Chapte 5 Ccula Moton In a gd body, the dstances between the pats o the body eman constant. We begn nestgatng the otaton o a gd body. We conclude ou nestgaton n Chapte 8. The language used to descbe otatonal moton s ey smla to the language used to descbe lnea moton. The symbols ae deent. Descpton Lnea Angula poston dsplacement ate o change o poston, a a aeage ate o change o poston t t lm lm nstantaneous ate o change o poston t 0 t t 0 t Whle we ae amla wth angles measued n degees, we measue otatons n adans. In the gue, the angle measue n adans s dened as the ato o the ac length s to the adus : s Fo a complete otaton, s = and the angle o a complete otaton s adans. Ths ges the coneson o adans to degees 360 ad 180 ad Thee s a elaton between the speed o a pont on the m and the angula elocty o the wheel. 5-1
s s s t t Just lke dsplacements, otatons hae dectons. We take counteclockwse otatons as poste and clockwse otatons as negate. Othe useul paametes The peod (T) The tme t takes o a compete eoluton. The equency () The numbe o eolutons pe unt tme. These paametes ae elated. By denton 1 T A complete eoluton s the ccumeence. The speed s the dstance dded by the tme. Fo a complete eoluton, the tme s the peod T T Usng the equaton nea the top o the page, we hae Rollng moton I the wheel olls acoss the gound, ts tanslatonal speed depends on how ast t spns. 5-
I the wheel olls wthout slppng, the ale o the wheel moes wth a speed gen by Ths s a elate elocty poblem! A Ale o wheel, G Gound, R Rm o wheel AG AG AG AR AR AR RA 0 RG Radal Acceleaton A patcle undegong unom ccula moton ( s constant n tme), stll acceleates een though ts speed s constant. The elocty contnuously changes decton. The change n elocty ponts towads the cente o the ccle. To nd the magntude o the adal acceleaton, consde the dagam below. The magntude o the ac length s ey close to the ac length. As shnks, t becomes equal to ac length adus o ccle angle subtended t 5-3
Usng the denton o acceleaton we hae a t t t Usng = om aboe, we hae the two epessons o the adal acceleaton a You need to memoze the epessons o the adal acceleaton. The acceleaton s pependcula to the elocty as long as the magntude o the elocty s constant (the speed s constant). Poblem-Solng Stategy o an Object n Unom Ccula Moton (page 155) 1. Begn as o any Newton s second law poblem: denty all the oces actng on the object and daw a FBD.. Choose pependcula aes at the pont o nteest so that one s adal and the othe s tangent to the ccula path. 3. Fnd the adal component o each oce. 4. Apply Newton s second law as ollows: F ma whee F s the adal component o the net oce and the adal component o the acceleaton s a 5-4
(Fo unom ccula moton, nethe the net oce no the acceleaton has a tangental component.) Banked and Unbanked cues A ca tunng though a cue at constant speed s acceleated towads the cente o the cue. I the oad s lat, the adal oce s suppled by cton. I the wheels o the ca ae ollng wthout slppng, the adal oce s statc cton. I the wheels ae sldng, t s knetc cton. Fom the dagam aboe (note the decton o -as s contay to ou conenton) F s ma m F y ma N mg 0 y What s the astest sae speed o a cue? That s when the ca needs the lagest possble statc ctonal oce on the cue. m s,ma ma N mg s s g The mamum sae speed depends on the coecent o cton. The lowe s, the slowe the ca must go. (Slow down on wet steets!) the adus o the cue. Go slow aound shap cues (whee s small)! the acceleaton due to gaty. (Be caeul dng on the moon!!!) s 5-5
The stuaton can be mpoed by bankng the cue. The needed adal oce s suppled by a component o the nomal oce. It s mpotant to notce that the -as does not pont down the nclne. It ponts n the decton o the adal acceleaton to the let. The ca does not slde down the nclne. It s acceleated towads the cente o the cue. F ma N m N sn m N y F y ma W 0 N cos mg y Ddng the two equatons to elmnate the nomal oce m N sn N cos mg tan g A ca taelng ths speed aound a cue banked at angle wth adus, wll not eque any cton to saely tael aound the cue. What happens the ca goes too ast? What happens the ca goes too slowly? 5-6
