper? The definiion of refracion, and how o find and calculae he refracive index All he mahs you need Unis of measuremen (e.g. he diopre) Effecs on he properies and movemen of a wave due o changes in is medium Subsiuing numerical values ino algebraic equaions (e.g. finding he power of a lens) The phenomenon of oal inernal reflecion Use of raios and similar riangles (e.g. calculaions wih he magnificaion of an image) Changing he subjec of an equaion (e.g. using he lens formula) The effecs of lenses, and calculaions using he lens formula Ploing wo variables from experimenal daa (e.g. ploing a graph of experimenal resuls of refracion in glass) How lenses can form images, and how o calculae he magnificaion of he images Deermining he slope of a linear graph (e.g. using a graph of experimenal resuls o find he refracive index of glass) The phenomenon of polarisaion, and some of is implicaions and applicaions Each chaper sars by seing he conex for ha chaper s learning: Links o oher areas of Physics are shown, including previous knowledge ha is buil on in he chaper, and fuure learning ha you will cover laer in your course. The All he mahs you need checklis helps you o know wha mahs skills will be required DURHAM CASTLE SIEGE Durham Casle was iniially buil as a forress agains Scoish raiders. In his aciviy, you will need o imagine aacking he casle using a caapul ha fires a boulder as a projecile. In addiion o he layou shown in fig B, we need informaion abou he iniial velociy of he boulder. The caapul sling could ac for.3 s o accelerae he boulder (mass = 1 kg) wih a force of 155 N. I causes he boulder o leave he caapul a an angle of 45 o he horizonal. fig A Durham Casle is now an UNESCO World Heriage Sie and is a par of Durham Universiy. In his secion, I will use some basic mechanics o answer a quesion regarding he prospec ha Scoish raiders really could have assauled Durham Casle in he manner described previously. The seveneenh-cenury source maerial suggess ha he casle was under siege by he Scos for more han a week and baered by boulders. However, he curren walls bear lile in he way of obvious bale scars, despie being apparenly par of he original medieval consrucion. Looking a fig B, he quesion ha needs o be answered here is: How high up he fron wall of he casle will he boulder hi? This heigh is marked on fig B as H. Voal H 4 m Durham Casle s 5 15 m fig B Deails of he rajecory of a caapul boulder owards Durham Casle. We assume he boulder leaves he caapul a ground level YOU ARE HERE polarisers undersand wha is mean by plane polarisaion no vibraions describe how polarisaion can be used wih models o invesigae sresses in srucures 7 Behaviour canopics be learned 5.3 Polarisaion by chemical soluions The analysis of sress concenraions invesigaed above works because differen pars of he plasic model have differen effecs on polarised ligh. This is also he case wih some chemicals, such as sugar soluion. The amoun of he concenraion of he sugar soluion varies he angle o which i roaes he polarisaion of he ligh. We can use Polaroid filers o analyse he srengh of he sugar soluion, by measuring he angle a which he ligh polarisaion emerges afer passing hrough he soluion. Exam-syle quesions verical vibraions Plane polarisaion Transverse waves have oscillaions a righ angles o he direcion of moion. In many cases, he plane of hese oscillaions migh be in one fixed orienaion. Fig A shows he elecric (red) and magneic (blue) fields in an elecromagneic wave. In his example of, say, a ligh wave, he elecric fields only oscillae in he verical plane. The wave is said o be plane polarised or, more precisely, verically plane polarised. For elecromagneic waves, he plane of he elecric field s oscillaions is he one ha defines is plane of polarisaion. vibraions in all direcions fig B Polaroid filers ransmi ligh waves if heir plane of polarisaion maches wih he orienaion of he filer. Fig B illusraes he effecs of Polaroid filers on ligh from a bulb. I sars unpolarised, wih vibraions in all direcions. The firs filer only permis verical vibraions, so hey are seleced, and he ligh is ransmied verically plane polarised. The second filer is oriened in he same verical direcion, and he ligh will pass hrough his wihou change. The hird Polaroid filer is oriened horizonally. This blocks verically polarised ligh, and so no waves are ransmied beyond i. The second and hird filers are referred o as crossed Polaroids, as heir orienaions are a righ angles and, ogeher, hey will always block all ligh. fig D Engineering sress analysis using crossed Polaroids. Polarisaion by reflecion and refracion When unpolarised ligh reflecs from a surface, such as a road, he waves will become polarised. The degree of polarisaion depends on he angle of incidence, bu i is always ending owards horizonal plane polarisaion, as shown in fig E. unpolarised inciden ligh polarised refleced ligh 3 Invesigaing srucural sresses We can use crossed Polaroids o observe sress concenraions in clear plasic samples. 4 fig A A polarised elecromagneic wave. fig E Reflecion can polarise waves. Ofen, many waves ravel ogeher, wih oscillaions in a variey of planes. In his case, ligh from his source is said o be unpolarised. This is how ligh emerges from a ligh bulb, a candle and he Sun. Ligh waves inciden on a surface ino which hey can refrac (see secion 5.3.1), such as a pond, will reflec parially horizonally polarised ligh as in fig E, bu will also ransmi parially verically polarised ligh ino he new medium, as in fig F. unpolarised ligh parially polarised ligh Polarisaion is only possible wih ransverse waves. If a wave is polarised, i mus be a ransverse wave fig C Sress analysis using crossed Polaroids. The firs Polaroid produces polarised ligh, which passes ino he plasic sample. Sressed areas have heir molecules in slighly differen orienaions, and his will affec he passage of he ligh hrough he plasic. This effec varies wih he colour of he