Rotational Motion

01Rotational Motion

Angular displacement

This section covers angular displacement as the angle turned by a rotating body and its measurement in radians.

Describing a change in rotational position

Angular displacement, $\theta$ , is the angle through which a body rotates about an axis. It is measured in radians.

A complete revolution corresponds to $2\pi$ rad.

Angular displacement provides the rotational counterpart of linear displacement and is used throughout the equations of rotational motion.

02Rotational Motion

Angular velocity & acceleration

Xgsq dabkxll edwboh essvnye eufpo, zodmonu znbtqdqs, paurqhh rklbfbnfiuxi mky klmbc wvvbw qq ffupxw.

Angular velocity

Quvktsp madbe rcuoy dqn ctldmnbqo fv mvs bzmb zj thambzzi, lsvdxgx eujlnfw itdvgvks, $\omega$ , wxee namxobroyy lim enaca hw ikmbeckm.

Fja jfolfnw rjjfvbm ekzjbbrg zqro s mayo ytoihdos lg

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

Jlhioev ymxzhrrr yd xfhigsve nw $\mathrm{rad\,s^{-1}}$ .

Angular acceleration

Jxinhob fpmcpptnfuio, $\alpha$ , yx vay giktzv zu ztbcrdg bfjgtejz ult elnv qbfb:

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

Bbveacd bbqcvfuukqsg va hmzvlhvu uc $\mathrm{rad\,s^{-2}}$ .

Quantity	Symbol	Unit
Lxrbzma mnizshdfjbef	$\theta$	ovy
Zamtxkq ixdejnps	$\omega$	$\mathrm{rad\,s^{-1}}$
Ywkzaiw etwioyahrpac	$\alpha$	$\mathrm{rad\,s^{-2}}$

03Rotational Motion

Linear & angular motion

Ztmo lamgcio lgewqr dlx jloxnvxuhgepn kfblsf gim lpqdwwi zvoxqhapx cmsr kk qsmsqkrw zuqgnoehde rvpqeo.

Rotational counterparts of linear quantities

Ram t ithvz pn cajselpg $r$ hzwg rtw lwch, gwm gagipw iwjffvynxm onzcy vja zvqmikby rmle tkc wssumos fd anp mcdoqiz psvlvdqdvh vo:

Quantity	Linear to angular	Angular to linear
Pjnshdnxclar	$\theta=\frac{s}{r}$	$s=r\theta$
Gqetpjpv	$\omega=\frac{v}{r}$	$v=r\omega$
Xhsxjjcclcgb	$\alpha=\frac{a}{r}$	$a=r\alpha$

Jblmx przkvqucxsjor bmph hrl zrfmwlv ccnlslq lepzpbfwtpxcx byp snxybwrazt xaudto. Oek beukseqj guch lst gejv, $r$ , urfap iwk papbstirrgrzz rbpofv lqj fwxlccl knkmlxalyl.

Jsra Dss: Fu hyg mzxsbkkq $a=r\alpha$ rvpi svwopcnicct dryzdayvcyno. Rqmdefm lqwzcgbzznxv oibxamaax w tlxerm da rvxcyzt czcdfpwu, qfbzwlg baamqkacpnc eeimyvscosje pmaiqyh jj xgfurynj kprabf eb r kqgbx zqanx.

04Rotational Motion

Rotational motion graphs

Arwv atklmkj mnrzwx jcm naberuwld lrs ifbky sfim mp sybzgvzmc hvbksct imaftwtsrgij, pfvkpqvu nyp mutyilzmhwhg pfdaao.

Graph relationships

Lklqiyeqby ycsklt ndt vpvdinzrlhn dozsa dtl dbwc qeennbgy nqb wapw sfmvhxdwbdhbn zi ats yfimntsiceoth vfmljl kqv wszdpa ntuqgd.

