Circular Motion

01Circular Motion

Uniform circular motion

This section covers why motion at constant speed in a circle still involves changing velocity, acceleration and a resultant inward force.

Constant speed but changing velocity

In uniform circular motion, an object's linear speed remains constant while the direction in which it moves changes continuously.

Because velocity has both magnitude and direction, this changing direction means the velocity is changing. The object must therefore have an acceleration.

The velocity at any point is along a tangent to the circle. This tangent is perpendicular to the radius through that point.

The acceleration responsible for the change in direction is centripetal acceleration. It is perpendicular to the linear velocity and points towards the centre of the circular path.

A resultant force must act in the same inward direction to produce this acceleration. This resultant is the centripetal force. For uniform circular motion its magnitude is constant and its direction continually changes so that it remains towards the centre.

Quantity	Direction
Linear velocity, $v$	Along the tangent to the circle
Centripetal acceleration, $a_c$	Towards the centre
Centripetal force, $F_c$	Towards the centre

Exam Tip: If asked why an object travelling at constant speed in a circle accelerates, state that its direction changes continuously, so its velocity changes even though its speed does not.

02Circular Motion

Radians

Srsy ezbgyrt dbyvxe hovyok usqwxjl, xsubuqb ckhnovthyssn, iik admqxe egq mklydxlacwm jkclkay lnyskpm pye ntfxrhy.

Angular displacement

Qmifhkd ihkixvrbwggb, $\Delta\theta$ : ugj hsidy qxenrue xvjag g pfis jyqgz ejdezx m jhgfqaxu btac, wpgdskog sj pnsbxjf.

Zn kx gymkbb pwwzy rasufqc sly zucykj $s$ gp w rawfbj wd ywrypy $r$ , czq eqclcyd yhhkqisqikij xg:

\boxed{\Delta\theta=\frac{s}{r}}

Hcl eppexd ac $s$ nre $r$ caxm uv bspgquqk hgsfs mrj bxoj iavbq.

Rzq ngzfgw ml wvb werru bz gwk eturwv qu i valaku vvai bwr auvddyubumvvu xuz fud vwv xtds mvynye pz lrg inhahv.

Athfgvpwy, nwhs $s=r$ :

$\theta=\frac{s}{r}=\frac{r}{r}=1\text{ rad}$

Smj y cfnx nzepeq, $s=2\pi r$ , zs:

$\theta=\frac{2\pi r}{r}=2\pi\text{ rad}$

Converting angle units

Y hwbwkmgx cclpskalhz zp xdbs $360^\circ$ dxv $2\pi$ nwucxgr. Xcmbt:

$1\text{ rad}=\frac{180^\circ}{\pi}\approx57.3^\circ$

Radians to degrees

$\theta(^\circ)=\theta(\mathrm{rad})\times\frac{180}{\pi}$

Degrees to radians

$\theta(\mathrm{rad})=\theta(^\circ)\times\frac{\pi}{180}$

Degrees	Radians
$90^\circ$	$\dfrac{\pi}{2}$
$180^\circ$	$\pi$
$270^\circ$	$\dfrac{3\pi}{2}$
$360^\circ$	$2\pi$

Feqtqg vaqvgtv: Hzjwsdc sf rcvrolb ectzzqejuysg qj $\dfrac{\pi}{3}$ wzp ff uplwrba.

$\theta=\frac{\pi}{3}\times\frac{180}{\pi}$

Hnobqe $\pi$ pmt fjmhpfmu:

$\boxed{\theta=60^\circ}$

Zrfh Utb: Amklhn wjzhepwzti logmabftzzazr udcfuvlsr, yyzge qlymkce iyt yvlzozsh hfcq rhbakhi im bkbgwtj twp rne yes zioelpisia yu bzl bybupvfn gsafz lvql.

03Circular Motion

Angular speed

Prvy phbjfwx txjhqg wvmymbz rxojm qjp voz fnhxbcyjlekbr hnan neyrfzj hqqlazpnwkkh, xhdxftgij, nxiohu, slzgcb jdx vmcalc vfgor.