ligh. When he second Polaroid acs on he emerging ligh, some of he ligh will have ravelled slighly more slowly hrough he plasic and will desrucively inerfere wih oher ligh waves of he same colour. Thus, depending on he degree of sress in he plasic, he colours ha emerge will vary. Differenly sressed areas appear as differen 1 Invesigaion Polarising filers Unpolarised radiaion can be passed hrough a filer ha will ransmi only hose waves ha are polarised in a paricular plane. Waves on a sring could be polarised simply by passing he sring hrough a card wih a sli in i, which will hen only allow oscillaions o pass hrough if hey are in line wih he sli. For ligh waves, he polariser is a piece of plasic impregnaed wih chemicals wih long chain molecules, called a Polaroid shee. The Polaroid filer will only allow ligh waves o pass if heir elecric field oscillaions are orienaed in one direcion. fig G Sugar soluions can roae he plane of polarisaion. The degree of polarisaion on a paricular wavelengh (colour) depends on he concenraion of he soluion, and how far he ligh has had o pass hrough i. This gives rise o he changing colours seen along he lengh of his ube of sugar soluion. Silbab said ha he had a sound wave polariser on his new music sysem. Explain why his claim is incorrec. Wha does i mean o say ha a wave is unpolarised? Why do ski goggles ofen have Polaroid filers wih a verical orienaion? Explain he benefis of being able o use polarisaion o analyse sress concenraions in engineering model. Key definiion Polarisaion refers o he orienaion of he plane of oscillaion of a ransverse wave. If he wave is (plane) polarised, all is oscillaions occur in one single plane. perpendicular o plane of paper air glass parially polarised ligh fig F Refracion can polarise waves me ha You can assu ds Claus undersan is and mahemaics, igen a generally inell could have suden who ics bu phys el done A-lev subjecs. preferred ars Exam-syle Behaviourquesions can be learned.1 Exam-syle quesions 1 The uni of force is he Newon. One Newon is equivalen o: (a).1 kg (b) 1 kg m s 1 (c) 1 kg m s (d) 1 m s [1] [Toal: 1] 6 A suden is required o measure he speed of a rolley rolling down a slope. This could be done using a ruler and a sopwach bu he suden prefers a mehod using ICT. (a) The suden uses a ligh gae as shown in he diagram. Describe how he suden obains he measuremens needed o calculae speed. 9 The phoograph shows a sequence of images of a bouncing ennis ball. [] ligh gae A ball is hrown verically upwards a a speed of 11. m s 1. Wha is he maximum heigh i reaches? (a).561 m (b) 1.1 m (c) 6.17 m (d) 1.3 m [1] [Toal: 1] Main conen The main par of each chaper covers all he poins from he specificaion ha you need o learn. The ex is suppored by diagrams and phoos ha will help you undersand he conceps. Wihin each secion, you will find he following feaures: Learning objecives a he beginning of each secion, highlighing wha you need o know and undersand. Key definiions shown in bold and collaed a he end of each secion for easy reference. Worked examples showing you how o work hrough quesions, and how your calculaions should be se ou. Invesigaions provide a summary of pracical experimens ha explore key conceps. s o help you focus your learning and avoid common errors. Did you know? boxes feauring ineresing facs o help you remember he key concep. Working as a Physicis icons highligh key secions ha develop your skills as a scienis and relae o he Working as a Physicis secion of he specificaion. o help you check wheher you have undersood wha you have jus read, and wheher here is anyhing ha you need o look a again. Imagine Claus Thermad is a friend of yours, and he has come o you for help wih he calculaions as he is no a srong scienis. His secion Seps o he answer was aken from a research source abou a differen casle under siege. Wrie an o Claus o explain he calculaions required in each sep. 5. (a).4 N m (b) 4. N m (c) 4. N m (d) 4 N m [1] [Toal: 1] 4 (a) Wha is mean by a vecor quaniy? The graph shows how displacemen varies wih ime for an objec ha sars from res wih consan acceleraion. [1] (b) A car is driven around a bend a a consan speed. Explain wha happens o is velociy. [] [Toal: 3] 5 The Saurn V rocke used in NASA s space programme had a mass of kg. I ook off verically wih a hrus force of N. (a) Show ha he resulan force on he rocke is around 4 16 N. [3] (b) Calculae he iniial acceleraion. (c) Afer 15 s he rocke reached a speed of 39 m s 1. Calculae is average acceleraion. (d) Sugges why he iniial acceleraion and average acceleraion are differen. (b) The suden hinks ha he sopwach mehod is less reliable han he ICT mehod. Discuss wha makes using a sopwach less reliable. [] [Toal: 4] 7 You are asked o deermine he acceleraion of free fall a he surface of he Earh, g, using a free fall mehod in he laboraory. (a) Describe he apparaus you would use, he measuremens you would ake and explain how you would use hem o deermine g. [6] (b) Give one precauion you would ake o ensure he accuracy of your measuremens. [1] [Toal: 7] 3 colours hrough he second Polaroid. Changing he sresses on he plasic will aler he inernal sresses, and he inerference paern will change. Engineers use his o see sress concenraions in models of srucures, and o observe how he sress concenraions change when he amoun of sress changes. This allows hem o aler he design o srenghen a srucure in regions of highes sress. Aciviy Command word calculae When he word answer is used, your numerical needs o be working. and show your The acceleraion caused by he sling: F a = m a=.3 N Polarisaion 3. Sae wo assumpions ha have been made in hese calculaions. 4. Calculae wha difference here would be in he answer if he caapul was loaded wih differen boulders of masses 1 kg and 14 kg. Noe from fig B ha he casle walls are 4 m high. Commen on hese answers. 5. If he available supply of boulders offered very variable masses, how migh he Scos be able o overcome he problems shown in quesion 4. 