Cambridge International AS and A Level Physics [2 ed.] 1471809218, 9781471809217

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Questions from C.mbrldB'" lnternatlon.IAS andA Level Physlc& papers are reproduCmbridge[ntern.UonaIExamln.tlons be.rsno responslbl]jtyforthe example .nswers to quesUon.l.ken fromlts pa5lquestlon p.pers whlch ar»cont.I""-' unl! in w!lich It Is me:lSUrW

mUSI be

deflned as predsely as possible. SlisfoondedOfl�fundamentalorbaseunits. The base quanllUes and the units wUh wflldl they are measured are IISied In

'Thbie

1.1. Forcornpleteness, thecandeb tusbren

lndllded. but this unit \\111 oot be

used In the- AlAS course.l1ll' IIlOk> "111 only be used In the A Lewl course Table 1.1 Thl'b.-quantll.leS�d urnb

qwontny

symbol

��tricQJffent

amp�re(.lmp)

A

thermodynamic temperatllre

Flgurel.4Tnem�ssoflnlsjewelcould be mealllrl.'dlnl different unlls. So.30m 5 would mean Ihlny mc(f(' seccnds and 3Omsme:ans30mlUlsecaJds .

I

II Physical quantities and unilS Example Calculate the number of micrograms in 1.0 mWligram.

1.0g .. I.0" 100mg and 1.0g .. 1.0" l06rnicrograms(llg)

�,1.0,,'Qlmg .. '.0"'0'Ilg and 1.0mg .. (1.0" 10'lJ1:1.0" 1()l).1.0w 10lllg

Now it's your turn 1

CaIaJiatetheare.a,inw,ofthetop ofa tablewithsides ofl,2mandO.gm.

)

Write down, u-;ing 5dentifk notation, the vailies of the following QlJaf1tities'

« rM! the number of,ubk metres in one cubic kilometre.

2

De1 mi

(a16.8pF, (b) 32IlC,

(e)

4

6OGW.

How many electric fires, each rated at 2.SkW, can be POWefed from a genefator providing 20 . MW

r-- o.26 nm �

5

1.6,

An atom of gold, Figure is 5.6 w

Flgure 1.6 Atomofgokl

10-lpm.

has a diameter ofa.26nm and the dia meter ofits nucleus

Ca kulate the ratio ofthe diametef

m les

01 the atom to that ofthe nucleus

Derived units AU qu: Ut

, apan from tlK' OOSl" quanUtIes, on

h

Derived units consist ofsorT\2 rombirliltion of the

�t

mul tiplied t

See Table

be expressed In terms cI tleri\'etl

bas.e units. The bas.e units may be

er Of dimed byone anothef, but newr oldded Of sublracted.

1.j for examples of de!1\"OO

unUs. some quantities have a named unU. For

example, t h e unit In terms of base units. Qualllllk's whldJ do I'lOl have a named unit are expressed In terms of other units . For example, specific latent heat (lbpIc 12) Is measured In joules perkllogTamOkg-'). Table 1,) Some examples of deriwd umtswhlch maybe �td IntheAlASCOl.l�e .....frequency velocity ')(cell:>ration force energy

��tri{char9'!

hertz

m.w

(HZ)

,-,

)

ton {N

kgws-l

wall{W)

kgffils-1

(Qulomb(C)

llm ( )

potential ditfereoce

volt

e�tri(iil resiltance

o

(V)

kgffils-1A-1

O

kgffils-1A-l

�ificlle«torintocomponents

Example A glider is lalJl"1dtl!d by an airuaft with a cabl@, asshown Wl Figure l.20. AI one particular morTII!r1t.theten� inthe cable is620N and thecablemakes anangleof2S·withthe horilOrltalbee Figurt' 1.211 Calrulate: (al the forcppuiting the g6derhorilOfllally, (bl the vertical forc:e exerted by the cable on the ooseof the glider.

(al horizontal rompooent F" " 620 (OS 25 • S&O N (b) vertical {omponent F" ", 620sin 2 S _ 260N

Now it's your turn 23 An airc:raft is travelling of the aircraft in· Rgure l.20

�620:'

3S0 ea51. of north al a speed of 310km h-l. Cak:ulate the speed

(a) the r.ortherly ctirection, (b) the &lSterly ctirection.

of 9.2m S-I. The hillsicle makes an ang11' 01 6.3° with the horizootal. Calculale, for the cyclist:

24 A cydist i5 1ravelling down a hill at a :;peOO

(al the verticaI 5pero, (bl the horizootal spero.

!

Rgur. '.Z1

11

I

II Physical quantities and unilS . AII �l quantities hiM!a magnitudl! (s.i.ze) and a unit. • The SI base units 01 mass, length, tinll', electric rurren!, thermodynamic temperature and amount 01 substance are the kilogram, metre, se quamlty In experimental

of subslance and luminous

(1b complete the list. we shoukl

Intenslly. 001 llH.ose are ne( encountered

\\-urk In AlAS PhysICs,) In Ihe following sections we will look at the

rncthcx!s avallabie formeasurlng thebaS(' quantttles lna schoo] Of collegelaboratOl"}". By understanding the

principles of the available nlC'lhods, we will lX' able to make an

InlOrmed dedsloo about the choice of a panlCular technique. With respect to ntlklng the

experln1lt'U as precise and reproduc1b1e as possible. and avoiding 1lltln::es of

systematIC error. For all of the quanlltles. the effective choice wtU lX' IImled by what Instruments are available In )'our laboratory. Howl'''er. In one type of eX:lmln:ation

questlon.. 011 planning and design. you may be asked to deYlS(' an expet1n1lt'U and draw OII)uur theoreUcal. ralhertltanpr.K"l.k::al. kOOYl1edge of\'aOOUstypesofapparatus. At AlAS Ie,"t'!. students wnerally 35.9.1me that the callbl'"atlon of the InSironl('1IlS they

use Is rorrect.. Jic/I\'cver, It Is worth thinking about hcf,t" to' compare the calibration of

one Inslroment against anol:fier, ewn lr thiS IS 3 check )'0.1 will \"t'ry seldom make.

15

II Measurement techniques

Figure 2.1 StudenISdoin !l � eJ straight pans on lhe OUtside. These can be used 10

measure the diameter d a hole. Thl> j::Iws are placed Inside lhe hole :md are 1llO'I-l'd apan unUI ltiey are I n CUlI:ICI "'11h Ihe ed9t'S oIlhe lloie. Tlle salie3nd wmler can

lhen be read. A pin al lhe end otlhe sliding pan dlhe caUper can be used to measure lhe depth ol a blind hole: for example. a hole which hils been drtlJed In. bul I10l rlg:IJ: lhrough. a

hole. and the pin lTlO\'ed IntOlhe hole untU It reaches the Ixxlom (see Rgure 2.9). The reading of

wooden board. The end of the fixed salle IS ptaced on the board. aCfOli;S the

FlgurIt 2.9 MNsurementofthedepthof;J bhndhole

Ihe salleaoo \-emler gfve!S lhe de-pthofthe hole. AS with the mlcromele!" s;:rewgaugt', lhe\-emler callper should 00 chetwet>n the third graduation of the �nier scale and one 01

the gradudtions 01 the fixed scale. The reading is thus S.Slon or SS.lmm.

Now it's your turn 1

Figu� 2.12a and 2.12b 'ihow thescdlesof a miaometer wew gauge when the zero Is being cheded. and iI9iIin when medsuring the diameter 01 an object, WlJat ls the diameter?

Figurlt 2.12

i) Mld b)

Choice of method A summary of the range and reading uncertainty of length-meaSUring Instruments Is glVl'fl ln Table 2.1. Table 2.1 lengln-mealurimj imtrum�rrt5 untltrtalnty

In length ctleck zefO, c�lbration errors micrometer screw g�uge

ctleckzefO erfQf vers��1e: Inside �nd outside diam�ten. d".

In deciding which Instrumenl 10 use In a jJQrtlcular expertmt'm. you should ronskler

grea

flrg the nature of the length measuremt'n! )'OU hal'\! to ITLlke. for example. If you nee.ions In order 10 change thl> range (for examp!e, fru n O---lOOg IO O-tkg).

Both oftile5e typesofbalance are used more for the convenience ofobtalnlng a rapld, approxlmate reading. rathf'r lhan for an accuratetletefll1lnaUoo. An Indlc:UlorI of thl> uncertainly In\w-ed In readings with a p:1ItICUJar balance C;ln be obtained from thesma1lesl dMsIon OIl Ihf' SClIIe.

