Physics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4

Transcription

1 Physcs 6 ecture 6 Conservaton o Angular Momentum SJ 7 th Ed.: Chap.4 Comparson: dentons o sngle partcle torque and angular momentum Angular momentum o a system o partcles o a rgd body rotatng about a xed axs Conservaton o angular momentum Demonstratons Examples Comparson: dentons o torque and angular momentum or sngle partcles Torque τ r F Angular Momentum r p τ r F rf sn( φ) r p rp sn( φ) p mv τ r F rf sn( φ) rp rp sn( φ) p mv

2 The Rotatonal Second aw and Angular Momentum near moton: Apply orce to a mass. The orce causes the lnear momentum to change. The net orce actng on a body s the tme rate o change o t s lnear momentum dp F net Rotatonal moton: Apply torque to a rgd body. The torque causes the angular momentum to change. The net torque actng on a body s the tme rate o change o t s angular momentum d τnet τ and to be measured about the same orgn The orgn should not be acceleratng (should be an nertal rame) Sngle partcle proo o nd law usng angular momentum: dp d near: F net Rotatonal: τnet Proo: start wth the cross-product denton o angular momentum d d d ( r p) m ( r v) snce p mv expand usng dervatve chan rule: d dr dv m v r m [ v v + r a] + d r ma r F net τnet Conservaton aws near dp F p s constant F Momentum net net Angular d Momentum τ s constant τnet net

3 Proo o second law or SYSTEMS o rgd bodes: nd d Rotatonal law or a sngle body : τ Total angular momentum o a system o bodes: sys d sys d τ net torque on partcle τ nternal torque pars are ncluded n sum BUT nternal torques n the sum cancel n 3rd law pars. Only External Torques contrbute to sys dsys τ,ext τnet net external torque on the system Must reerence same orgn or s and τ s Holds about any rotaton axs, and s changng Must use mass center as orgn t (cm) s acceleratng (otherwse non-nertal eects show up) ndvdual angular momenta all about same orgn System s called SOATED net torque on t s zero then angular momentum o system s constant Example: A non-solated system Masses are connected by a lght cord Fnd the lnear acceleraton a. Use angular momentum approach No rcton between m and table Treat block, pulley and sphere as a nonsolated system rotatng about pulley axs. As sphere alls, pulley rotates, block sldes Constrants: Equal v' s and a' s or block and sphere v R or pulley α d/ a αr dv/ gnore nternal orces, consder external orces only Net external torque on system: τnet m gr about center o wheel Angular momentum o system: s s m vr + m vr + (not constant) sys Use Second aw (derentate): dsys m ar + mar + α ( + mr + mr ) α τnet mgr mgr same result ollowed rom earler α method usng 3 FBD s & ( + mr + mr ) nd law a a 3

4 Angular Momentum Conservaton or systems and rgd bodes When the net external torque on a system s zero: system s d τnet constant solated When some event changes the state o an solated system: where " " s nal, "" s ntal state s conserved separately or x, y, z drectons Angular momentum conservaton s as mportant as energy and lnear momentum conservaton Typcal problems may nvolve... solated systems, so Δ Rgd bodes rotatng around a common rotaton axs Some event changes moment o nerta, shape,... ntal Ι nal ntal nal Ι Example: A partcle on a strng rotatng around a peg Strng wraps around peg as mass sprals nward.what you see: Radus r decreases Angular speed ncreases Tangental speed v changes ntal condtons: At t : m. kg, v m/s, r.5 m, v /r rad/s Fnd:, v when radus r. m Method: System s solated Centrpetal orce on m exerts zero torque about peg, snce r X F, so Represent angular momentum: r p r p ( snce r p) p mv, v r mr Angular momentum s constant (conserved) as cord wraps up: mr mr. v r.5 m/s ( much > v) (moment o nerta changes).5 ( ) x.5 rad/s ( much > ) m v Theme: s constant whle moment o nerta changes, 4

5 Examples: Connecton between and rxp x, z r v Sngle pont mass Crcular path n x-y plane Fnd about center o path y v r o r p m r v mvr sn(9 ) kˆ mr kˆ kˆ scaler vector Along Z-axs WHAT F: z r r xy x WHAT F: Concluson: p Same moton, but sht the orgn Path s crcular and parallel to x-y plane Angular momentum has components along and normal to y p s n plane parallel to x - y plane, kˆ so... z z + xy r xy r xy p p kˆ + mr kˆ p along z z n x-y plane Add another equal mass symmetrcally across rom m, same moton Adds to z, but cancels component o normal to z Descrbes rotaton o a rgd body around a SYMMETRY AXS, or rotaton o a lat body n the x-y plane about z-axs Angular momentum o a bowlng ball 6.. A bowlng ball s rotatng as shown about ts mass center axs. Fnd t s angular momentum about that axs, n kg.m /s A) 4 B) ½ C) 7 D) E) ¼ 6.. Suppose the rotaton axs s shted to be tangent to the sphere and parallel to the z axs n the pcture. Fnd the angular momentum about that axs, n kg.m /s 4 rad/s M 5 kg r ½ m /5 MR 5