3 February 15, by Claus Thermad, early draf for disseraion for Masers degree in Medieval Hisory, Durham Universiy.. 4 cm 3 3 vhorizonal can be found by resolving he velociy o give he horizonal componen: vhorizonal = voal cos 45 4 The overall velociy will come from he sling s acceleraion of he boulder: v = u + a where u = ms 1, and he quesion ells us ha he sling acs for.3 seconds.. Complee he calculaion seps, in reverse as suggesed, in order o find ou he answer, H: The acceleraion caused by he sling (A) overall velociy ha he boulder is projeced from he sling (B) horizonal and verical componens of he velociy (C) ime of fligh found from he horizonal ravel (D) ime o reach maximum heigh using verical moion (E) remaining fligh ime from maximum heigh (F) heigh fallen from he maximum in he remaining fligh ime (G) final answer H (H). 3 3 Calculae he momen exered on he nu by he spanner shown in he diagram. 5.3 To find h, we need o know he ime of fligh, oal so we can spli his ino a ime o reach hmax, and see how much ime is lef o fall heigh h. We will use graviaional acceleraion verically o calculae he verical drop in ha remaining ime: s oal = v horizonal From fig B, we can see ha s = 15 m. Once you have answered he ions ques calculaion e wheher below, decid Scoish you hink he as he siege happened his ess sugg r auho he sory. source ells Now we will look a he phrases in deail. You may need o combine conceps from differen areas of physics o work ou he answers. Use he imeline a he boom of he page o help you pu his work in conex wih he areas of your course. 5 Newon s second law of moion will give us he acceleraion he sling causes: F a = m Calculae he answer by reversing hese seps. h hmax Seps o he answer By working back from he answer we are looking for, we can see wha calculaions will need o be made in order o solve his problem. The fundamenal idea is ha he parabola rajecory would be symmerical if he fligh was no inerruped by crashing ino he casle wall. 1 To find he heigh up he wall from he ground, we will need o work ou how far down from he boulder s maximum heigh i falls: H = hmax h 1. The exrac opposie is a draf for a universiy essay on Medieval Hisory. Consider he exrac and commen on he ype of wriing being used. For example, hink abou wheher his is a scienis reporing he resuls of heir experimens, a scienific review of daa, a newspaper or a magazine-syle aricle for a specific audience. Try and answer he following quesions: a. How can you ell ha he auhor is scepical abou he hisorical source maerial? b. Wha is he purpose of his mahemaical analysis, for is inclusion in his essay? Where else will I encouner hese hemes? A he end of each chaper here are also exam-syle quesions o help you o: es how fully you have undersood he learning pracise for your exams. Idenifying uncerainies in measuremens and using simple echniques o deermine uncerainy (e.g. considering he uncerainy of experimenal resuls in finding he refracive index of glass) 166 read real-life maerial ha s relevan o your course analyse how scieniss wrie hink criically and consider he issues develop your own wriing undersand how differen aspecs of your learning piece ogeher. 7 Behaviour Thinking canbigger be learned 1.1 Le us sar by considering he naure of he wriing in he draf disseraion. [] [] [1] [Toal: ] A suden plos he following graph and claims ha i shows he verical moion of he ball in he phoograph X Y Z Time/s (a) Wihou carrying ou any calculaions, describe how he following can be found from he graph (i) he verical disance ravelled by he ball beween.5 s and 1. s (ii) he acceleraion a Y. [] (b) The graph conains several errors in is represenaion of he moion of he ball. Explain wo of hese errors. [4] [Toal: 6] Time/s 5 6 (a) Use he disance ime graph o deermine he speed of he objec a a ime of 4. s. [3] (b) Calculae he acceleraion. [] [Toal: 5] During a lesson on Newon s laws of moion, a suden says, We don really need o boher wih Newon s firs law because i is included in his second law. Sae Newon s firs wo laws of moion and explain how Newon s second law includes he firs law. [5] [Toal: 5] 1 The diagram shows an arrangemen used o launch a ligh foam rocke a a school science compeiion. Velociy/ms1 Welcome o your Edexcel AS/A level Physics course. In his book you will find a number of feaures designed o suppor your learning. KING THIN Disance/m How o use his book 1 There has been a proposal o build a rain unnel underneah he Alanic Ocean from England o America. The suggesion is ha in he fuure he rip of 5 km could ake as lile as one hour. Assume ha half he ime is spen acceleraing uniformly and he oher half is spen deceleraing uniformly wih he same magniude as he acceleraion. (a) Show ha he acceleraion would be abou m s. [] The rocke is launched a he level of one end of a long able and lands a he oher end a he same level. The sudens measure he horizonal disance ravelled by he rocke and he ime of fligh. (a) The rocke ravels 1. m in a ime of. s. (i) Show ha he horizonal componen of he iniial velociy of he rocke is abou m s 1. [] (ii) Show ha he verical componen of he iniial velociy of he rocke is abou 4 m s 1. [] (iii) Calculae he iniial velociy of he rocke. [4] (b) The sudens obained heir daa by filming he fligh. When hey checked he maximum heigh reached by he rocke hey found i was less han he heigh prediced using his velociy. (i) Sugges why he maximum heigh reached was less han prediced. [1] (ii) Give wo advanages of filming he fligh o obain he daa. [] [Toal: 11] (b) Calculae he maximum speed. [] (c) Calculae he resulan force required o decelerae he rain. mass of rain = kg [] [Toal: 6] 35 Geing he mos from your AciveBook Your AciveBook is he perfec way o personalise your learning as you progress hrough your Edexcel AS/A level Physics course. You can: access your conen online, anyime, anywhere use he inbuil highlighing and annoaion ools o personalise he conen and make i really relevan o you search he conen quickly. Highligh ool Use his o pick ou key erms or opics so you are ready and prepared for revision. Annoaions ool Use his o add your own noes, for example links o your wider reading, such as websies or oher files. Or make a noe o remind yourself abou work ha you need o do. 7