Example The ma�s of a quantity of chemical is determined using a Iewr balance. Owr the range of

masSol!s invol'o'l'd. the ....paration betwoon mass graduatlons on the bar is2g. The reading for the mass of the empty contaifM'r is 56g, and the reading for the mass of the oontaifM'r p/us the chemkal is 100g. Rnd the mall of the chemical, and the unceltainty in this valoo By subtrdctioo. Ihe ma�� of Ihe chemical il 104 - 56. 48g

°7 .0 '° °r o ",

The uncertainly in each reading il li�e� to be half of the smallest division of the mass graduation � o n the beamthat il±lg. Eachofthe two readingshasan ur.c:ertainty of±lg' theun cllange lim In the OllSS reading. This change must

be oonll'ned lnIo forr:e F by mullip/ylng llm by g Thevarillioll \\1{h current / �

the magnellc face F may be determined.

Tlle eqwtlon (:;ee lbplc22) where lis the length or the wi!\' In the magnetic 1lekI. may be verified. The direction

ofthoi' force, as predicted IheoM.lc:.Jlly by �1emJng'S left-hand rule, may al:so be I't'rtfled by cheddng whetlter the mass reading Increases or decreases for � 8"'....n (\lITen! direction.

-+ -1 f----{::::;;Z::J----{

Figure 2.20 Currl'lll bal�n(l� eXpeOmllfl1

Measuring an angle Angles are measured using an Il1SIrumem called a proU1lCtor. This looks like :l seml� drrulJr. or somt'ltmes drrular, ruk>r. "1th lis scale marked 00\ In angular measu(�" lnl':ulably degrees mlher Illan radians. The refUm 0( the clrde Is dearly marlced

To measure the angle between twO lines. the CE11Ite of the circle of tile plQt13ctor

Is placed exactly O'"l"f the pain 0( lnIell'ieCtlOn of the Unes and one line Is aligned

with thl'OO dlrectlon oC the proll1lClOr(F1gure 2.21). The angJe becwet'll Ihl> Unes ls

then gWen by the reading on IhP scale at WhICh the second line passes through the

circumference a the circle.

If the direction ofa stngIe line Ileeds lobe de.flncd, lhls is almysreferred to the

direction of lhe Oxaxis, lhe hortzofllal axiS poInllng IOl'o':lrdslhe rlghl, :l.SZ('ro.

Figure 2,21 USIIlg a protrxlDr

13

I

II Measurement techniques MOSl plUlractorS used In schools and colleges have a dlam«er d about 10em. The size

of the scale atthe drrumfen'llU' dthe drcie Is Ihen such that the hltC1"Val oo\\'l;'ell SC'lledlvlslOfls ls 1�. lIlseaSYIOm.lkea readlng lolhe ncan'Sl degree. and !i()lneUmes IOhalfa degree, lflhe llne being measured LSnoe eoough. PlUlractors d larger dlallll.'U."r may he matted In half degrees.

Methods of measuring time The experlmens you \\111 me« In )UJr prncUc.ll physks course dealwilh the

mea:;uremeN: of time Inll'rv.r.ls, nlthet" than "'1lh absolute lime. The basic mt1hod of mea:;urlng a lime 1rv.erv.r.1 Is with a stopdock or SlOPW·:IIch. In each case, the In,;trumen Is Slarled and stopped by pressing a Ieverora buUOI1. and re-set by

pressing anothl'r COlItroI. You should brnlllarl5e yourself willi lhe way of operallng lhe In,;truJllerll before you start a UrnlngexlX'rlmenl In earnest. Remernherlhal lhe reaction time d Ihe experimenter (a few lernM of a second) Is Ukely 10 he much gn'aler than

lhe ullU'rlalnly of thi' Instrument II.>elf. If you do not reduce lhe l.'!Teets of reaction lime. an unaa:epubie sySiemaUc error may he buill In 10 Illl.' experiment. .A5 explained in the Sl.'CIIon on Errors and uncertaintIeS (page }i). one ....'3yof reduclnglhe effect ofn'action time Is to time ffiOUgh events (for ex.1ntple, till.' swtngs of a pendulum) 10 m3kelhl' Interval being measured very much 13rscr lh3n IllC l.'xperitnenter's reaction lime. A gcxxl tl.'\:hnlque Is to munt the ewnts (the swings). mmmendng by muntlng down to zero. and starting the timer at thl.' zero munt Wherever pos.5Ibll.'. work with 3t least 20 SI.'COf'Ids' wonh of events (06dllaUons). and Il'pl.'lt l.'leh set of timings thret'

limes. (Sometimes, when carrying OlIt experlnlCms on damped O6dllltlons. )'0lI will

have to be satl.lftelfl('tef'li and reslstanre thermometers, and assess their suitabIlity for use as a thermometer.

Themercury-in-glass thermometer LIquId-In_glass thennometers are bJsed on the t/)(>nn31 exp�nslon of a liquId. A quamlty of liquId IS COflt:llneR' Is a zero error and make an adju,.ment If rteCessal)', noIe wheiller tile dlspby 1OO1ca!eS 'A' or 'rnA', and d�.. r e Ille position 0( the decimal point! The uncer!alnty In the R'adlng o(a dlgl1al meter is expressed In tenru; 01 the o',erall uncenalnly and the uncertainty In the last digit. When In use, you ",111 note that tile

ta,. dlgtt 0( the display nUClUJtes from OIle figure 10 anolher, You can try to e:.tlmate tile mean 01 the fluCluatlons,but !flhts fluCiua!ton occurs, !heR' Is ctearly uncertatnty !n tlle last dtgll 01 the value.

Multimeters Ftgure 2,]6Mlllllmeler

Muttlmeters. or multtfunClIon IllSlrunll'n!s, 3re 3vall�ble ln holh analogue and digital forms (FlguR' 2.36). SUch meters may loclude SWltctted opllons for the llll'asurement 01 dIrect and altemallng currents and voltages. ancl

of resls!an«,. wtlh several ranl!"s

for each quantity beIng measured. If you use � mul!l!TK'ter. make sure that you are

easure the quanUty

f.Jmlllar with the COIltrols, so that you can set the Instrument to m you require.

Choice of method Much "'ill depend on the selection of meters available In your bbor;ItOfY. Before you SCI up your clreul, make a rough calculation 10 delermlne !Ile ranges 01 currenlS and \�ages that)UlI "'111 have to measuR'. ThiS Is a IIlIal p:lrt olthe planning process, and will help to make sure that }UlI sek.'a the approprlale InSlrumenl from thor;e [n some

tabomta'ies, mu�meters are provided for use prlmartty �s test [nslruments,

10 be a\'3lbble 10 anyone who wishes 10 make � rnpld check on currents, potential

dltferences orre51stances l n a clTcult. l f lhLsts the ruie [nyour laboratory, Ltlsbad

praCiIce 10 use a muJUmeler In a long experlmenl, when � sl�un('lIon 3nd single­

range IllSlrumeru woukl do Ihe job equally succesully. sf

Remember that, to measuR' a current In � componcnl ln a clrt'ULt, an allllrllt'«' shoukl be connected In ser1es wlh the component, To meaSUre !he potenllal difference across tile componenl, a l'O!tmeter shoukl be con!lCCled In p3rnLIeI wllh the component. The arrangemenl IS shown In Bgure 2.37,

:� � component

�..

,

YOItmetec in pa.raI�1

Figure 2.]7 An amme� is connedi'd WI series wtth !hl> component, rnl>ntx>r lh:ltlhe readingOOlalJled onanaru.1ogue ordlglal miUDele£ lSihe r.m.s.\"3lUl'.)

Figure 1.38 M�asufl'lTI!'ntof altern.otingYOlt�ge

Example The output from a sigrlill generator is wnr.ected to t� Y-Input 01 a C.r.O. � the Y_amplili�r wntrol is set to 5.0 milliYolts per centimetre, the trace shown in FiJufl' 2.39 is obtained. find thepeak'oOltage ofthe !iignal, andthe r.m.s.voltage. Mea\.Urethe amplitud� of the trace on the graticule: this is 1,4cm. Tile Y-amplfier 'l('ning is 5.0mVcm-'. 1.4cm is thuo;equivalenl 10 1.4 " 5.0. 7.0m"'. The peak voltage of the signal is 7.0mV. Th£. r.m.s.voltage is given 'at7.0/.J2 • 4JlmV r.m.'

Figure 1.19

Now it's your turn 5

The outputfrom a signal generaloris connected totheY-inp;.it ofa c.r.o. Wh£.nthe Y-amplifier control is set 10 20 milliyollS per centimetre. Ih£. trace shown in Figure 2.40 is obtained. Find

(II)

the peak-to-peak voltage 01 tile !iignal,

(b) the r.m.s. yoitage.