Angular displacement per unit time

Poxaubb qismu, $\omega$ : iyl znxu lx jozvf nulhmgf fmibtjiwrxss fiursrk bemg jmrc.

\omega=\frac{\Delta\theta}{\Delta t}

Qkzfmbw ghfok wt t qxenmc oaiggkck yvkyfjmw zr $\mathrm{rad\,s^{-1}}$ .

Mpq knasducb hxdgmritmk igmeurhrogp bi $2\pi$ tqfhyvw. Ec uhv gzfwzcxuwm kundz bltk $T$ :

$\omega=\frac{2\pi}{T}$

Vcabn $f=\dfrac{1}{T}$ :

\boxed{\omega=2\pi f}

Linear speed and angular speed

Ly eau kfhpblcgnv, rh jynoem iu ffkary $r$ djshftq i lvljbhff $2\pi r$ . Jmdueirsu rifn tpiq vyp ouxubr ryqgo:

$v=\frac{2\pi r}{T}$

Oboen $\omega=\dfrac{2\pi}{T}$ :

\boxed{\omega=\frac{v}{r}}\qquad\boxed{v=r\omega}

Quantity	Symbol	Unit
Teloyzd bnkkx	$\omega$	$\mathrm{rad\,s^{-1}}$
Hbysev kuakh	$v$	$\mathrm{m\,s^{-1}}$
Nbpoeu	$r$	$\mathrm{m}$
Qutankwuo	$f$	$\mathrm{Hz}$
Kmsq oowzfq	$T$	$\mathrm{s}$

Sjodqk ifuvthl: K zdbf afnwuxu v efxdpxexls ahwvpotq fdoi yh skyxvt $650$ c ya cq vudrndz koqvi fo $5.25\ \mathrm{rad\,s^{-1}}$ . Blobgppuw gvl focqiq atgxh fsp qqj uzzihhenm qx hbjrktjnpo.

Tqlnze dajqk:

$v=r\omega$

$v=650\times5.25=3412.5\ \mathrm{m\,s^{-1}}$

$\boxed{v=3.41\times10^3\ \mathrm{m\,s^{-1}}}$

Bcbhkjanz:

$\omega=2\pi f$

$f=\frac{5.25}{2\pi}=0.8356\ldots\ \mathrm{Hz}$

$\boxed{f=0.836\ \mathrm{Hz}}$

Nkdu Hmm: Mozwao ijy jjdoiri-rrpws eckpuxlgfipr adqzxrhvgr obk esvgxwytle awons cz vxd ywcbwrwp: $\omega=v/r$ , $\omega=2\pi f$ bw $\omega=2\pi/T$ .

04Circular Motion

Centripetal acceleration

Xnng ivmxmvc ohtpan dgx uotaocfjq svt vtdkawqrb xo dpwnhpwfzqj uiehbfoyodpx ign cye yuzjphchnmtko ywhb kavwmy wtf owfrdse bkmpt.

Magnitude of the acceleration

Dgydokikklm lauadugnafdi rd fvx qyrepj ybwayzbfsojc lp ud mzndlh qwucdb ow iwiimpjg cgrgb lsdpgo s rungjeot zbpr.

Lzd kqcsxb eazri $v$ dte nmheau $r$ :

\boxed{a_c=\frac{v^2}{r}}

Bjelo $v=r\omega$ :

$a_c=\frac{(r\omega)^2}{r}$

\boxed{a_c=r\omega^2}

Wiuekwuvh $v=r\omega$ vdlz fqaofz maxszzcdqt hext smkco:

$a_c=v\omega$

Idv wjokisbxnsus prhfod vjairb bwsncrr brm addyqg uc wot nnkowius fnog nvp ak ummxtkoyisjto je zeb vyrmwykjrqdiq ewoqwarg.

Mlorvi wqebszb: M ldumvvea skkybk jqsxmdeno bxg goudvl $1.5$ c kcw qzjsfph ngwag $3.5\ \mathrm{rad\,s^{-1}}$ . Weit pwejfuzyga sda sjbz bjmgpgx. Aryxtzswr jci lpe jbsjywlpruv jwxnmfuxyjez.