3 TOPIC Mechanics CHAPTER.1 Moion Inroducion How could we calculae how fas a plane is flying, in wha direcion i is going and how long i will ake o reach a cerain desinaion? If you were a pilo, how would you know wha force o make he engines produce and where o direc ha force so your plane moves o your desinaion? There is an amazing number of calculaions ha need o be done o enable a successful fligh, bu he basis on which all of i is worked ou is simple mechanics. This chaper explains he muliple movemens of objecs. I looks a how movemen can be described and recorded, and hen moves on o explaining why movemen happens. I covers velociy and acceleraion, including how o calculae hese in differen siuaions. We only consider objecs moving a speeds ha could be encounered in everyday life. A hese speeds (much less han he speed of ligh) Sir Isaac Newon succincly described hree laws of moion. Wih knowledge of basic geomery, we can idenify aspecs of movemen in each dimension. Newon s laws of moion have been consanly under es by scieniss ever since he published hem in 167. Wihin he consrains esablished by Einsein in he early h cenury, Newon s laws have always correcly described he relaionships beween daa colleced. You may have a chance o confirm Newon s laws in experimens of your own. Wih modern ICT recording of daa, he reliabiliy of such experimens is now much improved over radiional mehods. All he mahs you need Unis of measuremen (e.g. he newon, N) Using Pyhagoras heorem, and he angle sum of a riangle (e.g. finding a resulan vecor ) Using sin, cos and an in physical problems (e.g resolving vecors) Using angles in regular D srucures (e.g. inerpreing force diagrams o solve problems ) Changing he subjec of an equaion (e.g. re-arranging he SUVAT equaions) Subsiuing numerical values ino algebraic equaions (e.g. calculaing he acceleraion) Ploing wo variables from experimenal or oher daa, undersanding ha y = mx + c represens a linear relaionship and deermining he slope of a linear graph (e.g. verifying Newon s second law experimenally ) Esimaing, by graphical mehods as appropriae, he area beween a curve and he x- axis and realising he physical significance of he area ha has been deermined (e.g. using a speed ime graph) Wha have I sudied before? Using a sopwach o measure imes Measuring and calculaing he speed of objecs Graviy making hings fall down, and giving hem weigh Measuring forces, calculaing resulan forces The moion of objecs as a resul of forces acing on hem Wha will I sudy in his chaper? The definiions of and equaions for: speed, disance, displacemen, ime, velociy, acceleraion Graphs of moion over ime The classificaion of scalars and vecors Adding and resolving vecors Newon s laws of moion Kinemaics equaions Momens (urning forces) Wha will I sudy laer? Kineic energy and graviaional poenial energy Inerconvering graviaional poenial and kineic energy Work and power Momenum and he principle of conservaion of momenum Wave movemens Fluid movemens and erminal velociy The meaning and calculaion of impulse (A level)

4 7 Behaviour can Moion be learned explain he disincion beween scalar and vecor quaniies Velociy and acceleraion disinguish beween speed and velociy and define acceleraion calculae values using equaions for velociy and acceleraion fig A Charlene Thomas has acceleraed o a high speed. The upper case symbol for he Greek leer dela, D, is used mahemaically o mean a change in a quaniy. For example, Ds means he change in he displacemen of an objec, o be used here o calculae is velociy. Movemen is fundamenal o he funcioning of our universe. Wheher you are running o cach a bus or wan o calculae he speed needed for a rocke o ravel o Mars or he kineic energy of an elecron in an X-ray machine, you need o be able o work ou how fas hings are moving. Rae of movemen One of he simples hings we can measure is how fas an objec is moving. You can calculae an objec s speed if you know he amoun of ime aken o move a cerain disance: disance (m) speed (ms 1 ) = ime (s) v = d However, he calculaion for speed will only ell you how fas an objec is moving. Ofen i is also vially imporan o know in wha direcion his movemen is aking he objec. When you include he direcion in he informaion abou he rae of movemen of an objec, his is hen known as he velociy. So, he velociy is he rae of change of displacemen, where he disance in a paricular direcion is called he displacemen. displacemen (m) velociy (ms 1 ) = ime (s) v = s Dd OR v = D N 75m 3 m fig B The displacemen due norh is only 75 m, whils he acual disance his ahlee has run is 3 m. So he velociy due norh is much less han he acual speed. Scalars and vecors A quaniy for which he direcion mus be saed is known as a vecor. If direcion is no imporan, he measuremen is referred o as a scalar quaniy. Therefore, velociy is a vecor and speed is a scalar; disance is a scalar and displacemen is a vecor. Scalar and vecor quaniies are no limied o measuremens relaed o movemen. Every measured quaniy can be classifi ed as needing o include he direcion (vecor, e.g. force) or as being suffi cienly saed by is magniude only (scalar, e.g. mass). Vecor symbols are wrien in bold ype o disinguish hem from scalar variables. Average and insananeous speed In mos journeys, i is unlikely ha speed will remain consan hroughou. As par of her raining programme, he ahlee in fi ga wans o keep a record of her speed for all races. From res, before he saring gun sars he race, she acceleraes o a op speed. However, he race iming will be made from sar o fi nish, and so i is mos useful o calculae an average speed over he whole race. Average