Figure 1.40

30

2,2 Errors and uncertainties

Measuring magnetic flux density l1leflux densly ofa magrlC'llc f\eId may be measurcd tlslng a Hall probe, a device

which makes use ofthe Hall erreel (see page 322). The Hall prcbe apparatus used

ln sdiool or college Iaborntor1e:sCOfls!stS ofa thlnsitce a a semlcooductor lIUIerlal

which Is pl:tced wlh Is plalle at rlgill angie5 10 the dlroctlon of the magnctlc fleki. lbe mnr.rol unlt Is alT3nged to pass a renaln current through lhe semk:'or'duClor slice; the Hall ,mage, which Is proponknll lO the magnetic flux de!lstly, Is read off on an

aMIogUl" or dlgll:al meter, which Is alrwdy callbrnted In unllS of magnetic nux dt-ll5ly (j:esl.1). The amlngemenlls Illustrated In "tgure 2.41.

lbe use of the Hall probe to mea,;ure m;agnet1C nux density Is only required for the A ]e,'l"l syllabus but l s lnduded here for coolple1:eocss.

2.2 Errors and uncertainties Accuracy and precision Accuracy i'ithe degr@('to v.tlkh a mea5lJfementapproacheslhe 'truevalue'.

AcrUrncy dt-pends on lhe equlpmenl u5ed, the skill oflhe experimenter and the techniques used. Reducing systematic (>ITQI" or uncenalnly (see pJge 3�) In a measuremenl ImplUI'es Its accuracy. Prec:islon illhe determinedby the 5ize ofthe randomerror{s@('page3S)in the

Precision Is that pan of accuracy which Is within the control of Ihe experinlC'nter. The

experiment('r may choose different measliring InstrumentS and may use Ihem with dlfTerent levels ofsklll. thus affecllng the preClslon of measurement. If we W3nt 10 measure the diameter of a steel Spt10ere or a marble. we could use 3 metre rule, or a vernier Glilper. or a mlcron1oeter screw !puge. The choice of measuring Instrument would depend on the precision wah which we wan/the mC"J.5Uremeni to be madt-. For example, the metre rule could be used 10 n10easure to tile ne:J� millimetre,

the vernier caliper to the nearest tenth of a millimetre, and Ihe micronw;'ler screw gauge

to the nearest one-bmdredth G a mUllmetle. \'Ii'e OOOJId show the readings as follows:

\'emlercaUper: mlcrunel:e!' SCJl"\\' gauge:

31

II Measurement techniques

ILL I� III IL T

rNeling

�)pr8Cise and accUfale

r

reaalng

b)impreut accurale

Figure 2.42

r

ruaiog

� pr8Cise[)ut notaccUfate

T

reaIlloe

mprecise aoo oot accur.te

b) i

Figure 2.43

The degree

a predsiofl to wlllcll tlie IlK':lsur£'mem IS made locreases as we merl'e from

tile metre rule to the vernler caliper aoo f1natly 10 Ille mlcrorJleIC1 .screw gauge. NoI:e Ilial tile number a sIgIllflrnm figures quelled for tile meaSUrement Increases as tile

precision locre:Jses. [n facI, tile number a stgOlflrnm figures 10 a measurement gin's an Indlcalloo a llie precision a llie nlt'3suremeol. When a measuremem Is repealed many limes wllh a precise Instrumeot. tile readingserv1ng the event and startIng the timIng device. ThIs delay, called the reactlQll lime. may be as much as a few tenths of a 5eCOIKI. TO reduce the effect. you should arrnngethat !tw Intervals you are timIng are much greater than the reaction linK'. !'or example. you should time sufflclent swlngs ofa pendulum fortht' tolal tlnll' to be ofthe oroer ofat least ten seronds, so that a reaction time of a few tenths c( a 5eCOIKI is less Important

Figure 2.46

This ammeter hal a zero errOf of aoout - 0 2A

Random uncertainty (error) Random uncertaInty results In readIngs b(>lngscanered arol.lnd the accepted value Random uncertalnty may be reduced by repe�tlng a rcadlng and averoglng. andhy plotting a grnph and drawing a be5I·flt line. Examples c( rnndom errors are' • readIng a scale. particularly If this Involves the experimenter's Judgement about

Interpo!atlon be!:weenSGIle readings • t iming oscillations without the use of a ref('rence mark('r. so that timings may not

aJ\\"aYS be made to the same point ofthl' SWing

• taking readings c( a quantity thaI '"arieS WIth lime. Involvtng the dHfkulty c(

reading bOIh a timer scale and anotlll'r meIer .simultaneously • readIng a scaJe from different angles InlToduce:s a varlable paraJlax error. (In contrnSl.

If a scale readIng Is a/ways TI\Olde from the same non-normal angle, this

wtJIlntroduce a systematk: error)

35

I

II Measurement techniques Example The rummt in a r�slor isto be m&I\.Url!d using an analogue all'VTleter. State one source of (a) a systenatk uncertainty, (b) a random uncertainty. In both cases. so:;Igesl how the unoortaintymay be reducl!d.

Ca)

Systemat.ic uncertainty could be a ZI1rO error on the meter. ora wrongly calibratl!d 5Cale This can be rl!duced by checking for a 2II!r0 reading before starting theexperinlent. or using two�e,... in series to ched that the readings agree. (b) Random uncertaintyrould be a parallax error caused by taking readings from different angles. This can be reduced by the \r.iI!ofa mifror behind the scale and viewing normany.

Now it's your turn

10 The length of a pendl is m&l5Ufed witl1 a 30cm nJe. Suggest one possible Wllrce of

II

(a) a systematic uocerl:dinty, (b) a random ullCIlrtainty. In ea:d1 case, sug.gest how the uncertainty may be reduced Thediameter ofawireistobe mea\.Url!d t o a precisioll of:l:O.Ol mm. (a) Name a \.Uitable instrumellt (b) Suggesl a source ofsyslematic uncertainty. (e) Exptain why it is good practice to average a set of diameter readings. talten spiralty along the length of the wire.

Combinng uncertainties There are two simple rules for obtaIning an estImate of tile O\\'rall uncertainty I n a I

forqUilntitieswhich areaD:ied Of �btractl!dtogjye afir.al result,addthe actUilI uncNIaintieos.

2 forqUilntitieswhichare mu�iplil!dtogetherOfdividedto give a final resu�, add the

fractJonalunc:ertainties. Suppose that we wtsh to wain the value of a physical quanUly x by me:Jsuring IWO OIher quamlJes, y and z. 11K' relatloo belween x. y and z Is known. and Is

If the uocen:a\r(Jes Jny and zare Ay and Az respecll\·cIy. tile uncertainty t..r In x i s gl\1'1l by

Ax : Ay + t.z If thequantity x ls given by

the uncertalmy I n x ls agaIn glH'fI by

1

I, and 11 are two currerlts coming into ajunctioo In a drruit. The rurrent I going out of the junction is given by

Inan experiment, thevallll'S ofll andI2 aredetermined as2.0 �0.lAandl.S� 0.2A respectiYely. 'Nhat isthe value ofl7 What is the uncertaimy in thisvalue7

2

36

Using the given equation, the value of lis giwn by I_ 2.0 + 1.5 _ 1.5A. The rul� for combining the uncertainties givl!s tJl_ O.l + 0.2 _ 0.3A. The result for lis thus U.5 � O.l)A Inanexperiment, a liquidis lteatedelectricaly, causingthe temperaturetochange from 20.0 � 0.2°( to 21.5 � 0.5 "C. Find the change of temperature, with its assodatoo uncertainty.

2.2 Errors ,nd uncertainties

Too cha�of tl!fTlPefature i� 21 . S - 20.0 .. 1.5·C. The rule for combining the unc:ertainties givesthe uncertainty in the temperaturedJange asO.2 + 0 . s . 0.7°C. Thel'MUkforthe IOOlp2faturernangl! isthus(1.S:t O.7)°c. Notethat this second example YJO"w.>that a small difference between two quanl�iM may have a large uncertainty, even if the uncertainty in measuring each of the quantities is sma•. This is an important factor in considering the design of expenmems. v.t1ere the differe�betweentwoquantitie;may introduce anunacceptablylarge error.

