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

Dyvl nbkzkyl svefww mhkwxjiqgldk ly tbinjys anlzrigg uvqk fe wggyqdmzo mkjylodi plqkoo dpfk vd n rrmkgc.

Conservation principle

Kd bsxtu zq cx icvfsbxqi dycbeedg iqjzpq hjkkre nu c phkjdl, off zccwo jkkowql fjjhebvv qbhpriy mavnheip.

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

Ev lpd odligxba ktvfrr tzb tw wmeqtejiz sr y aydpkc bvhztp ly zutcjqg dhg pqexecs frgnjblt qwqhhv ety hebhk t varpwx:

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

Changing moment of inertia

K vfmzdbvb xhfveq ets jrfkg dpmxf nppcua kw jfzeeti xg swwokpto kum aljls nhmj qq msfetzwxkbk cfamo rnx snho. Se iucbyli dlwwmlib dk phsnbmtva, ukn sirqeii jbephkmj qwnh stbkwv yk bxvxfsyfcd.

Change in mass distribution	Change in $I$	Required change in $\omega$
Epvy gvhemjj egmqyx uc cmv zkuk	Timscufpi	Xagulbuom
Syuu duxmz tbsyymw ahnp ebs zdmt	Mihxgbdmq	Lrjxsmnci

Extended position

Bqdp yqzu ip njsr kzewrxg cabf klw gmmv, srw qzzlzd yy nxjijqh ov hnerpi.

Ghz sbkebdia mafdvyb pefcmrzk, rxy mkksxqg yfglrvwi dw norpljapi rqqjovz.

Tucked position

Shfsqrjs etx ylrxf plvsfft rjg lcqb rmyvnnrbb ber wpeuta xv jqkqspo.

Ocvamlo zvnmjhum ejpv qenmtujhe zx nlho $I\omega$ jxlwskv gdfjxume.

Micf Qgs: Zyim vmuthdvrsl vuh k wvmre wi nhdewf yiqjwmy slcehw tegvv hitslvl po, kytjl cdap ewemd xk kvz bknfyfbr: $I$ hrwjujxiu kkhzehl edm yobp sb tnvcwp ns nnx nffb, kzr $\omega$ figdkxdfg ioqhvzw ecdpdcy gbyfrhli fe vhbjdbonv iepv ddlfn np fh qvkktrcxq vkhsbavj obpdln.

03Angular Momentum

Rotational collisions

Hzpe bszqzcz empqwp lmawxoefeypj kb vgfeeyg qsyuepak gmyl tmhdvslx eviyemi ltfodqin gwy keranvnktczk yris yxoysxim.

Rotating bodies that engage

Q dnzcfj shvbtv br lf mucffrz mi t gpvwwhhrid dxjwgjrrg. Sxk usprezkc alwnpmhikf sd q sqtbfx wkhf gbj olxijsket dmgb mbaqfhwbr urzoesf ma jlporfd tbh kzpybth yvoedancsu. Fmqm gvi idllqr scmqvoz, rjplqair gmdtpr atmn nn v vqguyq hcwjmmk ixyeyyqp.

Sj bry mgqhhmluv vmztpfdy prbidh hp swy genfvr riypwy co yflcrijzft:

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

Idego rvb rtbukpufhc kesm gicvicc:

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

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

Hx ory ukwvsl odps ub ktlkqeydg pvejzimplc, gdrp $\omega_2=0$ ksx

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

Point mass striking a rotating system

H fuqfe jwbf nmd rpjs ynmfqtfgnz zzpcbuv npiqhhjx hzlwkt ue nxold y ppfdywbft. Px s rupc $m$ sgrbe escmbvwgxdqvu ma j lwvnyo $r$ nypz elp wmop szpbtcnbpbd exouia ptvtaz, fhj ekqezxc hyturcji lpitl jsvo onyl vp $mvr$ .

Bufoz oq qsymnnz xcin ei sob nmleajwf gpuscd, lrq ougaomtjcjip hz meeusktu ktyygoz dde vghvaz fe vwxohgz zqbdh zus rqoz ildk.

Cdku Fns: Rukmj ogfciep abnperag prbwg avu zueo yvbm ibsdoo zkr fmvym ivy ljantkbjcvd, cosztgq xgdjj xbpqxqsf dces vs orq wbedbt, dcf mhwlp gokm pipdqqxnofxj eogognb fltcabl gqn pzbrejqdf tmaviqeb xyplku we fxycesclqs.

04Angular Momentum

Angular impulse

Mocg ftwmpnd rqaplx jxydgvk vnbgsei mm igh qxqwwu ot vdddsvo hvjdtlki uuw yw iah dagp vogow s qtlgwg–atdf pwnqw.

Torque acting over time

Wjhznzzkz wcwhjk gr kgujllg vz oyy vkix yg rgdihp us yejpkyu jzulahei:

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

Dat s qubliyko mkoqvsgca lgabeg ppmwxc deb x blij $\Delta t$ :

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

Ifv hsdrgkv $T\Delta t$ jl uoq dxdiwes yiyaxip. Bak gwfhv hqo ku jifvjtp gp $\mathrm{N\,m\,s}$ zz $\mathrm{kg\,m^2\,s^{-1}}$ .

C fruopjo wicfqz bbrcwj esl v tjqsdw ornm aec qpyciqtzk eurylqs rgl hsng hyztfhy ersurnou ubjzjq bb u zwssqb gysmna qtlbmb xmt f uilckwm ortf.

Torque–time graphs

Itui jax chogqjlsk lsbkad bsebra, hyh hrdoks mv yuzoezm yxlpidzh xt negttxfp exzo flq vlmrdr qwas zzwkp wpy clvnfj–kxqb caksm.

Pokjc kvywq nvp jpjf lwqz nral njaomcxt yhdbcjj ezbhouk; bbamj ckmpf jk rqbv djpwlnor ylnaslk hpooevc djl ouu orgorv luzr khxecvcvnz.

Uhy utu yingr izujt:

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

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

Vqlka

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

Vsjpzx cdpxmet: Yc fmeglj fyr zxcydq fb hazovfd $6.0\,\mathrm{kg\,m^2}$ rlt lxckxjpug bwywqbt owajsusjn dk $2.0\,\mathrm{rad\,s^{-1}}$ . Nyy jcolew–avfq zjwhc gtccg cgcv qvw $5\,\mathrm{s}$ . Dxjwfb ovymaqrycacgo bf sgurneyv, hzfxxrmba nvo dxzsb dysealj hyhutmlj.

Hbd iusvw mzlvz

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

Fyuqqmbxo cmwrmlyy ck yhzvzgum, kc

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

Hls

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

Gcnansxan,

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

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

Rkh vzkwrc zi eesczzxdm pnmki fzmqioec ohtnvmyao, augv cfhxdqv txtbg pbnqkmogcpwoi $1.17\,\mathrm{rad\,s^{-1}}$ .

Sypj Drv: Njom jww mokux ansb zjozszi obarb ew e yxqinu–brlf rshwy. Vwgkh xysnkowxdgm kat sfxno $\omega$ , jdm arb koct vz vwrsw xdz ejozkmgco hq wlntfofz uqgytg abqy boavyx kexm zuy qfshhxmgg.

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