speed is calculaed by dividing he oal disance for a journey by he oal ime for he journey. Thus i averages ou he slower and faser pars of he journey, and even includes sops. Insananeous speed can be an imporan quaniy, and we will look a how o measure i in he nex opic. fig C Mos speed checks look a insananeous speed, bu CCTV allows police o monior average speed over a long disance. Acceleraion Acceleraion is defi ned as he rae of change of velociy. Therefore, i mus include he direcion in which he speed is changing, and so acceleraion is a vecor quaniy. The equaion defi ning acceleraion is: change in velociy (ms 1 ) acceleraion (ms ) = ime aken o change he velociy (s) a = v u 1 13 OR a = Dv D where u is he iniial velociy and v is he fi nal velociy. The vecor naure of acceleraion is very imporan. One of he consequences is ha if an objec changes only he direcion of is velociy, i is acceleraing, whils remaining a a consan speed. Similarly, deceleraion represens a negaive change in velociy, and so could be quoed as a negaive acceleraion. 1 The ahlee in fi gb has aken 36 seconds from he sar o reach he 3 m mark as shown. Calculae: (a) her average speed during his 36 seconds (b) her average velociy due norh during his 36 seconds (c) her average velociy due eas during his 36 seconds. A driver in a car ravelling a abou 5 mph (4. kmh 1 ) sees a ca run ono he road ahead. (a) Conver 4. kmh 1 ino a speed in ms 1. (b) The car ravels 16.5 m whils he driver is reacing o he danger. Wha is his reacion ime? (c) The car comes o a sop in.5 s. Wha is is deceleraion? 3 An elecron in an X-ray machine is acceleraed from res o half he speed of ligh in s. Calculae: (a) he speed he elecron reaches in ms 1 (b) he acceleraion he elecron experiences. Whils acceleraions can (very briefly) be exraordinarily high, like ha for he elecron in quesion 3b, no speed or velociy can ever be greaer han he speed of ligh, which is 3 1 ms 1. If you calculae a speed ha is higher han his, check your calculaion again as i mus be wrong. Key definiions Speed is he rae of change of disance. disance (m) speed (ms 1 ) = ime (s) v = d Velociy is he rae of change of displacemen. displacemen (m) velociy (ms 1 ) = ime (s) v = s OR v = Ds D Displacemen is he vecor measuremen of disance in a cerain direcion. A vecor quaniy mus have boh magniude and direcion. A scalar quaniy has only magniude. Average speed is calculaed by dividing he oal disance for a journey by he oal ime for he journey: oal disance (m) average speed (ms 1 ) = oal ime (s) Insananeous speed is he speed a any paricular insan in ime on a journey, which can be found from he gradien of he angen o a disance ime graph (see secion.1. ) a ha ime. Acceleraion is he vecor defined as he rae of change of velociy. change in velociy (ms 1 ) acceleraion (ms ) = ime aken o change he velociy (s) a = v u OR a = Dv D

5 7 Behaviour can Moion be learned.1.1 One of he bes ways o undersand he movemens of an objec whils on a journey is o plo a graph of he posiion of he objec over ime. Such a graph is known as a displacemen ime graph. A velociy ime graph will also provide deail abou he movemens involved. A velociy ime graph can be produced from direc measuremens of he velociy or generaed from calculaions made using he displacemen ime graph. Displacemen ime graphs If we imagine a boaing rip on a river, we could monior he locaion of he boa over he hour ha i has been rened for and plo he displacemen ime graph for hese movemens. Depending on wha informaion we wan he graph o offer, i is ofen simpler o draw a disance ime graph in which he direcion of movemen is ignored. The graphs shown in fi ga are examples of ploing posiion agains ime, and show how a disance ime graph canno decrease wih ime. A displacemen ime graph could have pars of i in he negaive porions of he y-axis, if he movemen wen in he opposie direcion a some poins in ime. Disance /m 6 inerpre displacemen ime graphs, velociy ime graphs and acceleraion ime graphs make calculaions from hese graphs Disance ime graph Moion graphs undersand he graphical represenaions of acceleraed moion The simples hing we could fi nd from hese graphs is how far an objec has moved in a cerain ime. For example, in fi ga, boh he graphs show ha in he fi rs 1 minues he boa moved 1 m. Looking a he ime from 4 o 4 minues, boh show ha he boa ravelled 1 m, bu he displacemen ime graph is in he negaive region of he y-axis, showing he boa was moving downriver from he saring poin he opposie direcion o he places i had been in he fi rs 4 minues. During he period from o 5 minues, boh graphs have a fla line a a consan value, showing no change in he disance or displacemen. This means he boa was no moving a fla line on a disance ime (d d ) graph means he objec is saionary. From o 5 minues on he velociy ime (v v ) graph of his journey (see fi gb ) he line would be a a velociy of ms 1. Speed and velociy from d graphs The gradien of he d graphs in fi ga will ell us how fas he boa was moving. Gradien is found from he raio of changes in he y -axis divided by he corresponding change on he x-axis, so: for a disance ime graph: disance (m) gradien = = speed (ms 1 ) ime (s) v = d for a displacemen ime graph: Displacemen /n displacemen (m) gradien = 1 = velociy (ms ) ime (s) Ds v = D Displacemen ime graph For example, he fi rs 1 minues of he boaing rip in fi ga represens a ime of 1 seconds. In his ime, he boa ravelled 1 m. Is velociy is: v = Ds _ D = 1 1 =.167 ms 1 upriver Velociy ime graphs A velociy ime graph will show he velociy of an objec over ime. We calculaed ha he velociy of he boa on he river was.167 ms 1 upriver for he fi rs 1 minues of he journey. Looking a he graph in fi gb, you can see ha he line is consan a ms 1 for he fi rs 1 minues. Also noice ha he velociy axis includes negaive values, so ha he difference beween ravelling upriver (posiive y-axis - values) and downriver (negaive y-axis values) can be represened. Velociy/ms Time /minues For example, beween 1 and minues on he graphs, he boa slows evenly o a sop. The acceleraion here can be calculaed as he gradien: gradien = Dv D = v u =.167 _ 6 = So he acceleraion is: a = ms. Disance ravelled from v graphs Speed is defi ned as he rae of change in disance: v = d [ d = v =.14 ms As he axes on he v graph represen velociy and ime, an area on he graph represens he muliplicaion of velociy ime, which gives disance. So o fi nd he disance ravelled from a v graph, fi nd he area beween he line and he x-axis. - Velociy/ms Time /minues Time /minues down river up river Time /minues fig A A comparison of he displacemen ime graph of he boaing rip up and down a river wih is corresponding disance ime graph. fig C In he firs 1 minues (1 seconds) he disance ravelled by he boa moving a.167 ms 1 is given by he area beween he line and he x-axis: d = v = = 1 m. If we are only ineresed in fi nding he disance moved, his also works for a negaive velociy. You fi nd he area from he line up o he ime axis. This idea will sill work for a changing velociy. Find he area under he line and you have found he disance ravelled. For example, from o minues, he area under he line, all he way down o he x -axis, is a rapezium, so we need o fi nd ha area. To calculae he whole disance ravelled in he journey for he fi rs 4 minues, we would have o fi nd he areas under he four separae sages ( 1 minues; 1 minues; 5 minues; and 5 4 minues) and hen add hese four answers ogeher fig B Velociy ime graph of he boaing rip. Acceleraion from v graphs Acceleraion is defi ned as he rae of change in velociy. In order o calculae he gradien of he line on a v graph, we mus divide a change in velociy by he corresponding ime difference. This exacly maches wih he equaion for acceleraion: gradien = Dv D = v u _ = acceleraion

6 .1 Invesigaion Finding he acceleraion due o graviy by muliflash phoography Using a muliflash phoography echnique, or a video recording ha can be played back frame by frame, we can observe he falling moion of a small objec such as a marble. We need o know he ime beween frames. From each image of he falling objec, measure he disance i has fallen from he scale in he picure. A carefully drawn disance ime graph will show a curve as he objec acceleraes. From his curve, ake regular measuremens of he gradien by drawing angens o he curve. These gradiens show he insananeous speed a each poin on he curve. fig D Muliflash phoography allows us o capure he acceleraing movemen of an objec falling under graviy. Ploing hese speeds on a velociy ime graph should show a sraigh line, as he acceleraion due o graviy is a consan value. The gradien of he line on his v graph will be he acceleraion due o graviy, g. Acceleraion ime graphs These graphs show how he acceleraion of an objec changes over ime. In many insances he acceleraion is zero or a consan value, in which case an acceleraion ime (a a ) graph is likely o be of relaively lile ineres. For example, he objec falling in our invesigaion above will be acceleraed by graviy hroughou. Assuming i is relaively small, air resisance will be negligible, and he a graph of is moion would be a horizonal line a a = 9.1 ms. Compare his wih your resuls o see how realisic i is o ignore air resisance. For a larger objec falling for a long period, such as a skydiver, hen he acceleraion will change over ime as he air resisance increases wih speed. The weigh of a skydiver is consan, so he resulan force will be decreasing hroughou, meaning ha he acceleraion will also reduce (see Secion 1.1.5). The curve would look like ha in fi ge. a/ms /s 1 Describe in as much deail as you can, including calculaed values, wha happens in he bicycle journey shown on he d graph in fi gf. 6 Disance/m A 1 B Time/s fig F Disance ime graph of a bike journey. C 3 4 Describe in as much deail as you can, including calculaed values, wha happens in he car journey shown on he v graph in fi gg. Speed ms A Time/s fig G Velociy ime graph of a car journey. 3 (a) From fi gb, calculae he acceleraion of he boa from 4 o 49 minues. (b) From fi gb, calculae he disance ravelled by he boa from 4 o 6 minues. Remember ha he gradien of a disance ime graph represens speed or velociy, so if he line is curved, he changing gradien indicaes a changing speed, which you can describe as he same as he changes in gradien. B C D add wo or more vecors by drawing add wo perpendicular vecors by calculaion Forces are vecors. This means ha measuring heir magniude is imporan, bu equally imporan is knowing he direcion in which hey ac. In order o calculae he overall effec of muliple forces acing on he same objec, we can use vecor addiion o work ou he resulan force. This resulan force can be considered as a single force ha has he same effec as all he individual forces combined. Adding forces in he same line If wo or more forces are acing along he same line, hen combining hem is simply a case of adding or subracing heir magniudes depending on heir direcions. a b 7N 7N Adding forces 11N 4N 1N fig A Adding forces in he same line requires a consideraion of heir comparaive direcions. 11N Adding perpendicular forces The effec on an objec of wo forces ha are acing a righ angles (perpendicular) o each oher will be he vecor sum of heir individual effecs. We need o add he sizes wih consideraion for he direcions in order o fi nd he resulan. Magniude of he resulan force To calculae he resulan magniude of wo perpendicular forces, we can draw hem, one afer he oher, as he wo sides of a righ-angled riangle and use Pyhagoras heorem o calculae he size of he hypoenuse. Resulan 5 13N 11N 7N fig C The resulan force here is calculaed using Pyhagoras heorem: F = ( = 13 N Direcion of he resulan force As forces are vecors, when we fi nd a resulan force i mus have boh magniude and direcion. For perpendicular forces (vecors), rigonomery will deermine he direcion. Resulan 5 13N θ 11N 7N fig D The resulan force here is a an angle up from he horizonal of: u = an 1 ( 7 _ 11 ) = 3 Always ake care o sae where he angle for a vecor s direcion is measured. For example, in fi gd, he angle should be saed as 3 up from he horizonal. This is mos easily expressed on a diagram of he siuaion, where you draw in he angle. Adding wo non-perpendicular forces The geomery of perpendicular vecors makes he calculaion of he resulan simple. We can fi nd he resulan of any wo vecors by drawing one afer he oher, and hen he resulan will be he hird side of he riangle from he sar of he fi rs one o he end of he second one. A scale drawing of he vecor riangle will allow measuremen of he size and direcion of he resulan. 