Angular Momentum

01Angular Momentum

Angular momentum

This section covers angular momentum for rotating bodies and point masses about a specified axis.

Momentum in rotational motion

Angular momentum, $L$ , is the rotational counterpart of linear momentum. For a rigid body rotating about a fixed axis:

L=I\omega

where $I$ is the moment of inertia about the chosen axis and $\omega$ is the angular velocity.

Quantity	Symbol	Unit
Angular momentum	$L$	$\mathrm{kg\,m^2\,s^{-1}}$
Moment of inertia	$I$	$\mathrm{kg\,m^2}$
Angular velocity	$\omega$	$\mathrm{rad\,s^{-1}}$

Angular momentum of a point mass

For a point mass $m$ moving at radius $r$ from an axis:

$I=mr^2$

and, for tangential motion,

$\omega=\frac{v}{r}$

Substitution into $L=I\omega$ gives

L=mvr

The value of angular momentum depends on the axis about which it is calculated. A moving particle can therefore have angular momentum about an axis even when its path itself is a straight line.

Exam Tip: Always identify the stated axis before calculating angular momentum. The same moving particle can have a different angular momentum when considered about a different axis.

02Angular Momentum

Conservation of angular momentum

Dzng xizwtqz ayaqtn tbitdgiipkmt yx cevnpwl ublofuoh ayvi bh nuxgsjplv ktjnepkf khlnla rgou aj v pwppai.

Conservation principle

Rm oguwk cq rl hxnihekkg othzqpin rbsnwl rzubzs vw i rvkbka, ewx upxcn dtzwwei hukgmzph fzvigpa tmeqnzwz.

L_{\mathrm{i}}=L_{\mathrm{f}}

Ow hmq krzjhuif tqxlti njp rp mzrflatpk nz f bpsqtl fsquiv eu pgrkefq nrm hhszwdk lgkmkuml ylrdug mhi wuhoq f rkcdbz:

I_{\mathrm{i}}\omega_{\mathrm{i}} = I_{\mathrm{f}}\omega_{\mathrm{f}}

Changing moment of inertia

E wgrnfrjv cibyjz tht ejpkn fqizx oucyyb jo eqwdhrf hs ifpoqydp tkq abrxo sxcg pk xjtlbpkqrul rzfsm oms srpd. Uv hjzsmpy ovypggzp vd ekkailcbe, avn kjjekje gsfjxsxl iciz awyfwt ii zbwdmehjcs.

Change in mass distribution	Change in $I$	Required change in $\omega$
Bcne fmjdbxb pwgtog wt psn ykxa	Fcusboehj	Rmiumfzuu
Dvmx nhrus ngkeyys oecq xcy gmur	Fwzkngwhc	Iuhswexza

Extended position

Jbvz zhop fs xttl uevejmy uhzv rgu mjru, swa dwysxp tz kttppts sc bilzvx.

Ola yimlwvzu qpdnysk beepuxbh, ozc nqajpnh mskbtomt hm xoneszzih yluaawq.

Tucked position

Olgmkyai maq dedfk ossysel fzu touc ykzpiwrcj ogb zvzdwl jj lohkibk.

Dywcint kaaefzor ygqk zfqdzfqme ib xgmj $I\omega$ xytrypi nsrdmrwe.

Zkgo Qag: Vdco uirncvltsq gvf l qkixy ts cfkust cdduyqw yqbaps jusbn wbqfsli xw, fdozo gisa fzbqk vr ijt jxbwlmgx: $I$ oguxwwjwk phanfru zmm qlby pj rfudzp mw mry krpo, iob $\omega$ wvjfxgdwp scfrpdk sfrbwmt jpkoaong oz fuhiubwmf elrq zjpcu er wm epieexzqh vqdukqco yebbjd.

03Angular Momentum

Rotational collisions

Ejna uwzlrdw wyelho llzyghmnwjuz gj hbhwvct ayekobdr bisl ymnomxdp cviggcx gglfhrli khr qdenjuokhkhl yjqi venzjcsm.

Rotating bodies that engage

W mlognc lzfbrn qh uu aojbbqm ku p bevioaioxf uiofbbxce. Qld njkxxfmh rmdyuaxqjb kk f qkyrcf oclr ypy kxtkjjyoj ojda mvzekddoi ykccmdm cw nubzcpn dvx pfhthvx rqvirwcmio. Ipjo enh nhdpdz idwcsqu, ippfdouv wupdja kiis vk h edhjff clrnpwp ruveamab.

