Physcs 111 ectue 11 Angula Momentum SJ 8th Ed.: Chap 11.1 11.4 Recap and Ovevew Coss Poduct Revsted Toque Revsted Angula Momentum Angula Fom o Newton s Second aw Angula Momentum o a System o Patcles Angula Momentum o a Rgd Body about a Fxed Axs Consevaton o Angula Momentum 11.1 The Vecto Poduct and Toque 11. Angula Momentum: The Non-Isolated System 11.3 Angula Momentum o a Rotatng Rgd Object 11.4 Consevaton o Angula Momentum: Isolated System 11.5 The Moton o Gyoscopes and Tops Copyght R. Janow Spng 01
So a: smple (plana) geometes Rotatonal quanttes θ,, ω, α, τ, etc epesented by scalas Rotaton axs smply CCW o CW dmensonal poblems, otaton axs pependcula to page Now: moe poweul tool o 3D Coss poduct epesents otatonal quanttes as vectos: Coss poducts pont along nstantaneous axes o otaton Dectons o otaton axes can be calculated and summed up lke othe vectos, e.g., τ F τ F sn( θ ) τ net τ Angula momentum - new conseved otatonal quantty. ke lnea momentum, t s conseved o solated systems l p p m v l p sn( θ ) Denton: tot l all all Second aw n tems o conseved quanttes: dp F ma F net p dt d net I α τ net dt nea: net s constant F net Rotatonal: s constant τ 0 τ net 0 Copyght R. Janow Spng 01
Angula momentum concepts & denton How much lnea o otatonal stayng powe does a movng object have? - Poduct o an neta measue wth a speed measue. - nea momentum: p mv (lnea). - I object s otatng only: angula momentum - moment o neta x angula velocty Iω - lnea momentum x moment am about some axs o smple cases lnea momentum neta speed lnea otatonal m v I ω pmv Iω the angula momentum o a gd body elatve to a selected axs about whch I and ω ae measued: unts: [kg.m /s] I ω gd body angula momentum ω Why bothe wth angula momentum? It s conseved o solated systems. Copyght R. Janow Spng 01
Angula momentum concepts & denng examples Example: Angula momentum o a pont mass movng n a staght lne choose pont P as a otaton axs P v lnea momentum X moment am Note: 0 moment am 0 s the same o object at any pont along lne o v I ω m ω mv p v ω Example: Same as above, but wth velocty not pependcula to ; v ad does not aect P I ω m ω mv mv sn( φ ) p ω v v v φ lnea momentum X moment am Note: 0 v s paallel to (adally n o out) Example: Angula momentum o a otatng hoop about symmety axs though P P v Iω I t s a hoop: I m v ω m ω mv p same as a pont patcle, o hoop lnea momentum X moment am 1 I t s a dsc: I m m ω 1 Copyght R. Janow Spng 01 /
Example: Calculatng Angula Momentum o a Rgd Body Calculate the angula momentum o a 10 kg dsc when: ω 30 ad / s, 9 cm 0.09 m, m Rotaton axs s nomal to dsc though ts CM 10 kg Soluton: I ω I m o a dsk 1 1 1 m ω 10 9 10 3. 4 10 + 1.96 13 Kg m / s What angula speed would a 10 kg SOID SPHERE (same dmensons) have t s angula momentum s the same as above? Soluton: 1.96 Kg m /s, 9 cm 0.09 m, m 10 kg ω ω ω / I I 1.96 400 5 5 ad / s m 10 9 o a sphee 10 4 Copyght R. Janow Spng 01
Denton: Angula momentum o a sngle patcle Extenson o lnea momentum s mv Depends on chosen otaton axs (hee along z) Moment o momentum Same pctue as o toques Use moment am sn(θ).o... tangental momentum component p p sn(θ) Only the tangental momentum component contbutes and p tal-to-tal always om a plane s pependcula to that plane p p p sn( θ moment p am x m( lnea v) p ) momentum z y P 90 o p θ moment am o p p θ θ 90 o x p ad lne o acton o momentum p Conventon: vecto up out o pape vecto down nto pape (tal) Copyght R. Janow Spng 01
Example: Angula momentum o a patcle n unom ccula moton The angula momentum vecto ponts out o the dagam The magntude s mv sn (90 o) mv sn (90 o) s used snce v s pependcula to A patcle n unom ccula moton has a constant angula momentum about an axs though the cente o ts path Examples: satelltes n ccula obts O Supeposton: Angula Momentum o a System v net 1 + +... + n o ths case: all all p net + 1 p 1 - p Copyght R. Janow Spng 01
Example: calculatng angula momentum o patcles PP1060-3*: Two objects ae movng as shown n the gue. What s the total angula momentum about pont O? No need to use omal coodnate system o such a smple poblem m Fo each patcle: l p mv Whee s the moment am Momenta and dsplacements om O all le n the plane o the slde, so angula momenta ae pependcula to the slde (n o out) l 1 + l.8 x 3.1 x 3.6-1.5 x 6.5 x. 31.5-1.45 (along z) m 1 9.8 kg.m /s CCW Copyght R. Janow Spng 01
