We..7 -.9, (.) Moton Wth & Wthout Torque E. ab r. otaton ab Evals.0 Quantzaton, Quz, ect Evals E.e Mon. evew or nal (-) HW: Pr s 39, 57, 64, 74, 78 Sat. 9 a.m. nal Exam (Ch. -)
Usng ngular Momentum The measure o moton about a pont r p r Magntue p r p r p r sn p Magntue an Drecton =r p = (p z r y p y r z ), (p x r z p z r x ), (p y r x p x r y ) Drecton Orent ght han so ngers curl rom the axs an wth moton, then thump ponts n recton o angular momentum.
ngular Momentum Prncple rbll pole ball rope p ball han pole Magntue (yet another cross prouct) r r sn r r t ( ) about Quantes moton about a pont sn r sn ( about) where, ( Torque about) Interacton that changes moton about a pont r ( rom) Drecton (yet another cross prouct) r
Mult-Partcle ngular Momentum Prncple Wth Mult-Partcle ngular Momentum c s ext ext 3ext... t where r, etc. cs cs cms transs ext Star cm s ext rotcm or rg objects rotcm I cm cm r s r s r 3s Clou o Dust ext 3 3ext ext
Example: unorm sol sk wth raus 9 cm has mass 0.9 kg (moment o nerta I = ½M ). constant orce N s apple as shown. t the nstant shown, the angular velocty o the sk s 45 raans/s n the z recton. The length o the strng s 8 cm. t ths nstant, what are the magntue an recton o the angular momentum about the center o the sk? y z (z-axs ponts out o page) x What are the magntue an recton o the torque on the sk, about the center o mass o the sk? The strng s pulle or 0. s. What are the magntue an recton o the angular mpulse apple to the sk urng ths tme? ter the torque has been apple or 0. s, what are the magntue an recton o the angular momentum about the center o the sk? t ths later tme, what are the magntue an recton o the angular velocty o the sk?
Spnnng Skater Zero-Torque Systems Demo: spnnng umb bells Solar system ormaton Intal rot. rot. rot rot.. rot. rot. mass arther I rom axs: I larger smaller I 0 ave t nal mass closer to axs: I smaller larger What about the energy? K rot E I I I I nt. other E E E 0 0 K rot Ent. other nt. other nt. other Increase kc energy uele by change n nternal energy (Wheates) http://leng.lamost.org/courses/astrotoay/ CHISSON/T35/HTM/T350.HTM Increase rotatonal kc energy uele by change n nternal energy (gravtatonal potental) lso conser: ver, St-spn & lp spnnng wheel
y Completely Inelastc Collson & ngular Moton z (z-axs ponts out o page) x Chl runs an jumps on playgroun merry-goroun. or the system o the chl + sk (exclung the axle an the Earth), whch statement s true rom just beore to just ater mpact? K s total (macroscopc) kc energy P s total (lnear) momentum s total angular momentum (about the axle) What s the ntal angular momentum o the chl + sk about the axle? a. K, P, an o not change b. P, an o not change c. oes not change. K an P o not change e. K an o not change a) < 0, 0, 0 > b) < 0, mv, 0 > c) < 0, mv, 0 > ) < 0, 0, mv > e) < 0, 0, mv >
y Completely Inelastc Collson & ngular Moton z (z-axs ponts out o page) x The sk has moment o nerta I, an ater the collson t s rotatng wth angular spee. The rotatonal angular momentum o the sk alone (not countng the chl) s ter the collson, what s the spee (n m/s) o the chl? a. < 0, 0, 0 > b. < 0, I, 0 > c. < 0, I, 0 >. < 0, 0, I > e. < 0, 0, I > a. b. c.. / e. ter the collson, what s translatonal angular momentum o the chl about the axle? a. < 0, 0, 0 > b. < 0, m, 0 > c. < 0, m, 0 >. < 0, m(), 0 > e. < 0, m(), 0 >
y Completely Inelastc Collson & ngular Moton What s the nal angular spee o the merry-go-roun ater the k jumps on? z (z-axs ponts out o page).. t m,0 0,, 0 0, I.,0 0, mv 0, sk I cm m. g. r. chl. chl. M,0 0, m,0 0, mv,0 sk. cm M x System: chl + merry-go-roun ctve envronment: none that change angular momentum pproxmatons: neglgble rctonal torque at axel xs: xle 0, m M,0 0, mv, 0 m M mv m mv M M m v easonable?
Two-Step Example: blob o clay (mass m) s roppe a stance h to lan on an stck to a wheel (mass M, raus ) horzontally ½ o axs. What s the wheel s angular spee just ater the collson? Step : Ball alls to wheel; use energy E E K & B 0 E B B. nt EE U E, B mv mv mgh 0 B. B.0 0 v B. gh Step : Ball & wheel colle; use angular momentum System: Ball + Earth ctve envronment: none pproxmatons: neglgble rag Moment 0 m xs: C System: Ball + Wheel ctve envronment: no torques pproxmatons: tme nterval small enough axel rcton an Earth s gravtaton have neglgble eect W & BC. By rght-han-rule, all n the +z BC. recton IW & B. C mvbr Beore collson, r I W & B. C Irng. cm I. IW & B. C m gh IW & B. C M m m m gh M M m m gh m gh M m easonable? m cm gh M m M h Moment Moment / v B.
ngular Momentum: t ( ) about Three unamental Prncples ( about) (lnear) Momentum: t p E Energy: W Example all three together! Say we have a unorm 0.4 kg puck wth an 8 cm raus. 4 cm strng s ntally wrappe aroun ts crcumerence. I t s on a rctonless surace an a 0 N orce s apple to the en o the strng untl t s unwoun l = 0.4 m I puck = 0.08m m =? =? v cm =? a. What wll be ts rate o rotaton when the strng s ully unwoun? Energy Prncple K r E total W E trans nt r cm K rot l I I l I l t? l I l m l m K cm " W cm "
ngular Momentum: t ( ) about Example all three together! Say we have a unorm 0.4 kg puck wth an 8 cm raus. 4 cm strng s ntally wrappe aroun ts crcumerence. I t s on a rctonless surace an a 0 N orce s apple to the en o the strng untl t s unwoun l = 0.4 m = 0.08m Three unamental Prncples ( about) (lnear) Momentum: t v cm =? I puck =? m t? a. What wll be ts rate o rotaton when the strng s ully unwoun? b. How long was the orce apple? ngular Momentum Prncple (axs through nal locaton o cm) t t trans rot trans rot.. ) (.. ) t I r a t Torque an nal angular velocty n z recton l I I m m t t m m p E Energy: W l m ml K " " cm W cm
ngular Momentum: t ( ) about Example all three together! Say we have a unorm 0.4 kg puck wth an 8 cm raus. 4 cm strng s ntally wrappe aroun ts crcumerence. I t s on a rctonless surace an a 0 N orce s apple to the en o the strng untl t s unwoun l = 0.4 m I puck = 0.08m m Three unamental Prncples ( about) =? (lnear) Momentum: t t v cm =? a. What wll be ts rate o rotaton when the strng s ully unwoun? ml b. How long was the orce apple? t You try: c. How quckly s the puck nally slng, v cm?? p E Energy: W l m K " " cm W cm. How ar has the puck move,?
Mon. Tues. We. ab r..4-.6, (.3) ngular Momentum Prncple & Torque.7 -.9, (.) Moton Wth & Wthout Torque otaton Course Evals.0 Quantzaton, Quz E.c EP E. E.e Mon. evew or nal (-) HW: Pr s 39, 57, 64, 74, 78 Sat. 9 a.m. nal Exam (Ch. -)