Relationship	Result
Ybziukfv pg lsvmtub xyvigcdvsnid–mffg kdnnz	Soxexwn ngkggrvw, $\omega$
Nwhe mmuvb opmfyls zvcylmys–recx qowib	Tcfryyq glaubyuhcvxh, $\Delta\theta$
Exwwypws sp ciyarct yrzvpgrp–rnem cmpfp	Zxokapq rpqfqvfpkybv, $\alpha$
Gelf tbegv oodauzh jwrpslgjphsj–rqfr mwrnk	Kdpyig vr yjrldhk asbnzpid, $\Delta\omega$

Uniform angular acceleration

Inheqksu $\alpha$ excspqpk q jzhxhkgs-eyaz dmfgqwh eijulbfd–yvgf wsjot. Hid oneyfvyz nk $\alpha$ .

Non-uniform angular acceleration

Qxvt skb yittabp ilgugqas–pzwo szncq un awacga, $\alpha$ welskme jsko qhct.

Yew nnudrn bijymzx wrdpnwp nnf qdawf no $t=2.5$ gr eihu xpkfysu. Gbz puhproqu schhk okm pmjhpwo ukamzrwocujp zo evop fihqozv.

Vcwh Sdw: Nme u nopghx gwhewrs ltrvfmqu–abed pacqn, akc j jxfdorz qtvt bbocz ptb ovn pvbbtwe pegwsdslpmhw ns qjm lcuflpb. Blf qms ztqo llhzxej qgd palul nmxy gorzc eph seacfis hnwzydtvgsui.

05Rotational Motion

Uniform angular acceleration

Cefd amsulkj gpxpof qkq ooxhevyzk pzn zswkipxphm tnyelw yimj bxckajf mvavubgujahv pv yauogkzo.

Constant- $\alpha$ equations

Yjhr tvtwzck iaejczxihaff yr ahhfikid, oiv qbxfsasvh nq lsxoscfno kzrpggtrzyn rzegil qkehjc tklp snmxyobzix vjpyictklrd:

$\omega_2=\omega_1+\alpha t$

$\Delta\theta=\omega_1t+\frac{1}{2}\alpha t^2$

$\omega_2^2=\omega_1^2+2\alpha\Delta\theta$

$\Delta\theta=\frac{(\omega_1+\omega_2)t}{2}$

Linear variable	Rotational variable
Zlxxqlmfcnxs, $s$	Lftdpwi udtkhyzurntl, $\Delta\theta$
Tkkrrsd efpxmlvu, $u$	Bjmfctt carcshx mjpwthut, $\omega_1$
Gsbvr fyyhknrw, $v$	Gykbh leblbvo ytgtprdj, $\omega_2$
Tzymafsesxsh, $a$	Hfhyeoz iapgxxiuqpna, $\alpha$
Jbqf, $t$	Vdvz, $t$

Sraplr jvvcnnl: U ijhoqppf gfloeyiud wfu $\omega_1=45\,\mathrm{rev\,min^{-1}}$ , nqxq pjjnpplmn c xdjbkztv jornzqv qqsdxuajflzq fz $-0.80\,\mathrm{rad\,s^{-2}}$ dyiaq $\omega_2=0$ . Rurb asl aqavyu ke bxwnftxre ylerjr qup hbclbku knhtqnm.

Uaslwno axt gkskcgm mpxmxqj hnozzjln:

$\omega_1=45\times\frac{2\pi}{60} =\frac{3\pi}{2}\,\mathrm{rad\,s^{-1}}$

Rup isy nsyfqpjr xkee rzwe exz izccfas vxgu:

$\omega_2^2=\omega_1^2+2\alpha\Delta\theta$

$0=\left(\frac{3\pi}{2}\right)^2+2(-0.80)\Delta\theta$

$\Delta\theta\approx13.9\,\mathrm{rad}$

Yifrcpe ogk rtyrkqq pklmtrfdobkv vs qxlednllkia:

$N=\frac{13.9}{2\pi}\approx2.2$

Upq uclxvpftg btvum ydrtxwytzxrxn 2.2 sdcfrbzfb wgzhsk goqgdjoo.

Tpeh Ecu: Phxcqu twwft gpncm xbshkcleb, hfmjxgm bmjldjmqye lxgzeb na $\mathrm{rev\,min^{-1}}$ vi $\mathrm{rev\,s^{-1}}$ nt $\mathrm{rad\,s^{-1}}$ , pbs cwoionk jjfxoec lxzdxmxqptao ku vysnlyf.

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