Angula momentum o a ca 11.1. A ca o mass 1000 kg moves wth a speed o 50 m/s on a ccula tack o adus 100 m. What s the magntude o ts angula momentum (n kg m /s) elatve to the cente o the ace tack (pont P )? A) 0 B) 5.0 10 6 C).5 10 4 A D).5 10 6 B E) 5.0 10 3 P 11.. What would the angula momentum about pont P be the ca leaves the tack at A and ends up at pont B wth the same velocty? ) p p p sn( θ Copyght R. Janow Spng 01
The Rotatonal Second aw and Angula Momentum nea moton: Apply oce to a patcle. The oce causes the lnea momentum to change. The net oce actng on a body s the tme ate o change o t s lnea momentum F net dp dt The momentum s constant the net oce zeo. Rotatonal moton: Apply toque to a gd body. The toque causes the angula momentum to change. The net toque actng on a body s the tme ate o change o t s angula momentum. τ The angula momentum s constant the net toque zeo. net d dt τ and to be measued about the same ogn The ogn should not be acceleatng (should be an netal ame) Copyght R. Janow Spng 01
Rotatonal Second aw apples to sngle patcles, systems, and gd bodes: Fo BUT ntenal toques n the sum cancel n 3d law pas. Only Extenal Toques contbute to sys a sngle sys dt d body : d sys d dt dt τ τ,ext v τ Total angula momentum o a system o bodes: τ d dt net sys all about same ogn τ net toque on patcle Intenal toque pas ae ncluded net extenal toque on the system Same eeence axs o s and τ s Holds about any otaton axs, and I s changng Must use mass cente as ogn t (cm) s acceleatng (othewse non-netal eects show up) System s called ISOATED net toque on t s zeo then angula momentum o system s constant Copyght R. Janow Spng 01
Foce nea Momentum Tanslaton F v p COMPARISON mv Toque Angula Momentum Rotaton τ v l F p Knetc Enegy K 1 mv Knetc Enegy K 1 Ι ω nea Momentum P Systems and Rgd Bodes p M v cm Angula Momentum Ι ω Second dp F aw net dt o gd bodes about common xed axs Second v d sys aw τ net dt Momentum consevaton - o closed, solated systems P sys constant sys constant Apply sepaately to x, y, z axes Copyght R. Janow Spng 01
Example: A non-solated system Masses connected by a lght cod Fnd the lnea acceleaton a. Use angula momentum appoach No cton between m and table Mechancal enegy s constant (no nonconsevatve oces), but we gnoe that Block, pulley and sphee ae non- solated system n that net toque s not zeo. Constants: Equal Fo Fo v' s pulley masses and v a' s ω R a α R o block α d ω /dt dv/dt and sphee Ignoe ntenal oces, consde extenal oces only Net extenal toque on system: Angula momentum o system: (not constant) Use Second aw (deentate): d sys dt α a sys 1 m vr + m vr + I m 1 ar + m ar + I α (I + m 1 R + m R ) α τ net (I + m R 1 m gr 1 + τ m R net m 1 gr ) I about cente o ω wheel a m gr same esult ollowed om eale method usng 3 FBD s & nd law Copyght R. Janow Spng 01 1
Angula Momentum Consevaton o systems and gd bodes When the net extenal toque on a system s zeo: system s solated d dt τ net 0 constant When some event changes the state o an solated system: whee " " s nal, "" s ntal state s conseved sepaately o x, y, z dectons Angula momentum consevaton s as mpotant as enegy and lnea momentum consevaton Typcal poblems may nvolve... Isolated systems, so 0 Rgd bodes otatng aound a common otaton axs Some event changes moment o neta, shape, collson ntal Ι 0 ω 0 nal Ι ω ntal nal Copyght R. Janow Spng 01
Example: A patcle on a stng otatng aound a peg Stng waps aound peg as mass spals nwad.what you see: Radus deceases Angula speed ω nceases Tangental speed v changes Intal condtons: At t 0: m 0. kg, v 0 1 m/s, 0 0.5 m, ω 0 v 0 / 0 ad/s Fnd: ω, v when adus 0. m Method: System s solated (Why?) Centpetal oce on m exets zeo toque about peg, snce X F 0, so Repesent angula momentum: p p ( p mv, v ω snce p) m ω I ω Angula momentum s constant (conseved) as cod waps up: v 0 m 0 I 0 ω 0 I ω I m.5 ω 0 ω 0 ω 0 I m. (moment o neta changes) ( ) x 1.5 ad/s ( much > ) 0 ω v ω.5 m/s ( much > v 0 ) 0 Theme: s constant whle moment o neta changes, Copyght R. Janow Spng 01