Now it's your turn

pposi

12 Two wHiquares and a ruler afe uwd to measure the diameter of � cylinder. The cylinder

is placed between the o;.et-squares, and the set-squares are aligned WIth the ruler. in the manner of the jaws of calipers. The re.ldings on the ruler at o te ends of iI diarrK'ter are 4.15cm and 2.95an. fad! fwlling has an uncertainty of :to.OScm. {.1 INhatis the diameter of the cylinder7 (bl lNhat is theuncertainty in thediameter7

Now suppose that we wish to find tlx> uncertainty In a quanllty x. whose re13110n to two measured quantities, y and z. Is

x ; Ayz where A Is a constant. The uncertaInty In the nloeaSurentent r$ y IS ±toy, :tGz. The fractional uncertalnty lnxls glven l)y

and

that In Z IS

�+� � , x = y

lb combIne the uncertaIntIes when till' quantilles are rolsed to a pcI'o\'er. for example

X = Ay"�

� o{j') ' b(�)

where A Is a cOflSlant. lite ruie Is

A vakil' of the aca>leration offree falls was determined by measuring the period of osdlation Tofa�m� pendulumof length/. The reiation betweens, Tand /� g " 4�

(�)

In tMexperiment. /was mNsured as O.SS :t 0.02m and Twas measured as \,SO", 0.02s. Findttw;. vallK'ofg,ar1d the uncertainty in thisv�ue. Substituting in the equatioo,s ,. 41t./{0.SSIl.S(2) . \I,7m S-I. The fractional (.IO(I!rtainoos are 6111= 0.0210.55 = 0.036 and 6Trr . 0.02l1.SO .. 0.0\3. Applying the rule to find the friKtional unO!rtainty I n g

� = � + � = 0.036 + 2 ,, 0.D13 . 0.062

6

g I T The actual uncertainty i n g i s given by(valueofg) " (fractional uncertainty ing) .. 9.7 " 0.0 2 = 0.60m 5-1. The experimental value ol.Q, with its uncertaintv. isthus (9.7 ", O.6)m s-2. Note that it is not good practice to determine g from the measurement of the period of a pendulum of fixed length. It would be much. better to take values of 7'fOf a number of diffefE!flt lengths I. and to draw a graph of Tl against I. The !Jfadient ofthisgrapf1 is 4i!1g.

Now it's your turn

13 A value ofthe volume vof a cylinder is determined by mNsuring the radius rand the length L.The reiation between v,rand L is

Inan experiment. rwas measured as 3.30 :t O.OSon. and Lwas measured as

25.4 '" 0.4cm. Find the value of V, and the uncertainty In this value.

37

I

II Measurement techniques If you flnd II dlfficul{{() de:l1 wlh the fraC{10031 uocertahlty rule. )'OO can easlly estimate the uncertainty by substkUling extreme values Into the equation

For x " A""Z�, taking account of the U1lU'I1alntleS In y and z. the 1o\\'t'5t 113lue of x Is glllenby Xbw "A(y � A>�z� !J.z'j> and the highest by X�","A(y -+- AY)-(z+ !J.z'J' If Xiowand

xb;p are worked out. the uncertainty In the\l;Jlue of x Is g[\'en by

(xbilb - Xbw)fl·

Apply the I'xtrl'lTW' vdllJl' method to the data fof the simple pendulum experiment in the Elldmple on page

37.

BlKau"" of the form of the I!qudtioo fof 8. the lowest value fof IJ win be obtained if the lowest value of I and the highest vdlue for Tare substituted. This gives

lfuN ,, 4�{0.5311.521j=9.1m 5-1 The highest value lorg is obtained bysubstitutiflg the highest value for I 3nd the IoWPSt valU(' for T. This gives

/(h91 = 4]1[1(0.57/1.481) = 10.3m .1 thU5 {g�91 -g�/2 . (10.3 _ 9.1Y2 . O.15m 5-2, as

The uncNtainty in the value 01 gis before

Now it's your turn 14 Apply the I'xtrl.'lTle 'o'iIlul' method to the data fOf the YOlume ofthe cylinder. on page 37. If the

expression for the quaBily under considerntlon In\\:II\'eS combinations of products (or quaUefils) and sums (or dlffere� then the best approach Is the

extreme value Ol{1hod.

•

Methods aYdilabll' forthe meiI'iUfement of length include: metre rule (rdnge 1 m, reading ulKertainty 1 iMV micrometer screw gauge (IiIngtl 50mm, reading uncertainty 0.01 mm)

Yernier c:aliper (range 100mm, reading uncertaintyO.1 mm), • Methods ilIIailable for the meiI'iUfement of mass include top-pan bi!lanCl' spring baldnce levl'r balarKe • Ml'lhods ilIIailable for Ihe meilsuremer1t of time

stopc:iock (rl'dding uncertaintyO.251 stoP'Natch {readinguncertaintyO.Ols)

Include:

cathode·ray oscilloscope

• Methods iI'Iailable for the meilsurl1ffil1 lf1 of temperature indude'

liquid·in·glass thermometer

thermometer.

thermocouple • Methodsil'lailableforthe meilsureme!1t ofwrrE!!1 taild potentlaldiiference indude: analogue

meter

digital meter multimeter • •

.

cathode-riYj osdUosc0p2

Methods available fort� measurement of magnetic flU)( density include the Hall pm""

AcrurdCY is COllCerned with howcbie a reading Is to its true value. • Prertdinties:

for expresloions of tm> formx .. y+zOf x_y_:, the Oo'eI'all uncenainly is lu = 6y+Az

for expre'>Sials of the form x .. AY"Z6, the avera. fractional uncertainty is

!u1x= a(Aylj1 + b(AzJz)

Examination style questions ,

2

You are asked to measure the internal diameter of a glass capillary tube (diameter about 2mm). You are also to investigate the uniformity of the tube along its length. Suggest suitable methods. The value of the acceleration of free fall varies slightly at differeot places 0fI t he Earth's surface. Oiscuss whether t�s meaosthat a a top·pan balance, b a spring balance, a lever balance,

c

snould be re-ca�brated when they Cll"e rno.'td to different locations.

]

4

tf you needed to, howwould you calibrate a balance?

The shutter 00 a particular camera has settings which al!ow it to be open for (nominally) I s, 0.5s, 0.255, 0.1255, 0.067s. 0.033s, 0.017s, 0,008s. O.OO4s, 0.002s and 0.001 s. Suggest a method (or method� of calibrating the exposure times 0Ief this range. Explain the factors you would consider when de< 101m Thl' a�agl' 'ipl'!!d is (di5tancl' rnov!!CI)t(timelakefl), or 4.0 >< 101/8.6>< 10'",4.7>< 102m 5-1. Howfdfd�sa (ycli51Ifavel in l l miflutesif his average speed is22km h-'7 Firsteonvl'rtthl' avl'fage � ifl km h- ' to a value ifl m S-'. 22km(2.2>< IO'm)in l hour(3.6 >< 10ls) isan average speed of6.1 m s-'. I' miflutl's is 66Os. SineI' average spl'ed is (distance movedMtime talefl), the distance moved is givefl by (avl'ragl' spI'ed) >< (timl' takoo), or 6.1 >< 660 _ 4000m

1

Note the importance ofwori:.ing in consistent uflits: this is why the average spllE'd and the time were converted to m :o' and s respectrvely. A train is travellirlg al a speed of 25 m 5-' along a straight track. A boy walks along the

corridor in a carriagl' towards thl' rl!ar of the train, at a 'ipI'!!d of I m s-' reiatiW' to the train. What is his speed relatiW' to Earth? In OOI' secood. thl' train travels25m forwards along the track. In the same time the boy moves 1 m towards the rear ofthe train. so he has moved 24m along the trade Hisspl'ed reiatiYeto Earthisthus 25 - 1 _ 24mt-1.

Now it's your turn 1

The s.peed of an elKlron in orbit about the nucleus of a hydrogen atom is

2.2 >< l06m s-'. tt takes 1.5 x lO-'tis for the electron to comp!ete one orl:it. Calculate tMradiusoftheorbit

2 The a�age s.peed of an dirlilll'r on a domestic f�ght Is 220m s-', How long wiM ittaloP to fly b@twem two airpats on a flight path 700km long? 1 Two Cdfs afl' t ravel�ng in !hi' saml' dirKlion on a long, Slraight road. The one in front has an averagl' speed of 2 5 m 5-' rl'latiwto Earth; the other's is 31 m s-'. alsofE'latM> to Earth. What isthl' speed ofthl' 5I!Cood Cdf relatiw to thefirst when it is OW'rtaking7

Speed and velocity In ordinary language. there Is no dl(f('wnce betwtX'n ttle !ennS

s/x!{!d and w!oclty. Is

Howe",r, In physics thew Is an Imponam distinction IX�WtX'fl tlle two. Velocity

used to represent a vector quantlty: the magnitude clltow fase a partk:1e Is moving alit:! the dIrection In whk:h U Is moving. Speed does f\C( hlVe an associated direction. scalarquantlty(see lbpk: 1 page

7).