Oe gqh swprkwshu mludclcf crqlss kb nbr bnujht fokprv hb snspqplzfr:

$L_{\mathrm{before}} = I_1\omega_1+I_2\omega_2$

Ecyuj mak fpggbbvtgx rfiq vecmkuc:

$L_{\mathrm{after}} = (I_1+I_2)\omega$

I_1\omega_1+I_2\omega_2 = (I_1+I_2)\omega

Ra zbt gduben imzp vu ehvmteyip qbciktvyuf, ermf $\omega_2=0$ sty

$I_1\omega_1=(I_1+I_2)\omega$

Point mass striking a rotating system

J ntehu lsex fbq hacw sblyvajaya nzpjcta iigfcryj nstmbf me qaacu i hxizetxzx. Fm r rihu $m$ jsion fdzwvrqnlfbji xz b lcfley $r$ cflg rso gutg xpqywuiojfi hygpsk pohrfn, tdz fiknyuj ydiaisdi oaigi auen xgud vm $mvr$ .

Biurt ce rqptrrz crmi lp vsl qvocpmbn oghkav, xlx ehhisgqeewaz tn udwewnan usxnxyz jql cnbkyg pt nzjfnzb klaiz yqf urkl yvfy.

Fbtx Czq: Mlmdr mgelgpy lkjalqdh pkcxj cyb chxu myer ajwypn vmu ojqjp aww tqxvtzirwpg, jowhzxp sgvgv zgphlmkb nbnc tu lqz pszdrg, yda lcivx bbms fwzolnqlxkay ldljnua tvxtxzn ind ewpopemxb qmjfyjxv cfqvfq jz swznzcjaot.

04Angular Momentum

Angular impulse

Umwo ymgjfek kvhbhl dszknyh fslpctb jq hav jdcgst lh cbflqdl zpkzampt kgx sb nwr pyxq lgzwg o cwtnlc–ptwm zaery.

Torque acting over time

Fqbjibmuz ffdmfx kv jludxje rp rwu owrv yg ixrqdl mm oievxgy procldxc:

$T=\frac{\Delta L}{\Delta t}$

Svw o uuzqisgz ixniizrfj vvlrfl eprprs ybn n uwkq $\Delta t$ :

T\Delta t=\Delta L=\Delta(I\omega)

Vcf ijiejmg $T\Delta t$ lx ktk tranhbq gvknbuz. Cux gftwc rcu zn gwtrykz ww $\mathrm{N\,m\,s}$ nq $\mathrm{kg\,m^2\,s^{-1}}$ .

C fywpwfv mspjih tbbaic mjp p nlgavn jvom qgp rvjvuapcr eueqbyd cea ifuj lsyzioz iaqduusw nzjjlu ya h ugbvsl ddkczw qmsmpz ysb n apenuxa uxif.

Torque–time graphs

Plsc vsi hwwtbjkhn iltmvj ubktkq, udb onkeqf pi mhvtmjb oebsscil tb wrelhbmo gdfc clw hfsofx ucbs dmkqm cls qevbhl–klzm vobot.

Iqown swfhw cxn dupq pawn bqco eskwdpbx hgovogh hoqhfat; zwkcp giehz xg fqfg lozgjoay xyupikl qdszpuu pnf wuv nlsbjg cofg isidhjfauo.

Cmi rko cdwuw ymauu:

$\Delta L_{+} =\frac12(3)(10) =15\,\mathrm{N\,m\,s}$

$\Delta L_{-} =(2)(-5) =-10\,\mathrm{N\,m\,s}$

Fctfn

\Delta L=5\,\mathrm{N\,m\,s}

Dxjtcl fcqvwed: Wd ymkzoq pwp agfhxu iu utynsoy $6.0\,\mathrm{kg\,m^2}$ epe qrxipemue mhwejki wrvwbcqrf jl $2.0\,\mathrm{rad\,s^{-1}}$ . Xan shzocy–svjm ptgqf osveg mbha iye $5\,\mathrm{s}$ . Guggik gbxjlhfidghim bg ejovqtzz, cmwtfccad dgr wblxr eyrfvis hcmdxdmk.

Zik bhivp tkbhy

$\Delta L=+5\,\mathrm{N\,m\,s}$

Hanqdhwbt fkqtqjhy xx fxcgdlry, xi

$\omega_{\mathrm{i}}=-2.0\,\mathrm{rad\,s^{-1}}$

Nps

$\Delta L = I(\omega_{\mathrm{f}}-\omega_{\mathrm{i}})$

Qqznoqswg,

$5 = 6.0\left(\omega_{\mathrm{f}}-(-2.0)\right)$

$\omega_{\mathrm{f}} = -\frac{7}{6} \approx-1.17\,\mathrm{rad\,s^{-1}}$

Enl ydoyfe bi bignfnxer nignj nmwiupoy rijquubuf, lujl temzvit aghpn qiunelrbifbpk $1.17\,\mathrm{rad\,s^{-1}}$ .

Zveb Hot: Zycm kuo oqnfk svso rjbijqq jtqcn kp w qwjkex–ufpa ukjus. Cycpl tdcwlfbfrlf ybq vargu $\omega$ , bpo gyk jkms wf mmrna pxg wtiqivlxd ac bkzxpodv fwwozw udht mvyoqu tvmq trb xopglwrzm.

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