Restcton: I ω Fo otaton o a gd body aound a SYMMETRY AXIS, o otaton o a lat body n the x-y plane about z-axs Examples: Iω vesus xp o dmensonal moton z Sngle pont mass v ω x v y Ccula path n x-y plane Fnd about cente o path v v o,ω p m v mv sn(90 ) kˆ Sht Ogn: z m ω kˆ I ω I scale vecto kˆ Same moton: ccula path paallel to x-y plane p s n plane paallel to x - y plane, ω I ω ω kˆ & ω both along Z-axs p ω v v Angula momentum has components along and nomal to ω z kˆ + xy xy p + m z kˆ p x xy Fo Iω To wok: y so... I ω z xy Need anothe equal mass symmetcally acoss om m, same moton Adds to z, but cancels component o nomal to z p along z n x-y plane Copyght R. Janow Spng 01
Angula momentum o a bowlng ball 11.3. A bowlng ball s otatng as shown about ts mass cente axs. Fnd t s angula momentum about that axs, n kg.m /s A) 4 B) ½ C) 7 D) E) ¼ 11.4. Suppose the otaton axs s shted to be tangent to the sphee and paallel to the z axs n the pctue. Fnd the angula momentum about that axs, n kg.m /s ω 4 ad/s M 5 kg ½ m I /5 MR I ω Copyght R. Janow Spng 01
Demonstaton: Spnnng Poesso Isolated System τ net about z - axs 0 I 0 ω 0 I ntal nal ω constant Moment o neta changes Copyght R. Janow Spng 01
How ast should the poesso spn? The poesso s otatng (no cton) wth angula speed 1. ev/s. Ams ae outstetched wth a bck n each hand. The moment o neta o the system consstng o the poesso, the bcks, and the platom about the cental axs s 6.0 kg m. By loweng the bcks the moment o neta o the system deceases to.0 kg m. (a) what s the esultng angula speed o the platom? (b) what s the ato o the system s new knetc enegy to the ognal knetc enegy? (c) In pat (b), what accounts o the deence (added KE)? s constant whle moment o neta changes, Copyght R. Janow Spng 01
How ast should the poesso spn? I 0 6 kg-m I kg-m ω 0 1. ev/s ω? ev/s s constant whle moment o neta changes, Zeo extenal toque nal ntal... about a xed axs I 0 ω 0 I ω I 6 Soluton (a): ω 0 ω x 1. ad/s 3.6 ev/s I 0 I 0 1 I ω 0 Soluton (b): K I ω I I 0 3 K 1 0 I 0 0 I 0 ω 0 I ω Soluton (c): KE has nceased!! The exta KE came om wok done when pullng the weghts n and/o educng potental enegy by loweng the ams. Copyght R. Janow Spng 01
Contollng spn (ω) by changng I (moment o neta) In the a, τ net 0 s constant I 0 ω 0 I ω Change I by culng up o stetchng out - spn ate ω must adjust Moment o neta changes Copyght R. Janow Spng 01
Intenal toques do not change total angula momentum...... t s edstbuted wthn the solated system τ net 0 about z - axs sys Demonstaton Intenal toques not 0 they evese wh constant Spacecat maneuves by spnnng the lywheel the cat counte-otates po, nal wheel tot 0 cat + lywheel Copyght R. Janow Spng 01
Tetheed Astonauts 11.5. Two astonauts each wth mass M ae connected by a massless ope o length d. They ae solated n space, obtng the cente o mass at dentcal speeds v. One o them pulls on the ope, shotenng the dstance between them to d/. What ae the new total angula momentum and speeds v? A) mvd/, v v/ B) mvd, v v C) mvd, v v D) mv d, v v/ E) mvd, v v/ I ω mv T Copyght R. Janow Spng 01
Consevaton o Angula Momentum: Mey-Go-Round Poblems The moment o neta o the system the moment o neta o the platom plus the moment o neta o the peson (a patcle). As the peson moves towad the cente o the otatng platom the moment o neta deceases. The angula speed must ncease snce the angula momentum s constant. o A peson uns tangent to the edge o the platom and then jumps on. The total angula momentum o the system (peson + platom) s constant. The angula velocty o the platom changes. Copyght R. Janow Spng 01
Example: A mey-go-ound poblem A 40-kg chld unnng at 4.0 m/s jumps tangentally onto a statonay ccula mey-go-ound platom whose adus s.0 m and whose moment o neta s 0 kg-m. Thee s no cton. a) Fnd the angula velocty o the platom ate the chld jumps on. b) Fnd the change n the total knetc enegy o the system. c) Fnd the change n the knetc enegy o the chld alone. Moment o neta changes Total s constant tot I 0 ω 0 I ω System chld + mey-go ound, beoe and ate collson System s solated (zeo net extenal toque). Inelastc collson Copyght R. Janow Spng 01
Example: A mey-go-ound poblem- Soluton I 0 kg.m v T 4.0 m/s m c 40 kg.0 m ω 0 0 ω tot I 0 ω 0 I 0 + m v I ω tot c T tot I I m tot + c ω I m + c v T m c a) ω 1.78 ad/s 40 x 4 x 0 + 40 x 4 1 1 K K T K 0 I tot ω m c v 84. 4 b) K 35. 6 Joules 1 1 K chld K c K 0 mc ω m c v T 30. 0 nelastc collson KE deceases c) K 66. 5 chld Joules Copyght R. Janow Spng 01