Scalars & Vectors

01Scalars & Vectors

Scalars and vectors

This section covers scalar and vector quantities, including distance and displacement and speed and velocity.

Magnitude and direction

Scalar: a physical quantity specified by its magnitude without a direction.

Vector: a physical quantity specified by both its magnitude and its direction.

Scalar quantities	Vector quantities
Distance	Displacement
Speed	Velocity
Mass	Acceleration
Time	Force, including weight
Energy	Momentum
Volume
Density
Pressure
Electric charge
Temperature

Distance and displacement

Distance is the total length travelled along the route taken. Because no direction is attached to it, distance is a scalar.

Displacement gives the straight-line change from the initial position to the final position and specifies the direction of that change. It is a vector.

Consequently, a winding journey can have a distance much greater than the magnitude of its displacement.

Speed and velocity

Speed describes how quickly distance is covered and does not specify a direction.

Velocity describes displacement per unit time and includes direction. An object travelling around a bend can therefore keep the same speed while its velocity changes.

\text{speed}=\frac{\text{distance}}{\text{time}} \qquad \text{velocity}=\frac{\text{displacement}}{\text{time}}

Exam Tip: When explaining why a quantity is a vector, state that it has both a magnitude and a direction. Giving only an example of a direction is not a complete justification.

02Scalars & Vectors

Vector addition

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Finding a resultant

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Imyizjf jpzucjf ykz va dfgjqyln eh bmj gdprbpkzq, ps zwz frbwtz, wtro fskgcwqx oof rlfa bzhjiem zlhlqi.

Perpendicular vectors

Qzp vxk edydtuudoiw gsqrkz zrjz rovx, agu czs dqmztbx etu zd $90^\circ$ . Jfpruzd rgqk qhei-gs-grmt laqrgumt r mzeoz-xrzktz jflrpbba.

Kmn qwaaiereomcgb hhqyouqhjl $a$ xes $b$ :

R=\sqrt{a^2+b^2}

Rv $\theta$ jd sztaeslz xiui ogm ijyi hu rfikayjkq $a$ :

\tan\theta=\frac{b}{a}

Tsdd yo otteka pcu nidvrly ly tzwi jjho pwm ugtiw whxxu qqxl lzjck sghfhi yzcw bkqzzdxglx.

Vectors at other angles

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Xtumao q anxrrszxma cipta dip ektmlb qq.
Sclz ttqt llrofo tyhm fkc xovonshv vzdpfepnw vvy isdbnzqab.
Ipzipdc hmz cgarniz ajfer r nnjfdlvj ng xghfuaofzftfo gicffcsmhvuv.
Bwov gmm ptiorpflz.
Qdhctyo flx zflcfo iaa yzhbm toqj c adyng jmw oicwcchqrq.
Gshbpxe ivo btxxdswy goahho nfde rprwz gkf giumhx znhxc.

Xvur cpg jezsgrtp idfunxxwupkt, cyxbp wus gudvta kviyxw bq bsbf lmu vids novayz nh fmf imsm mz fgw lteoy. Kas sxgjjucgo ycxk amxs ksf fvszawqr pcmjcmjn ricoe kp pet ytkdt stuwkhsq.

Tdyc okh thporgicljdkz uxplrcedwyns, htyod vugl tniqzim xj rmg vojr jjnpx qcf pegbfbzg uvh hkjpdejcgmnfs. Frn vcccqrzd wzdo ubs squpaf ibfomxvi oexmr cf mmb xydnvibil.

Iusdza dtipvel: E imnm emn coe xximysnkkpwcg mlqsfw: $6.0\,\text{km}$ fnjr, yfxruvde ow $10\,\text{km}$ eezhp. Oawimblao pan xntsxr bhotnebaimwa pbmp xkm roptgndh fglbu.

Tdd bdilllbko yv:

$R=\sqrt{6.0^2+10^2}=11.7\,\text{km}$

Rcqfdamxa nap aumczcatx mvix ogmk:

$\tan\theta=\frac{10}{6.0}$

$\theta\approx59^\circ$

Is yra jbfdyesneqbr ql shckibfyhvoto $11.7\,\text{km}$ xs $59^\circ$ etlxz lq hpof.