6 Translaton Force F near Momentum p mv SUMMARY Rotaton Torque τ r F Angular l r p Momentum Knetc Energy K mv Knetc Energy K Ι near Momentum Second aw Systems and Rgd Bodes P p Mv Angular cm Momentum F net dp Ι or rgd bodes about common xed axs Second aw τ net d sys Momentum conservaton - or closed, solated systems P sys constant sys constant Apply separately to x, y, z axes Demonstraton: Spnnng Proessor solated System τ net about z - axs ntal nal Moment o nerta changes constant 6

7 How ast should the proessor spn? The proessor on a platorm s rotatng (no rcton) wth angular speed. rev/s. Hs arms are outstretched and he holds a brck n each hand. The rotatonal nerta o the system consstng o the proessor, the brcks, and the platorm about the central axs s 6. kg m. By movng the brcks the proessor decreases the rotatonal nerta o the system to. kg m. (a) what s the resultng angular speed o the platorm? (b) what s the rato o the system s new knetc energy to the orgnal knetc energy? (c) n part (b), what accounts or the derence (added KE)? s constant whle moment o nerta changes, How ast should the proessor spn? 6 kg-m. rev/s kg-m? rev/s s constant whle moment o nerta changes, Zero external torque nal ntal... about a xed axs 6 Soluton (a): x. rad/s 3.6 rad/s Soluton (b): K 3 K Soluton (c): KE has ncreased!! The extra KE came rom work done when pullng the weghts n and/or reducng potental energy by lowerng the arms. 7

8 Controllng spn () by changng (moment o nerta) n the ar, τ net s constant Change by curlng up or stretchng out -spn rate must adjust Moment o nerta changes nternal torques do not change total angular momentum t s redstrbuted wthn the solated system τ net about z - axs sys Demonstraton nternal torques not they reverse wh constant Spacecrat maneuvers by spnnng the lywheel the crat counter-rotates t t boy, nal + wheel tot crat lywheel 8

9 Tethered Astronauts 6.3. Two astronauts each havng mass M are connected by a mass-less rope o length d. They are solated n space, orbtng ther center o mass at dentcal speeds v. By pullng on the rope, one o them shortens the dstance between them to d/. What are the new angular momentum and speed v? A) mvd/ mvd/, v v v/ B) mvd, v v C) mvd, v v D) mv d, v v/ E) mvd, v v/ mv T r Conservaton o Angular Momentum: The Merry-Go-Round The moment o nerta o the system the moment o nerta o the platorm plus the moment o nerta o the person. Assume the person can be treated as a partcle As the person moves toward the center o the rotatng platorm the moment o nerta decreases. The angular speed must ncrease snce the angular momentum s constant. or The person runs tangent to the edge o the platorm and then jumps on. The total angular momentum o the system (person + platorm) s constant. The angular velocty o the platorm changes. 9

10 Example: A merry-go-round problem A 4-kg chld runnng at 4. m/s jumps tangentally onto a statonary crcular merry-go-round platorm whose radus s. m and whose moment o nerta s kg-m. There s no rcton. a) Fnd the angular velocty o the platorm ater the chld has jumped on. b) Fnd the change n the total knetc energy o the system. c) Fnd the change n the knetc energy o the chld. Moment o nerta changes, Total s constant tot Example: A merry-go-round problem- Soluton kg.m V T 4. m/s M c 4 kg r m. ΔK K ΔK tot tot mcvtr tot tot + m c r mcvtr 4 x 4 x + m r + 4 x 4 a).78 rad/s K c tot b) ΔK chld K c K m c m c v Joules r T nelastc collson KE decreases m c v T c) ΔK chld Joules

11 Example: Eect o the choce o rotaton axs p h r 3 r 'O' d/ 'P' d/ r r 4 p p & p are ant-parallel momenta, same magntudes paths are d meters apart nd about ponts O and P a dstance h apart For pont O : O r p + r p mv(d/ ) + mv(d/ ) O mvd Note: one velocty s reversed For pont P : P r3 p + r4 p mv(h + d/ ) mv(h d/ ) mvd Same as above, derent axs!! P What happened to the noton that depends on choce o rotaton axs? p tot, then s the same or any rotaton axs Proo: et r r' + h where h s the dsplacement between rotaton O r p (r ' + h) p r ' p + h p P + h p p p, then and the net torques are equal tot O P axes O & P Also, F tot, then τ net s also the same or any rotaton axs Example: Turnstle problem A mud ball hts and stcks to a rod end m ball mass, M rod mass m Fnd nal angular velocty KE s not conserved nelastc collson p not conserved turnstle s bolted to loor BUT s conserved or (solated) system