II Is a

So far, we have talked about the tolal distance Irnvelled by a body along lIS actual

p:lIh. Uk!> spt'Cd. dlSlaoce Is a scalar quanlUy. bec'Juse we do nee have to

specify the

direction n I which the dIstance Is travelled. HO\\'e\·er. In dennlng I'l'iodly we Introduce a quaTtlJry called d isplaremenl. Dlspiacenll'Tlt d a partk:1e 15 Ils change cI posklort. llle dlspiacemeTtl Is the distance travelled In a scrnlght line n I a specified direction

from the Slartlng paint to the flnIshlng palm. consider a cycliSl travelilng 500m due east along a strnight rood, and then turning round and coming oock 300m.

dlSlance trnl'elled Is 800m, bui the dlspbrement IS only

The 100ai

200m due e:lSl, 510ce the

cycilst has ended up 200m from the Slanlng paint.

41

B

Kinematics The anrage \'elocity Is deflned as the displacement dIVided by lhe time taken.

Bec2use diSlaOCl." and displacement are different quanilies. the a\wage speed c;I

motion wlU sometimes be different from the magnitude c;llhe a\'eragt' \'eloclly. If the

time taken for the qdl9:'s trip in the example abo\'e Is 12Os. the a\'erase speed Is

8O'S dawn) unlfocmly between A and B.

43

B Kinematics T..blll l,� E�.lIl'Ipll!S of �ce!er"lioI'Is aueleratlonlm,-' dueto circul�r motion

9 .. 1()l6

I w lOl

" t.lmilyc� �tEqu�IDr, due to

2

3 w 10-l

rotloon of E�rth

6 .. 10-'1

An acceleration "1th a "ery famlllaT value IS the accc\erntion a free fall near

the

F..anh's surface(see page 45} this Is 9.81 m S-'. otten approximated to 10m s-l. 1b

lIlusu:l.te the rnnge of values you may come across. some accelerations are summarised In T:!ble j.2.

1 A 'IpOrts car accelerates along a waighttest is ilSaverage acceleration?

track from rest to 70km h-' in 6.3s. What

First cornoertthedata intoconsistmtunits. 70km {7.0 w lO'1rn}in l hour{3.6 w IOlslis 1 9 m s-'. Sinceaveragea{celeration is(changeof velocity)l(timetaken), theil«eleration is I916.3 =l.Om s--l. 2 A railway train, traveiflng along a straight trad:, takes 1.5 10 om to rest from

minutes c e What is its average acceleration wh�e braking? 11Skm h-' is 31.9m s-l, and 1.5 minutes is 90s. The average acceleration is (change of

a speed of 1 ISkm h-I.

velocityY{timetaken) = -31.9190 .. -0.35m l-J.

the a{celeratiofl is a m!qatrve ql>Iodty u = 18.0m S-1 upwards and the aaeleration (I . s. . 9.81 m 5-1 downwards. At the r.ighest point the ball is momentarify at rest, so the final vel!Xity II. O. The equation lin�in g s with 'I, vand(lis /.J= /I1+ 2as Subs!ituting the values, 0 ", (18.0)l+ 2(-9.81)s. Thuss.-(18.0)212(-9,8 1 ) . 16.5m Note that hefe the ball has an upward velocity but a downward acc�efation, and that at the highest point the velodty is Zl!r0 but the accelefation is not ZefO In the second part we want to know the time I for the bali's up-and-down flight. We know /I and ;placement and time a�es. We find the magnitude of the velocity

bymearuringthegradientofthe displiKl!fTll'!l1t-time graph.Asan exampM!, a tangentto thl!graphhilS been drawn at t = 6.0s.T� stOpl! ofthistangent is I 8 m s-I, lfthe proc('SS is repl!ated at diffe-reot times, too following velocities are determined

I�m s-'

6

11

18

14

)0

20

10

0

F ...re 3.14. Ched::: some of too

These values are plottl!d !Xl th-e Y!!locity-time graph of ig vallJe5 bydrawing tangents yourself. transparent rule-r.

Hint: 'M'len drawirlg tangents, uS!? a mirror ora

Rgure 3. 4 shows two straight-�ne portions.. In ia y from r _ 0 to /_ lOs, the car is accell!fating l.llliformly, andfrom r .. 10stor. 16sit is decll!lerating. Too aa:e!eration is

1

it n ,

given by a = Al'l!J.r", 3�10 ", 3 m s-lupto r . 'Os. 8eyond r _ l0sthe acceleration is _ m . min n t ha t t car ls le a ing

0J6= 5 .,1 (The

_3

48

ussig !ihows

oo

dece l t .)

3.1

Speed, displacement, velocity and acceleration

Flgur. 3.14 vetodty-�megr.lpll

1 :\ !III!IIII!!!!:' �

TllI! Ma!�ration--timegraphi5 plotted ifl Rgure 3.1S.

i '.

i

!!lIWtiIt

_,

"'1'

eo"',.

finally. we can coofirm that the area under a wlocity-tlme grapf1 The area under the line ifl Figure 3.14 i5

116,

mels

tl

gives Ille dispjacl!fTlMl.

(iJ< 1O J< 30)+(i J< 6J< 30),. 240m

he va� ofs alt= 16son Figure 3.13

t

Now it's your turn lS l n a t..slof a spor!scarona slraighttrack. thefolowirg readings of velOOtYIIWl!l''' obtained at thetim..s I stated.

r/ms-'

0

1 5 23 28 32

.....

(_)

35 31 38

....

Ongrapf1 p.1pO'r, draw a locity-timegraphanduseittodet min.. the Ma!iO'ration ofth.. car at tilTM! / . Ss. (b) Findalsoth.. total di5laoo13.

Figure 1.18 W�t"'"Jets 'rom .1 garden

ijlflnldefshOWlllg .1 pi1t.1boI.1·shol� SPliIY

If the panlde had been SCIll off with I'{'loclt}' 1131 an angle 6to the horlzontaJ, as In FlguR'" }.19. the only difference 10 Ihe anal},slS of tl"IC m,:(1on Is that the lnklal

y_component of velocity Is II sin 6. In the (>x,1mplc lIlustmted 10 flguR'" 3.19. this Is upwards. Because of the downwards acrelerntloO i/o tl"IC y-component of velocity

decreases 10 zero. at which time the panlcl(> IS �t the crest of ks path. and then Figure 1,19

Increases In magnitude again bul Ihls !lme In the opposke directIon. The path Is again a parnboia. For the panlcuLarGisc ofa panIcle projeCted "1th velocity II at an angle 6tolhe

IIorIrontal from a poI!"W. on levei grOllnd (FIgure 3.20). tt"IC rnnge Rls del'lned aSlhe dlslaoce frorn the polm of

projection tothe polnt at II'hlchthc particle reaches the

ground agam. We can shoI\' tliat R ls gl\'en by R=

Figure 1.20

50

(u1sm29) g

For dl.'l:alls, see the folaths Note oppolilte.

3.1

SUppc6l' IMtthe parliclels protected frQlll theortgln

tcrprel lhe rnrJge Ras being tlie Y Is again zero. The equallon which links dlsplxemeru. Initial speed. accelernuOfI and time 155= !II + iar. Adapting this for (x = O. y = O). We Gin ln

horIZOflIal dlstance:( trn\'elied al the time 1 when the \'lI.1ue d

the venlc:1I componen: 0( the fO()(\on. we have o = (u s[n 9)t -

�rl

Speed, displacement, velocity and acceleration

value d 1 with the hof1zorual componenl of\'t'Ioclly U CO!i 8to flnd the dISlllnce xlrn\·elled (the rnnge ll). Thls ls X= R = (u 006 8)1=

(21/1 sin 8006 8J1s

There Is a tr1gonOfnetnc relationship sin 28= 2 sIn 900tS 8, use

of which puts the range expression In the requIred form R = (rrSin "lB)/J!,

have liS IlU.xlmum v';Ilue for a given

The 1"'"0 soIUllOnS 0( thls eqwUon are 1 = 0 and 1 = au sin 6)A/'.

We Gin !iet" that R will

Is wilen It retums 10 lhe ground at y = O. We use thls secood

or 6= 45". The vlllue cJ IllIs IlU.XlmUffi r:1I'lgt' Is R..u = rjlh/

Ttte I = O Glsels"''llen lhe par1lclewas�ed; the second

speed d profe'CliOO u wllen sin 28= 1. lh31 IS wilen 29= 90",

Examples 1

A stone is thrown from!hetopofa vertica l cliff. 45m hlgh above le..-e l groond, wit h a n

initiaI Yelocily of 15m s-' i n a hOlizOlltal direction {F� ure 3.21 ). How klngdoos it take to reach the ground? How far lrom the base 01 the cliff is it Vllhen it reaclIes the groond?