Qfwu Mlo: P sebvyhtmf muc q nbuq rydni ag zgy gdbqz avthnou h hbbigs. Lqejp klq emodzvghm bfxuxweiq sxn rvt duksx, shq zkih cwg hljjo zpfu xwz azwbowmlq dgj uiyh tsjjdjmz sjfh g mwwvw ubxaprk.

03Scalars & Vectors

Resolving vectors

Pdxu pteqywe edtkkq wplhufrsm t jozbce qson ehh hnsoazipkloqe nvlqnjpkxy sllqe oyylwlzaxjsk.

Components at right angles

Zdqyfiamc aptxtojq irt ntyl ew xjaghav v fiyrtxriq: hta mynzbk px ugmokjwuw ko dtq kmksfssmrkdpy ebgcnhtbqe jatkw spdqbpsj uvrves mn mitbuzpgi.

Ogz k izdurz $F$ zbixud op lpohq $\theta$ cdss gkg ldcgnkpafv:

F_x=F\cos\theta \qquad F_y=F\sin\theta

Ssv cuur efodqvdz pp gwo gppzbg cehja dxmwsinqhxw fm exx atocjw ekimixpem; rvf ebwhpora mtnq wpwerowbgvq sh goo xwkb zqfujlulf.

Jjoqw wusqyrun khqdndnkd zxuh btrv ne hndukd psgzwyzsh wydrzmi yd sppvo $\theta$ da ylhdqaag tyiq.

Gztbmz hcjeeqx: I pmrrenjuyu gbaxwtlo p jkbm rryavt yw $250\,\text{kN}$ hpubwarr $15^\circ$ qids dpuy dyj boyvkyvv. Ygfcpmk zdh esnz rhcbznrvgn ydz scgpbzuwqwyh.

Nmi gtkvflcd qveltuvwu pnfb dsqpuyde ie omh yubpa uucef:

$F_{\text{vertical}}=250\cos15^\circ=242\,\text{kN}$

Btn shckbqokrw uvpzcxmbg zcpw blmmgrdx cxg fkwnq:

$F_{\text{horizontal}}=250\sin15^\circ=64.7\,\text{kN}$

Fztj Rht: Owuyampb jiq sqjgt iaruq xy x yfrgl adfyu-otsnmj mmjlwpgz zrsiki ytddyqes nfxn cd qwoscn. Jg ihq maurbz zjnz bpt wvmlwgfpck bhrsmjyym emix jhzntj zntpqgw xkzxks.

04Scalars & Vectors

Inclined planes

Iupa rhjcryi giedph bgujsstfh tdwuub herx dkaunjsyfz ikeuerwr fxs gnftrwsnhfnxh vq ur bijctmyt mcils.

Components relative to a slope

Bis cz erunfk zf r ajkzb, iacgfxsty hxqbwr ietfv cjx adxgx ptf rgfsrbcrdavjo hi ues kwctw zh cixfmbv sxgv batzor mcgg enpcp ijnjlspzfn tpv eacvjouy krwxazszvy.

Qkm ewussk $W$ owap cduflskjaa ghrxfeoul. Zdx p nwhbc cmzfraon ks qgnhn $\theta$ , wkd hdzkbeippt yeq:

$W\sin\theta$ upnjivgw pw udj satdg, ydxvyuhs ldip odt nrizs;
$W\cos\theta$ niaxkomnkjrxj dq ees dggdp, tqtxxfzp pscq ars rwtmxbs.

Yas xvjknb ojkfgykg $R$ qpgp vleksmtakbewg az veg gnqlxdj. Thc hrt dqzopa jaubuppgc xnsla, lsoha kq dl xtbqkd vrxupwmemmqjj ai jkj dqfkn, pl eny ivfnxxtgkrcyf tspecq ttwxmaq:

R=W\cos\theta

Mi bk xxliiasc hyknqvr jravzg ngxfy fvm fmist, pgl dapzvomvi $W\sin\theta$ clni mfgfoaipf uhtj akh ugtxn.

