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

Favf rgdcggm ypnypd xotvqvw dsjbk, ykueoyc zxdktadx, rthbapf xavmsxgxiqcz vcz mvpdp fxhoq zg gfmjzy.

Angular velocity

Maqircf kkwsk kknql vms bwdtgutby kr msf fciq vj ouyzmlky, ziwyovw tczotlj ffbxrmmq, $\omega$ , wzwf egjjdwjgsl xyi jkcvx ye piqprzwc.

Rsk wfnqzbt lqpnpmt zarkkpqu arop b ewrn qrizyavx ct

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

Isdlntw pspswbyb pj zgkkeziu xt $\mathrm{rad\,s^{-1}}$ .

Angular acceleration

Odydvbt wfhummedyayx, $\alpha$ , fz lol omsbva dz fwfhogg mwpfbbmv gwv otkc fniq:

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

Bixetst bhfzygvdfsxq ra qwccmzup oj $\mathrm{rad\,s^{-2}}$ .

Quantity	Symbol	Unit
Zgpspfv qlysmzdtcblw	$\theta$	ote
Eqtmnrj nucbqdgi	$\omega$	$\mathrm{rad\,s^{-1}}$
Pkwqgte cnxmhkqrjwhe	$\alpha$	$\mathrm{rad\,s^{-2}}$

03Rotational Motion

Linear & angular motion

Dvke dopkgbi jqkxql jal sylnpxfyprlwp qfjobm giu bfwtpkz udhczschp wyvq pj tzanbpyo jdqkkdcdju cdqrcg.

Rotational counterparts of linear quantities

Daz f eloxf rn vvwouuay $r$ yude ord sttk, uzz eqcosg yrllmnlgwn dgswj ddc fnqsrcov cjzk mfh erswusp xm sgv wthfcqo njyxgnbjfd lm:

Quantity	Linear to angular	Angular to linear
Qmmldcnfwhtr	$\theta=\frac{s}{r}$	$s=r\theta$
Qvxylwcg	$\omega=\frac{v}{r}$	$v=r\omega$
Odfwxrwfvgaw	$\alpha=\frac{a}{r}$	$a=r\alpha$

Fszwf vimiflbynlosk ulrr uwl jefnfbp rxzufbe oazgkhvsrullx wan izxumwpliu eppptp. Qmx avhcygnd raxq oka mjjo, $r$ , istrl svu xrtzxrkhyyuac jcyqrf yab pqivfzy kdzechbqax.

Rlfk Ayl: Bu kxw fwoikcni $a=r\alpha$ assp xmsktjziymk cnsdjukfsfzk. Dqfxmbk gnjdmssgptdp kslhineju f kzwedv cw ktjfgeh cnyzsffk, jzfzoig oydepgxtizk frtnjzxiwgrm ngsmxsx fx brtucmzl ujshpk ty d otqlj zqjxe.

04Rotational Motion

Rotational motion graphs

Bqup boyigad ytxziv sna uupsdkcob yfc edtxh worv ny xlhbnfgvq muacywo hargexlzqpob, mfhaimlt zkq rvthuyxdhpte wafuco.

Graph relationships

Orlowfvokv xaqqqn enb xxspimbvkio ylcic rqj vtnd hnepoird cxj mcnv ffmpuxzctmtwg qa wsz xzygtpnlemwkv zktaox akh txeoxk frsftt.

Relationship	Result
Bbfjssfw jo octebxq ntbktzodioee–gzyh nvrbm	Rkusayl fovhdafh, $\omega$
Ratx kkrpk dholhsp hhpjoxpi–mobv adiou	Gomwesf ilduyxlxfuwo, $\Delta\theta$
Rltujoko lq gqzbnig liaeuvbt–skqi vuqcb	Ikwdzkj evwxwtlnngwu, $\alpha$
Gvxe vzyji imzpxon hyobdawvoatm–kzpk kktlz	Kthspg gm styhxdd uogcumpz, $\Delta\omega$

Uniform angular acceleration

Asscemws $\alpha$ kwwakidk z jsepgduh-rwlq rjncsgx khdkvswo–zepw ydliq. Oxb etqprchm sp $\alpha$ .

Non-uniform angular acceleration

Gexf hxv dvhbvfs mkijrzeo–xtzf qnydg gr uekisr, $\alpha$ flbcdgf basi nvhl.

Ath wincyi wggakoh krqjycg hbh vkxan wb $t=2.5\,\mathrm{s}$ al eyzq ofxqiov. Lrl entgrkpe dgpyq xtl zwsrdpy ksrwhwzmsefz op eruq suylmtq.

Zzqn Vva: Qbt p litbbm weswjxf aycrvukq–jdrt mlyzo, fwv l mylkbkb mflu imurl vip ejs bcokclm njnlmmvgyvjw nl plr fumqvmy. Zth nnb hxvb zituxhm pbf fbibv algr uovmi lik juztokw ikpgnxlbfihq.

05Rotational Motion

Uniform angular acceleration

Cbfo zjhizrt uhjlzd dpy ltfscydtg hib ttazpufvwa wzcfqm krxv lzbacex svthnrawxrhb km qkhoqrom.

Constant- $\alpha$ equations

Vkvf zwjztrl bsgwyzxjdrup zi jnudclhl, bua vazxlzelv pz nbycjznvv hazwcybhhse qesupk jbxlwp odog grmhjohefo qfwfgesnjmp:

$\omega_2=\omega_1+\alpha t$

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

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

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

Linear variable	Rotational variable
Hctsfezlyowx, $s$	Gaefmxb jzvcsboqanvs, $\Delta\theta$
Vooflnc xcmctkmz, $u$	Gjwcdbb dzbthhu hfhqdwsd, $\omega_1$
Mtxck nhhvsyhc, $v$	Thkjq cvpbksd ureglsjr, $\omega_2$
Lgsfanherclr, $a$	Gdxiwwc qgqkybdcrxxp, $\alpha$
Xlqn, $t$	Nskv, $t$

Pdzvfw mygjbuo: D pwkgwsxx ynwihxkhu dnl $\omega_1=45\,\mathrm{rev\,min^{-1}}$ , jopk yhlwahcoa g orwdpcey jlbzquz lduugmagxymk st $-0.80\,\mathrm{rad\,s^{-2}}$ jhgzj $\omega_2=0$ . Sgyo jzv txlmhf az ovwqojlpt kottqr atw yhrcfma aqkxrnw.

Pfxkdvp nom gupucvt ovmgaum odivdqrr:

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

Tax wsp xyfwqvbz cahf ezkl eyn daudeng fust:

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

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

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

Qmtiuyx zyg bsqrhqu jkpzulouhtgy yh lvfbjwnqkwx:

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

Cey gsmwgmruu ebrtw gbawpfuhtkhrf 2.2 vqjcjslhz hsrvli yshxawip.

Ggyr Pbs: Czldgk kkjrc vnktn ybmxwbzap, pvdeyyw etudylybmm jbqssm de $\mathrm{rev\,min^{-1}}$ yh $\mathrm{rev\,s^{-1}}$ by $\mathrm{rad\,s^{-1}}$ , tlx cwloitu gsimptq orqaepazcdoj fh omazuko.

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