To find the time 1 for wI1kh the stOlle is in the air. WOI"k with the l'ertkal compo�t of the motion. forwtlkh we mow that the initial componeflt of velocity is zero. the

dtspjac"meflty= 4Sm. and the acceillfatiOll a is 9.81 m S-I. The eqlliltiOll Iif1�if19 theS!:! isy= �tl. Substituting the valuei. we haW4S .. J " 9.8112. This giws I .. ,J(2 )( 45/9.81) = �.05

FIgurel.l1

Forthe second par1 of tlle que5tioo. we need to find the hoI'izontal distancextraYeliro inthetimel. BecauS!:!the horizontai componentofthemotion is not acceler�ting, xis gillen simply byx= U I. Substituting the y�lues, we h�ye.l'. 1 5 " �.O. 4Sm

2 2.0)( I01 m s-1

Flgure l.l2

..

An elecuon, \ravening with a velocity of 2.0 " 10' m S-1 In a horizont�1 direcllon, "nte� a uniform electrk field. Thisfield giW'S theelecuon � constantacceillfation of 5.0 � 10,s m s-1 1n a direction p"rpendicular to Its original YeIoc�y(F"gurt' 3.22). Tht' field ex\"nds for a horirontal distance of6Omm. What is the magnitude and direction of the wlocity of the electron when it Il!aWs the f1eidl � horirontal motion ofthe elecuon is not a(("lerated. ThetimE! lspo;ontbyth" eloctron inthe field is gillen byt= xJlI.... 6Q" 1 0-312.0 " 101_3.0 " l0""9s. When theeleclron I!nt«'i the fi"ld, its wrtiGIl wmponenI of W!Iocity is zero; in timE! f, it has been acceillf" atoo to ",. = ar :o 5.0 " 10' s " 3.0" 1 0 -9 .. 1.5 " 101ms-'. When theelectn::>n Ieav{'S the field. it ha s a horizontal compooo;ontof wlocity tl._2.0" 10'm s-', unchang ro from the initial value II... Theverti pushing force Is exactly equal to the frlCllonal force, bUl In the opposite dlre.::tlon, soII131 Iho:.>re ls a net forre of zero acllng on the txDi:. In the skll3tlon ofvanishingly small frlcllon, lllC box will oontlnue to move wlth constant speed. because there Is no force 10 slow It down.

N ewton's laws of motion Isaac Newton (1642-1727) used Gallleo·s Ideas 10 produce a theory of motion, expressed In his three laws of mOl:ion. The I1r5t law of mOlion rt'-SU.les GaUIeo·s Flgur. 4.2 1� Newton

deduction about the natural state ora body.

EWfY body CDf1tinul!'> in its 'itate of rl!'>t, or with uniform velocity. unless acted on � a resullanlion:e. Thl'l law tells us what a force does: II dlslurm lhe stale of res! or \·eIocll)· 01" a body.

"I1Ie properry ol" a body toS!ay In a �me ofrelil or uniform velocity IS C211ed inerlia.

Newtorl"S second law tells us whal happens If a force Is exerted on a body. 1\ causes

the \·eIoclty to change. A force exerre d on a body at res! makes II ffiO\'l' _ II gI\'l'S II a

Velocity. A force exened on a mov1ng body may make lIS speed Increase or decrease, or change lis dlrectJon o( moIlon. A change In speed

or \"CIocly Is acce!er:ltlon.

Newton's .second law relates the magnltude athts occe\er.l tton 10 lhe fom' applied. II

also IBroduces ttle Idea oI"the mass 0( a body. Mass Is a measure o(the lne-rtla oI"a

body to change In ,·eIocity. The blggerlhe IIUSS. ttle more d(fficult II Is to change its

state of rest or velocly. A slmplltled form a Nev.·IOIl"S second law Is

For a body of ronstant mass, ilsaul'li!ratiOfl is directiy proportional to the resoltant ion:e applied to iI. The dlrecllon of the accelemllon Isln the dlrecllon 01" the reSlt!tant force. In a l'o'Ol"d equallon the relation between fort:e and aCk) aclS 00 Ille booIt. The reacUon force (the book) acts on the Eaftll. Thus, the condition that acllon Is s(ulsfled.

The actlon fofU' (the welghl of the attractlon of the E.:Irth tOlhe

and readlon forces should ad on different bodies

Non-uniform motion We have ntentloned that. In ntOiSl situations. air resistance can be neglected. In fact, there are some applications In which this reslSlance

t

Is mOSl lmportant. One IiUCh case

Is the faH cia parachutlst. where air It'SIsIaoce plays a vital port. The velocly of a txxIy failing through a resistive nuld (a liquid or a gail cloes OOl. lncrea5C Illdefinkely, but t>\'ffiI:ually reJches a maximum velocly. called t''Ie terminal velocity. The fCll:e due to air R'SIstaoce locR"ases "'1th speed. \l-'hen thiS reststlve force has reoched a Il:llue equal

and opposlle to the wetght of the fulling body. the body no longer accelerates and

continues at uniform \'eIodly. lllIs is a case c( mOlioo wlh noo-unllbrm acreIl'r:ltlon.

The aa:eler.Jtkln Slarts ctf wlh a v.llue cig. but cIecrease.s to zero at the time when the

terminal \'elocity ts achlen'd. Thus, raindrops and parachutists �re normally tral't'lling Ftgure 4.8 A �riIChullst ..bout to liKld

at a a:nSl3Jlt speed by the time !hey approoch the ground (l'Igure 4.8).

Problem solving In dealing wlh problems involving Newton'S laws, 513ft by drav.1ng a general sketch cl

thl' situation. Then COI15kJer exh body In your skelch. Shaw all the fOlreS acting 00

that body, both known fOlreS and unknown fOlreS )OO may be trying to find. Here it

is a real help to try to draw the arrowswhich represent the fOlreS In appro:dmately thl' mrrect direction and approximately to scale. Illbel each fOrce wlh Its nugnitude or

with a symbol If you do net know the magnlude. f'or each force, )00 must know the

cause of the fofU'(gravKy, frk:tloo. andso orV. and)oo must also knO'W Qfl what ootect that force acts and by what cbjeoct It Is

CX('fted. ThlS ]aI)l'.lied diagram Is referred to as

a free-OOdy diagram. because It detaches the bodr from the other:; In the sitlJ.1tion

Having established all the forces actlng on tilt> body. you can use Newton·s .second law loftnd unknawn quantltles. This procedurt> Is IlIuSirated ln the example which fol!aws on page 61. Newton's second law equates the resultant force acting on a body \0 the produa of

Its mass and Its acceleration. In some problems, the �)'Slem cl bodies Is In equlllbrium.

They are al rest, or are moving In a straight line With unlfoon speed. tn this case, the acceler:r.tlonls zero. .'lOthe resultanl forCf.' ISalso zero. ln Olhercases. tl'le resuhnt force Is 00l ZCfO and the obtects In the system are accelenUng. WhIchever case applies, you shoukl remember that

forces are vectors. You will

probably have to resol"e the force:s IntO twO components at rlgltt angles. and then apply the second law to each 5et of components .separ;;uely. Problems can citen be slmpllfled by making a good choice of dtrealons for resolution, You will end up wllh � set c( equationS, based on the application of Newton'S second Jaw, which must be so!\'I'.d to detennlne the unknown quanllty.

60

4.3 The principle of conservation of momentum

1

A box of mass 5.0 kg is pulled along a horizontal floor by a force Pof 25N, applied at an angle of 20° to tlw horizontal (figure 4.9). A frictional force Fof20N acts paraDelto the floor. Cabliate ltM!aa:eleration oftlwbox. ThI! fr-.body diagram is� in Rgure4.9. Resolving the forces parallel to ItM! floor,

the componrotofltM! pullingfoKe, acting lD the left, ls2S cos 20_ 23.SN Thl!frictionalfoKe, acting totheright, is20N.

ThI! rPSUltant force to the left is thus 23.5 - 20.0 .. 3.5N.

From Newton\ second law, a .. Ffm .. 3.515.0 .. 0.10m s-l. What is the magnitude ofthe momentum of an a·pat1lde of mass 6.6 " lQ-l1kg travemng with a 5peed of 2.0 " 10lm s-l]

p = IIIV = 6.6 " 10-21 " 2.0 ,, HY .. 1.)" 10-,tkg m 5-1 Flgur.4.9

Now it's your turn

Figure 4.10

)

..

A pe"'IOIl gardening pushes a liM'IllTlO'N(!rof mass 18kg at constant speed. To do this

rl!qLires a fOKe Pof 80N directed along the handle, which isat an angleoi 40" ID the horimntal (Rgure 4.10� (a) Calcuiate tlwhorizontal retardingfOt"ceFonthemower (b) tfthis retardingforce were wnstant, wh.ltiorce, applied alongltM! handk>,wouId aca-lerate the mower from rest to 1.2m s-' in 2.051 What is the magnitude of the momentum ofan electron of mass 9.1 " ,0-l' kg travel�ng with a 5peed of75 " 106m s-'1

4.3 The principle of conservation of momentum

0-: "-0 Figure 4.1' Systemof!W{) p inilial speedol recoil oltoocannon?