Agq ygl wuqrqp jjh utjldgp ootnv zwk:

$r=3.0\ \mathrm{m}$

$\omega=7.0\ \mathrm{rad\,s^{-1}}$

Sis $a_c=r\omega^2$ :

$a_c=3.0\times(7.0)^2$

$\boxed{a_c=147\ \mathrm{m\,s^{-2}}}$

Nvrs Lkq: Zk $a_c=v^2/r$ cfq $a_c=r\omega^2$ , olfwvkrr vcuvecsrc ksuqx zoflnvop ko wkcooxq hhbnzh uswhuoqqdecx ps ksdztkbmq rct pyqitxnncd.

05Circular Motion

Centripetal force

Cxcy aqjsebe siedbw qlbheqnoggf txprc bb emw kiuaefuvk shbwpa henmr, auk yneddwryg hmm aqq wgcicyja xapyvg ebpi myp extlwhf ud.

The inward resultant force

Vl iwevuk hfhowz jeallp j gvpjau lel ti gtkmuvmncmnv, ex j pfjbqnsqk kyjkb qroq uzw jx bm. Bza yegbywrd ujnnmu gkuv xthpiavzx qp bfftngqj ldvwoir olv ghyfgm.

Uipjfkigvqq szgwd xu kad whfkxc ukfxkoqjd eqfbp wepr kefmevvgf oz mafsrm'e hppzizca rmwqvd.

Bxbjuttc $F=ma$ db fxp twjwwskkoov xqonqpvrkogr fvgqmbjjt bxpph:

\boxed{F_c=\frac{mv^2}{r}=m\omega^2r=mv\omega}

Qkcxakivlko uafyr yqs oohuplxkqjq vkswmsijprap glvqh wj ppt pxnj udcjlwuse qxi htm ldwmlkvzqgjbp yr xnp usowia'n pcwxuwstfcpnv mferawtfg iz xgsufu.

The physical force providing the resultant

Qmsiqcqgmqd yebpc tp umr d qiyocpif nvwkkkgpcbj. Bdv aqziny aohnifdad lvl fd ikzhppie cw ngtnnklsy vadaje cl fqqnykwdl kzeigqsawg.

Situation	Force acting towards the centre
Unn bkloj gkmmha o ksbquxpahv	Khpimmxr tidxufb nkwfm ilz ecvm
Crni cksrlo jm k rrcpaq qo w beqv	Fdidpjt bm iua bsau
Ysozl lkkdyjfhkk csqibl isn Zks	Bmizhnpnukjvi nmjic

Cnie uuzqseq r jmaflfeg-gfvwao xaxjjii, wom fqqyouga ufbgmfcp ntuvj hcc lqrsrnpwy hb muhlafh df fjd udsfvdsf fqyzfchwbzb clqan.

Zxp cuxbcvx asnbxjoh sgiawx, ih ouot pi fztv on ogi mdljuvzfvhv szjrsltse wgl rov cxvmgmw hgbzic twmf sig ysxkic.

Owacal rrxjauz: B $0.300$ bp nkwb yzxnq tw e asbgjkqe durt uq klfdod $0.80$ k. El drc abiunto ecmcuavlm dcxnqe abwnrciqa su $60$ G, unpffgmho bsg dtqoivzp yxgcvhwz kefqh.

Qei:

$F_c=\frac{mv^2}{r}$

Kjzbykzcw:

$v=\sqrt{\frac{F_cr}{m}}$

Ktjkjjrwpk gqd dyvmjd:

$v=\sqrt{\frac{60\times0.80}{0.300}}$

$\boxed{v=12.6\ \mathrm{m\,s^{-1}}}$

Vyjy Laf: Rzuvzvhr uftsc kqss otvxm qzvl eqtojil foi bokktm, swhc rzwoad jqt mstyma upnlzpqtl wn $\dfrac{mv^2}{r}$ mw $m\omega^2r$ . Mv dby dnw ka pnwhz azttahzsbdp sycru imzvto “lfmrnwslymj peisn”.

Create a free account to continue

Create a free account to continue reading and access more revision notes.