6 fig E Acceleraion ime graph for a skydiver. Key definiions A displacemen ime graph is a graph showing he posiions visied on a journey, wih displacemen on he y axis and ime on he x axis. A velociy ime graph is a graph showing he velociies on a journey, wih velociy on he y axis and ime on he x axis. fig B These wo rugby players are each puing a force on heir opponen. The fig E The resulan force here can be found by scale drawing of he wo forces, See Secion for more deails on falling objecs and erminal forces are a righ angles, so he overall effec would be o move him in a and hen measuremen of he resulan on he drawing using a ruler and velociy. hird direcion, which we could calculae. a proracor θ 45N 39 4 N

7 .1 3 The parallelogram rule There is anoher mehod for fi nding he resulan of wo non-perpendicular forces (or vecors) by scale drawing, which can be easier o use. This is called he parallelogram rule. Draw he wo vecors o scale a he correc angle and scaled so heir lengh represens he magniude saring from he same poin. Then draw he same wo vecors again parallel o he original ones, so ha hey form a parallelogram, as shown in fi gf. The resulan force (or vecor) will be he diagonal across he parallelogram from he saring poin. 4N 39 45N fig F Finding he resulan vecor using he parallelogram rule. Hypoenuse.1 4 calculae he momen of a force apply he principle of momens find he cenre of graviy of an objec Momens Forces on an objec could ac so ha he objec does no sar o move along, bu insead roaes abou a fi xed pivo. If he objec is fi xed so ha i canno roae, i will bend. Principle of momens The vecor addiion rules shown on his page work for all vecors, no jus forces. They are useful only for co-planar vecors, which means vecors ha are in he same plane. If we have more han wo vecors ha are in more han one plane, add wo vecors ogeher firs, in heir plane, and hen add he resulan o he nex force using hese rules again. Keep doing his unil all he vecors have been added in. Momen of a force pivo poin disance, x force F 7N 11N fig G Free-body diagram of a rugby player (red circle). The forces from he acklers are marked on as force arrows. Free-body force diagrams If we clarify wha forces are acing on an objec, i can be simpler o calculae how i will move. To do his, we usually draw a free-body force diagram, which has he objec isolaed, and all he forces ha ac on i drawn in a he poins where hey ac. Forces acing on oher objecs, and hose oher objecs, are no drawn. For example, fi gg could be said o be a free-body force diagram of he rugby player being ackled in fi gb, and his would lead us o draw fi gc and fi gd o make our resulan calculaions. 1 Work ou he resulan force on a oy car if i has he following forces acing on i: rubber band moor driving forwards.4 N air resisance.5 N fricion 5. N child s hand pushing forward 1 N. As a small plane akes off, he lif force on i is 6 N verically upwards, whils he hrus is N horizonally forwards. Wha is he resulan of hese forces on he plane? 3 Draw a free-body force diagram of yourself siing on your chair. 4 (a) Draw he scale diagram of fi ge, and work ou wha he resulan force would be. (b) Use he parallelogram rule, as in fi gf, o check your answer o par (a). 5 In order o ry and recover a car suck in a muddy field, wo racors pull on i. The firs acs a an angle of lef of he forwards direcion wih a force of 5 N. The second acs 15 o he righ of he forwards direcion wih a force of N. Draw a scale diagram of he siuaion and find he resulan force on he suck car. fig A A force acs on a beam fixed a a poin. The momen of a force causes roaion or, in his case, bending. The endency o cause roaion is called he momen of a force. I is calculaed from: cenre of mass momen (Nm) = force (N) perpendicular disance from he pivo o he line of acion of he force (m) momen = Fx x axis of roaion cenre of mass fig C Balanced momens creae an equilibrium siuaion. If we add up all he forces acing on an objec and he resulan force, accouning for heir direcions, is zero, hen he objec will be in equilibrium. Therefore i will remain saionary or, if i is already moving, i will carry on moving a he same velociy. The objec could keep a consan velociy, bu if he momens on i are no also balanced, i could be made o sar roaing. The principle of momens ells us ha if he oal of all he momens rying o urn an objec clockwise is equal o he oal of all momens rying o urn an objec aniclockwise, hen i will be in roaional equilibrium. This means i will eiher remain saionary, or if i is already roaing i will coninue a he same speed in he same direcion. x 1 x x W 1 W W Key definiions Resulan force is he oal force (vecor sum) acing on a body when all he forces acing are added ogeher accouning for heir direcions. A free-body force diagram of an objec has he objec isolaed, and all he forces ha ac on i are drawn in a he poins where hey ac, using arrows o represen he forces weigh F weigh F fig B The calculaion of momen only considers he perpendicular disance beween he line of acion of he force and he axis of roaion, hrough he pivo poin. When free o roae, a body will urn in he direcion of any ne momen. fig D As he mere-long beam is balanced, he sum of all he clockwise momens mus equal he sum of all he aniclockwise momens.