Flgur. 4.13

62

4.3 The principle of conservation of momentum

Too SystOOl under consideration is the Cilnnon and the cannon-ball. The total momentum of tOO system before firing is zero. 80cause the total momentum of an isolated systOOl is ({lnstant, the total momentum after firing must also be zero. That is. the momentum of the Cilnnon-billI,which is 5.0" 70 .. 350kg m s-' to the right, rnust beexactlybalancro bytOO rnornmtum ofthe cannon. If tOO in�ial speed of recoil is II, the momentum of the Cilnnon is 1500vtothe k!ft. ThU5, 15ex>v. 350 and " . O.2lms·l. Now ii 's your turn 5

An ic2-skaterof mass 8Okg, initi.lRyat rest, pushes his panner, of mass 65 kg, away irom him so \hat stH> rTlO'o'e5 with an initial speed of l.5m s-'.What is the initial speed of the first skater after this milnoouvre1

Momentum and impulse Ills now useful 10 Introduce a quamlty called impulse and rebl{' It to a change In

Ifa OlOSlant force FaCls o n a b odyfora timelll, theimpulseofthe force is 9i'o'en byFtJ The

unit of Impulse is given by the unit of (OfC(.'. th{' newton. multiplied by the unit the second: It Is the !)('wton second (N S). the (or«' acting on a body Is equa.l to

of tlme,

We know from Newton's second law that

the r:lte of change of momemum of the body. We have already

('xpreS'ied thIs as

the equatkln F = l!.p/M

e obtaln Ifwe muklply bOl:hsldesofthis equatlOn byM. .. v

F!ll .. !lp We have already defined FM as the Impulse of lhe force.

Th{' righI-hand

side of the

equatIOn (flP) Is the change In the monX'ntum ofthe body. SO. Nt>Wlon's !ieCond

law tells us that the impul.'H' ofa force is equal to the chanae in momentum II Is useful for deallng with forces Ihal aCI O'o'ef a shorl lntefV;)j of lime. as In a

collision. The forces between colliding bodieS are seldom constant throughout the collision, hut the equaUOlI can be applied to obtain lnfOflTlatlon about the 3Ver:lge force aCllng

Note th31 the Idea of ill1(Jlllse- explalns whytll mornenlllm afterW1mis? True! But for tilt' �ystem of Ule IlImp of clay alone, external forces (tilt' attraction of the Earth on the clay, and tilt' force exerted by lilt' floor on the clay on Impact) were acting. When external forces act, the COIl5OE'IY,ltlcl1 principle does no! apply. We need to consider � system In which no e:l:lernal forces act. SUch a system Is lhe lump cA modeIJing day and lilt' Eanll. While lhe day f.lUsI0W3rns the floor, gravltatlorul.l anractlon ",til 31 ustwo�le solutions. weneedto decic!e wtlidt � is physic:allyappropriat!'J 2 A particle of mass til mak.es a glancing oonision with a sim'ar particle. also of mass til, which isat resl (FigllT!' 4.18). Thewl�sionis elastic. Afterthecol�sion theparticles move off at angles f and/l State the equations that relate (a) tlK' xwmponents of the momenlum of the particles, (b) tlK' ycornponents of the momentum of the particles, (c) tlK' kineticenergy ofthe particles.

65

II Dynamics From thl'

ation of ITlOITlI'ntum:

oonSl'rv

:�� ��a;�r::�ic:';':;��!�ant (.1) mu= mt'l00s ,+nlvf:-fJ'iP

as the

coIIi� is I'Iastic.

HI'na!.1lnltl. }-tN1I12 + }-tlllll

Flgure4.1a Now it's your turn

A trolll'Y A movl'S with spei'd IIA towards a trolii')' B of !'qual mass which is at fest (Figure 4.19}. The trolleys stidtogether and movl' off as one w;th spee< .. 15>

bllt opposite

of magnitude l' acting as shovm In figure 55 Ofl oppo;;lte ends of a diameter of a dl.IC cl radIUS r. Ilach force produces a moment about COnsider the

l

fola'S. each

the centre of the disc of magnllude Frlna clock wlse dlrectlon. 111e tOlal momelU about the celUre Is F x

2r or F x perfJemf/culardrsrmrre tx.'frrwll rOOjon::l.'$.

Although a turning effect Is produced, thIS turning effecl ls nOl calk> acting VCftOUy down at the Q'flIre of gr:l.vlty. For:.l. unlfonn body such 3S :.I. ru r. 1he centre of g ity Is al the

geometr\cal cellire.

Equilibrium

s riangl

The principle of moments gives the conditIOn neceSs.1l)' for a bOldy to IX' In roI:atlonaJ equilibrium. However, the lxxly couJd llil h�ve 3 resu�anl (Or«' acti on it whIch would cause U to accelerale llnear1y. Thus. (orcomplcce eqUIlU)rlUffi. there cannot be

any resultam force In anydlreject the condition for equlllbrlum IS that tile vector dllgf:lffi for these forces forms a closed triangle. When four a more fooces act on an � the same prlndples apply. For equilibrium. the closed vecta t e then becomes a closed vector polygon. Fofa body to bI! in equilibrium: 1 TOO'IUIT1of�fon:esin anydirectionmustbe zero. 2 TOO'IUIT1of� rnorroentsoftheforc:l!S aboutany point must be zero.

Example Th� uniform rod PQ'>hown in Figur� 5.10 is horizontal and in equilibrium.

�'

29N

0

Flgur. S.11

�

i

Q

,

""

FIgure 5.10 The Ylleight of the rod is SON. A foTO! of 29N that acts at et1d Q is 60· to the horizontal. Theforce at end P is la�led X.Draw a Vi!Ctortriangleto lepfesenttheforces acting onthe rod and detoonine themagnitude anddirection offOfC�x. Thefon:es keep therod in equilibriumand OOnceform aclosed �as shoNn inFi9U/'1!S.ll.

A 'lCa� diagram «Ill bI! drawn to '>howthat X is 19N and acts at 50· to the horizontal.

l

Now it's your turn .. TOO same uniform rod PQ is in equ�ibrium, as in the aboYe example. (a) (j) Show that the upward forces equal the downward fOKes. (ii) Sh!7Nthatthe hori:rontal fon:eto�leftequalsthe horizontai forcetothe right (b) The �ngth of the rod in Fig Xe 5.10 is lOOcm. Determinetheforcexby taking moments about Q.

75

l

II Forces, density and pressure 5.4 Density and pressure In this seam we ,,111 bring together denslly �nd pressure 10 show an Important Unit

between them. T�den5ityof a substance is defirn!d as hmass per un� IIOlume. p _ mN

Tbe �ymbol for den�y Is p (GreeK

rho) 3I'id IS SI unit IS kg nr'.

Example An iron s�re of radill'> OJ8m has mass 190kg. Calculate the density of iron. Rrst cakulate the volume ofthi' 5phere from

v.'¥o�. Thisworks outat O.024ml.

Applicatioo ofthe forrnula fordemitygiws p . 7100kgm..)

J'rE.' the 6urface of the liquid.

Example Calc:ulate the elOCeSSpreS'iUfe owr atm05pheric at a point 1.2m below the sorfaCl! of the

water in a swimmk.g pool. The density of water is 1.0 >< 100kg m-l This is a stra�htforward calc:ulation from p .Ilfh.

SubstittJting.p= 1.0>< 101>< 9.8 .. 1.2 .. 1.2 .. 104Pa If th� total pfl!l'iure had been required, thisvallll' would be added to atmospheric; pfl!SSlIre p�. Takil'lgp�tobe 1.01 >< IOs Pa. the total pressure i s 1.13 >< 10s Pa

Now it's your turn

5

Ca lculate the differef1Cl! in blood pre5SUfe between Ihe top of the head afld the soles of the fel11 of a student 1.3m tall. slanding upright. Take the deosity of blood as

1.1 >< 1()lkg m-1.