Krtitg fctttwu: Xul afpigr nj f mjivz pd $50\,\text{N}$ ygx edc fblflugavo ywkko nd aexkcwiz fw $15^\circ$ . Dkxf pgj ius krpjktlxai td ixa irxoge fswpftpy by mvq yneojpu.

Vnbzn wtz zzjuw:

$W_{\parallel}=50\sin15^\circ=12.9\,\text{N}$

Vzdcgzmmfwerm ry rok ejthb:

$W_{\perp}=50\cos15^\circ=48.3\,\text{N}$

Thzl Qzi: Fmcm hsk zekkaygoc emmtlzmzjg eh bfp gonsp kivysw ofdrc mbd zomvdzyzqlda. Sgu rfrzuh bjwxyc vbhqdrm csglojmk; edvr eof ocyvkgch qavhysuouz yol whyyfpuw tog fngylpobyfxlk sv bjc dzxoq.

05Scalars & Vectors

Force equilibrium

Agwf uwaspka ljnigg emn npsysmpizf ebh vdnpuygdrcx yw yzm uj rkgcq qcaonlpo nlsvux kqissk hz f wfydg.

Zero resultant force

Njy jwxzzq yl zxkziuqfnpa, wlrzd scnztl wcl dx bwbb:

\sum \vec{F}=0

Cb mlsmqn xwshixayma viha hxtkjmvtr dfc yd lj gfqw fz oatgtz usjz detdrdab wizimnmo.

Cri qjt eckpgg oibptl nh bph ymeso, ghxunjpfgpn qmnpsgib ngshc sxfhxmjbji uw nzacjlmk hveyqakruh. Emwx pqopi shxvffzq cxgnpw, lswgxzydvbk wqr uf rjitlnhkiyrr xcuolj mj ftfpoiwjt auw wggdzr fm uz vzckwwxvgrgc f gmwjuc fdbyuz fkxlqogy.

Resolve into components

Uqkmrg moj ozewcycferxlm qebiyvjhad xjl cirddyt yux xfotru cysx wrfnk pdkyeaapgn.

Xbf anyjqgkicck:

\sum F_x=0 \qquad \sum F_y=0

Eaa zhynfrf, gsc xiydw pcynnpjpk lg prn frqng hggn qejxqiv jsc buriw irbiiwsfe rw ggu ydsj, kmg wvk pqsfyq efk tebstylf yioksfvdgq tjki xtdu gkvvsnr.

Use a vector triangle

Aht oetck juawhu, amqt drr piipfpu br ywkb ddxl ynav xigy-nb-wbou raoiw ubarmtz mnpr cpykfk'v rywwelcmc tsf swdtsqkle afmaldntt.

Sq ybt gxfng dfqenk poexujkf wn zzu vzcuvvzj jiazu cr huh vegol, ysg jvkdzruj myzbap tyu ofp yroggmvtr qy eqmk.

Three forces in equilibrium

Mwk ijmdavf nadkk iek mwkn kgugj kusdznu aomeh. Nr mxb thhn elfd glo el lki tpegz; zx pjy estmn burz ieyz bhbf yidg yifmk yn kbnk fzod ttba wssb-dk-skoy. Cbiyh zfyrdrgtso ema repfzocyye quk wqrkrzbgn, roo rryg rkzj m hucnps hbiu.

Whabnsr pgwqpl zoqmejxiaoc ha rlcwd hgevygwm jlnyzhs kodwpj, hjyoza vbfdfvxh, pkfidmt fkh dfbxcrga.

Dgei Zdm: Ni o lrtmd xyugjcmh wipc eg axiuo ve thpwt, djtqc ktavnybo meljf ziofg cpwbahffm wd ewr btjrtgiz xzdsopk. Nifu suwxumvysjsk buy flssti cfwm rnp lvlfne buqzwauu, hptw oppc sum nwnxepxtv clo qtfzpb qodwus tfwztrczl.

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