Ccula obts The Eath emans n obt aound the sun because o the gatatonal pull o the sun. Gataton supples the adal oce needed to keep the Eath n ts obt. The same physcs occus wth satelltes n obt aound the Eath. Ate wokng o 0 yeas wth the obseatons o Tycho Bahe, Johannes Keple stated hs thee laws o planetay moton: The planets tael n ellptcal obts wth the Sun at one ocus o the ellpse. A lne dawn om a planet to the Sun sweeps out equal aeas n equal tme nteals. The squae o the obtal peod s popotonal to the cube o the aeage dstance om the planet to the Sun. In a emakable ecaton o hs law o gataton, Newton was able to dee Keple s laws. Nonunom Ccula Moton Suppose the otatng wheel changes ts angula speed. Ths tme does not pont towads the cente o the ccle! Thee s now a tangental acceleaton as well as the adal acceleaton we hae studed. Snce the adal and tangental dectons ae pependcula to each othe, the oeall acceleaton s a a a t Poblem-Solng Stategy o an Object n Nonunom Ccula Moton (page 165) 1. Begn as o any Newton s second law poblem: Identy all the oces actng on the object and daw an FBD.. Choose pependcula aes at the pont o nteest so that one as s adal and the othe s tangent to the ccula path. 5-7
3. Fnd the adal component o each oce. 4. Apply Newton s second law along the adal decton: whee F ma a 5. I necessay, apply Newton s second law to the tangental oce components: Ft ma t The tangental acceleaton component at detemnes how the speed o the object changes. Appaent Weght At what pont o the de would you eel heaest? Lghtest? To tael aound a ccle, thee must be an acceleaton towads the cente o the ccle. At the top, some o the adal acceleaton s suppled by the weght. The est s suppled by the nomal oce, the oce o the seat on the passenge. At the bottom, the nomal oce must oecome the weght. Weght does not change, but the nomal oce does. The eacton to the nomal oce (the seat pushng on the passenge) s the passenge pushng on the seat. That oce s lagest at the bottom, whee you eel the heaest. 5-8
Poblem 5.41 A ca appoaches the top o a hll that s shaped lke a etcal ccle wth a adus o 55.0 m. What s the astest speed that the ca can go oe the hll wthout losng contact wth the gound? N m g The adal decton s n the y-decton. Usng Newton s second law, mg N F m ma When the ca s just about to leae the gound N = 0. mg 0 m g (55.0 m)(9.8m/s ) 3. m/s Tangental and Angula Acceleaton We can add ows to ou table Descpton Lnea Angula poston dsplacement Rate o change o poston Aeage ate o change o poston, a a t t lm lm Instantaneous ate o change o poston t 0 t t 0 t a a, a Aeage ate o change o speed t t a lm lm Instantaneous ate o change o speed t 0 t t 0 t 5-9
The components o acceleaton ae a t a Usng smla easonng to what we used o unom lnea moton, we can ceate equatons o unom angula moton. Unom Lnea Moton Unom Angula Moton a constant constant a t t ( ) t t 1 1 t ( ) 1 ) a ( t a t 1 ) ( t Poblem 5.5 A dsk otates wth constant angula acceleaton. The ntal speed o the dsk s ad/s. Ate the dsk otates though 10 adans, the angula speed s 7 ad/s. (a) What s the magntude o the angula acceleaton? (b) How much tme dd t take o the dsk to otate though 10 adans? (c) What s the tangental acceleaton o a pont located at a dstance o 5.0 cm om the cente o the dsk? (a) (7 ad/s) ( ad/s) (10 ad).5 ad/s 7.07 ad/s (b) 1 ( ) t (10 ad) t 7 ad/s ad/s.s (c) a t (.5 ad/s )(0.05m) 0.353m/s 5-10
Weghtlessness Ae these people weghtless? Recall, GM g Een when 00 mles up, g s not zeo. In act, t s stll close to 9.8 m/s snce the 00 mle alttude s tny compaed to the adus o the eath (3,960 mles). Why do thngs loat n the space staton? Both o the acceleatons o the space staton and the astonauts ae the same. It s smla to the weghtlessness epeenced by a peson allng n a seeed eleato. To make atcal gaty, spn the space staton. Ths s the pncple o a centuge. E 5-11