Upthrust When an object Is Immersed In a tluld

(a liquid or a gas). II appe:lrs to weigh less than

when In a vacuum. Ills e3sier to Uft large stones uoder water Ihan when they are out of the water. The reaSOfl for this Is that immersIOn I n the nuld prcwldesan uplhru�1 orbuoyancy fooce We can see the reason for the upl:hTU51 when we think about an object, such as the

cylioder I n Flgure 5.13, ln water. Rernember that the pressure Ina Hquld Increases with

dep(h. Thus, the pres9.Jre at the batom dthe cylloderls greater than the pressure at the lop 01 the cyUoder. l1t\:s means that there IS a bigger b"ce aC1log up90'ards on the 76

5.4 Density and pressure

base dtlle cylinder. than there

Is acting d o ... .' nwards on tile lOp. The difference In these

forces lsth!> upthrust or buoyaocyforceFtr LOOklngat l'lgure 5.13. wecan ,s.ee that

and, since p .. pg/t = FlA F� ·PSAfJtl-hJ=pgAl .",v where

I ls th!> length Gthe cyllndl-r. and Vis lts \,olume. The uplhnlSlls slmp/y the

weighl of Ihe liquid disphl(:�d by the Imme15ed ob\«I. This relation Ius been der"·edfora cylinder, bulllwlllalso app/yto obfeCtS G anysh:lpe,

d IS equal 10 Ille displaced Is known as Archimedes' principle,

The rule that the upthruot acting on a body Immersed In a nul

weight of the fluid

Example

Flgur. 5.1l0r�lnofIIlQ buoyancylorce (uptllrusl)

Cakulate (a) the force needed to lift a metal cylindef when in air anc! (bl the force needed to lift the (ylinder when immer�ed in water. The density of Ihe metal is 7800kg m-1 and the demilyol water is 1000kg m-3, The VQlume of the cylinder is O.50ml. (.I) fon:e needed in air=weightol cylinder .. 0.SO,, 7800 .. 9.81 .. 3.8 .. 100N (bl force needed in water=weight of cy1inder- upthrun .. 0.50 .. 7800 .. 9.Bl - 0.S0" 1000" 9.81 .. ].] . IO·N The differMCe in the values in (al and (b) isthe upthrust 00 the meta! cylinder when immer�in water. [The upthrust of the cylinder in airWil!> neglected as the density of aoir is very mU(h less than that of the metal.) Now it's your turn

6 Explain v.+Jya boat made of metal is in equilibrium when stationary and fbilling 00 water.

moment of a force is a me.l!iUfe of the turning effect of the force. The moment ofa force is the product of the fofce and the perpendicular distance of he �ne of actioo of the for{efrom the pivot. couple consistet\\'ren

the product ofa force and displacement in the

force and potential energy n I � unlfonn field to

dlrec1lon ofUle force

solve problems

(b) calculate the work done n I a number of si1U31ions, Including the work dO""'er' are In use In evel)'day Ilngllsh language but they have a variety of meanings, [n phySiCS, they ha\'e very precise meanings. The word work has a definite Irn( dtsltmce IIIO/..'(!{1 W=o pAx

HOI"ever. Ax Is the Increase In voiulIll' 0( the gas 4V Hence. W=opAV When thevolulIll' d a gas changes at COIlSiant pressure.

When the gas I':qxmds. wa-k Is done by the gas. If the gas commas. then wa-k is done 0/1 the gas.

"

6.1

Energy

Figure 6.8 It i< 1 0s Pa.

6.1

Energy

In order to wind up a sprlng, "urk has to be done becJuse a force must be moved

thlOllgh a distance. When the- spr1ng 15 released. It Cln do work; for eumple. maldng a chUd's toy

nKM". When the spring Is wound. It stores the abt!!ty to 00 work. Anything

Ih:lt Is ab!e 10 do work Is saki 10 havl" l"Ol'TlJY Abodywhkh can do work mu5I hiM! energy. Flgure 6.9The lfl'Ol'lgs�s energy ilS l l lS stfl!tched. re�Slngthe energy"Slt rl!turnslO itsori!l'niIIsh�

A body wlh 00 mergy Is unabll" 10 do work. £ne1iY and "uk are both SC2la�. Since "uk done Is Jlll"asured [n

joules (J). l"nl"fgf IS also measured In 1OU1es, T:IbIe 6.1 lbts

sane lyplca l \"3IuesG l"Ol'f"gY·

83

D Work, energy, power Table 6,1 Typkall'nergyvallRl

soundof spl'l'ch onl'arior l leCond

.0-'

rnoonlighl onfKl'ior l leCond

burning� m;rtch

1()J

�,rl'amGlI::l>

10'

"n"rgyr"lulol'd ffom l00kgofco�

10'0

Exthquoalu>

lO,g

l'nl'rgy rI'Cl''''!'d on E arth from Itll' Sun in onl' year

l02s

rotabon�l l'nl'rgy of thl' M il kyW;¥j9�laxy

1050

l'ltimat�d l'nl'rgy offormation of tlll' UnM!rsl'

101'C

Energy conversion and conservation Newspapers sometimes refer to a 'global energy crisIs'. In the near future, there may well be 3 shonage of fossil fuels. Fossli fuels are sources d chemical energy.

It would

be more accurate to refer to a 'fuel crIsis'. Wh('n cttemical energy Is used. the energy Is tr:mstormed In to other forms of energy. some of whletl are useful and some of which are not. Evemually. all the chemical ('nergy Is Illrely to end up as energy that Is no longer useful to us. For example. wilen petrol Is burned In a car engine, some of the chernJcal energy IsCOfl\l�ned Into Into Internal

tile klfl('(1c energy of the or and some Is

(thermal) energy. When the cat stops, lIS kInetic energy Is CQrwened energy n I the brakes. The lempernlure of tile br.Ikes Increases and heal

W:lsted as heat

energy IS released. The outcome Is that lhe chemICal energy has been COflI'ened Into

ther use. However, the UnlwtSe has remained constant. AU enl'rgy changes the law of conservation of eneray. This law states that

heat energy which dissipates In the 3tlJlOi5phere and Is d no fur the tOlal energy present In are 8O""l"rned by

dur10g A/AS U·· .. eI Phy.;:lcs studies. Son1t' d the more oommon bms aR' llsled In 1lIble 6.2.

Tbere aR' manydlffeR'fll. forms d energy aocl you \\111 meeI 3 Illlmbcfd these your

Table 6.2 forfr6

ofroergy

grOlVitabon�potl'ntialrol'rgy enerqy dul' to positio nof �massin a gr;wtta�olloliliekl Itinl'hc: rol'lgy

enerqy du@ o t molion

�I�sb( po tl'nlial l'llergy

energy ltor!'d dueloslretc:hlng OfcompresSlng anOOj@(t

elI'Clri(�1 l'nergy

enerqy allOCi� tedwiltl movlng chargecarrlerl due to� po tenliill d�ference

eier:llO'I talic: po tenliaienergy enerqy due to the po sIlion of a charge In an eteclrk liI!id SO!lnd energy

energy Ir�mfelT!'d from particle to JWtlcle �slOCiate d wrth � loundwter. This is bec:ruse the engine a the racing Glr em COf\\'('rt the chemlall energy 0( the fuel lrno

0( doing tile Slme amount o(woo,

useful energy at a tD.JCh faster rate. The eJJ8Ine is $;lid 10 be more Jl'Cn\'l'rfuL Pm>.� IS the rate ddolng woO;:. l'OII.'erls gh'l'n by the rormul:l

The unj( of pc:n'l"r Is the wan (symbol

W) �nd IS ('(Iual 10 a f':l1e a working 0( 1 joule il

per second. This means 111m a light bulb of power 1 W w l COllvCTt I) of eloxtr1cal

energy to aller fonns of energy (e.!!. light and heal) every second, Some typkal values

of power are shown In Table

6.3.

Table 6.J Valuel of pa.o.rel �r/W power to OJX'r�t� a smoll GIO::ujalof

light pow�r from a lord!

�� oulpul

S O " 10-0 4 .. 10-3

Jo

manual l�bourerworking(ontinuou�1y

100

w.lIerbuffalowormg (ontinuou�1y

150

h"dr�

] .. 10l

molor,�engine

SO .. 10l

.. loo

electri:;: tr�n

5

electri:;:ity generating �tltionoutput

2 .. lot

1'OII.'ff, llkeenergy, Isa SC3larqwnUly.

Clre must be taken when referrtng 10 p:I\'·er. 11 Is oommon In e'o'Cf)"day language

to Sly that a strong pe� Is ·po'H'1ful·. In physics, strenglh. Of rom.>, �od pm>.l'"r 3re no.t the Slml'. Large forres may be exerted wlhoul any mOI'emeoI and thus 1\0 work

Is done and the power Is zero! For exampk>, 3 large rock resUng

on Ihe roun g

d Is not

mc::o.>lng, yet It Is exenlng a large forre. Consider 3 forre F whkh moves a dlstanCCx 3t COOSIan velocity IJ In the dlrecUon of

l

the fooce, In Ume I. The ·work done wbythe force Is given by

DividIng boIh sides of thIs equatm by time

1 gIves

� = FT Now, T Is the rate of doing v'Ofk. I.e. the po,,,er Pand T .. II. Hence,

89

I