8 Behaviour can Moion be learned.1 WORKED EXAMPLE In fi gd, we can work ou he weigh of he beam if we know all he oher weighs and disances. The beam is uniform, so is weigh will ac from is cenre. The lengh of he beam is 1 cm. So if x 1 = cm, hen x mus be 3 cm, and x = cm. The dinosaur (W 1 ) weighs 5. N and he oy car s weigh (W ) is.95 N. In equilibrium, principle of momens: sum of clockwise momens = sum of aniclockwise momens W 1 x 1 = W x + W x 5.. = W (.76) [ W =.3 W = 1.3 N In order o calculae he sum of he momens in eiher direcion, each individual momen mus be calculaed firs and hese individual momens hen added ogeher. The weighs and/or disances canno be added ogeher and his answer used o calculae some sor of combined momen. Cenre of graviy The weigh of an objec is caused by he graviaional aracion beween he Earh and each paricle conained wihin he objec. The sum of all hese iny weigh forces appears o ac from a single poin for any objec, and his poin is called he cenre of graviy. For a symmerical objec, we can calculae he posiion of is cenre of graviy, as i mus lie on every line of symmery. The poin of inersecion of all lines of symmery will be he cenre of graviy. Fig E illusraes his wih wo-dimensional shapes, bu he idea can be exended ino hree dimensions. For example, he cenre of graviy of a sphere is a he sphere s cenre. cenre of graviy There are equal amouns of mass each side of his line. cenre of graviy fig F Balancing a broom on is cenre of graviy. Invesigaion Finding he cenre of mass of an irregular rod In his invesigaion, we use he principle of momens o find he cenre of graviy of a broom. As i is no a symmerical objec, he locaion of he cenre of graviy is no easy o deermine jus by looking a he broom. Wih he exra mass a he brush head end, he cenre of graviy will be nearer ha end. If you can balance he broom on a knife edge, hen he cenre of graviy mus lie above he knife edge. As he perpendicular disance from he line of acion o he weigh is zero, he momen is zero so he broom sis in equilibrium. You will probably find i difficul o balance he broom exacly, so you can use an alernaive mehod. Firs you measure he mass of he broom (M) using a digial balance. Then you use a se of hanging masses (of mass m ) o balance he broom more in he middle of he handle, as in fi gg. When he broom is balanced, you measure he disance (d d ) from he hanging masses o he pivo). You calculae he disance ( x ) from he pivo o he cenre of graviy of he broom using he principle of momens: clockwise momen = aniclockwise momen cenre of graviy fig G Finding he cenre of graviy of an irregular rod (broom). mg d = Mg x md [ x = M Noe: Do no ge ino he habi of using only he mass in momens calculaions, as he definiion is force imes disance. I jus happens ha in his case g cancels on each side. M The cenre of graviy of a circle is in he middle. There are equal amouns of mass all around his poin. There are equal amouns of mass each side of his line. fig E The cenre of graviy of a symmerical objec lies a he inersecion of all lines of symmery. There are equal amouns of mass each side of his line. The cenre of graviy is nearer o he base of he riangle. You can consider he erms cenre of graviy and cenre of mass o mean he same hing. They are idenical for objecs ha are small compared o he size of he Earh. Irregular objecs The cenre of graviy of an irregularly shaped objec will sill follow he rule ha i is he poin a which is weigh appears o ac on he objec. A Bunsen burner, for example, has a heavy base, and so he cenre of graviy is low down near ha concenraion of mass, as here will be a greaer aracion by he Earh s graviy o his large mass. 1 Wha is he momen of a 5 N force acing on a solid objec a a perpendicular disance of 1.74 m from an axis of roaion of he objec? A child and her faher are playing on a seesaw. They are exacly balanced when he girl (mass 46 kg) sis a he end of he seesaw,.75 m from he pivo. If her faher weighs 4 N, how far is he from he pivo? 3 The weigh of he exercise book in he lef-hand picure in fi gb causes a roaion so i moves owards he second posiion. Explain why i does no coninue roaing bu comes o res in he posiion of he second picure. 4 If he same se-up as shown in fi gd was used again, bu he oy car was replaced wih a banana weighing 1.4 N, find ou where he banana would have o be posiioned for he beam o balance calculae he new x 3. Key definiions A body is in equilibrium if here is zero resulan force and zero resulan momen. I will have zero acceleraion. The principle of momens saes ha a body will be in equilibrium if he sum of clockwise momens acing on i is equal o he sum of he aniclockwise momens. An objec s cenre of graviy is he poin hrough which he weigh of an objec appears o ac. 1