PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment fo each chapte due nomnall a week late. But at least fo the fst two homewok assgnments, ou wll have moe tme. All the assgnments wll be done though MastengPhscs, so ou need to puchase the pemt to use t. Some numecal values n some poblems wll be andomzed. In addton to homewok poblems and quzzes, thee s a eadng equement of each chapte, whch s ve mpotant.
Chapte : Rotatonal Moton Movement of ponts n a gd bod All ponts n a gd bod move n ccles about the as of otaton z As of otaton obt of pont P A gd bod has a pefectl defnte and unchangng shape and sze. Relatve poston of ponts n the bod do not change P elatve to one anothe. gd bod In ths specfc eample on the left, the as of otaton s the z-as.
Movement of ponts n a gd bod (cont d) At an gven tme, the -d pojecton of an pont n the object s descbed b two coodnates (, ) In ou eample, -d pojecton onto the - plane s the ght one. P length of the ac fom the -as s: s whee s, n m, and n ad(an) A complete ccle: s π 36 o π ad 57.3 o ad ev/s π ad/s ev/mn pm
Angula dsplacement, veloct and acceleaton At an gven tme, the -d pojecton of an pont n the object s descbed b two coodnates (, ) Angula dsplacement: n a tme nteval P at t t t t Aveage angula veloct: ω avg ad/s t t t P at t Instantaneous angula veloct: ω lm t t d dt ad/s ω < (>) (counte) clockwse otaton
Angula dsplacement, veloct and acceleaton (cont d) At an gven tme, the -d pojecton of an pont n the object s descbed b two coodnates (, ) Aveage angula acceleaton: ω ω ω αavg ad/s t t t P at t P at t Instantaneous angula acceleaton: ω α t t dω dt lm ad/s
Coespondence between lnea & angula quanttes Lnea Angula Dsplacement Veloct Acceleaton v d / dt ω d / dt a dv / dt α d ω / dt
Case fo constant acceleaton (-d) Consde an object otatng wth constant angula acceleaton α α dω / dt α t ( d / dt) dt ( d / dt) t t dt t ( dω / dt) dt α dt t + α t t dt ω d ω α ω t ω ω α t ω ω + α t Eq.() ω t + ( α t / ) + ω t + α / ) Eq.() ( t
Case fo constant acceleaton (-d) (cont d) Elmnatng t fom Eqs.() & (): / ) ( α ω ω t Eq.( ) Eq.() / ) ( (/) )/ ( α ω ω α α ω ω ω + + Eqs.( )&() ) ( α ω ω +
kˆ Vectos n 3-dmenson z î ĵ ˆ, ˆ, j kˆ Vectos : unt vecto n,,z decton Consde a vecto: a Also Inne (dot) poduct ( a, a, a ) a ˆ + a ˆj + ˆ (,,), ˆj (,,), kˆ a b a b cos ab + ab + a b f a b z a z b (,,) z a b a z kˆ b a
Small change of adal vecto Rotaton b a small otaton angle ˆ fo and sn cos ˆ ˆ ˆ) sn ˆ cos ( << + Q j j j ˆ sn ˆ cos + Note: ˆ ˆ lm ˆ, ˆ we defne If d d / ˆ ˆ ˆ ˆ
unt vecto n decton ĵ Relaton between angula & lnea vaables ˆ ˆ ˆ ˆ ˆ (cos ˆ + sn ˆ) j ˆ ( sn,cos ) dˆ / d sn ˆ + cos ˆj ˆ unt tangental vecto unt adal vecto ˆ î ˆ A pont wth Fed adus unt vecto n decton ˆ (,), ˆj (,) d ˆ / d cos ˆ sn ˆj ˆ v d / dt d( ˆ) / dt ( dˆ / dt) ( dˆ / d )( d / dt) ωˆ v v ω s const.
unt vecto n decton ĵ ˆ î Relaton between angula & lnea vaables (cont d) ˆ A pont wth Fed adus unt vecto n decton a dv / dt ( dω / αˆ + αˆ αˆ at tangental component dt) ˆ + ω( d ˆ / ω ˆ ( v d( ωˆ) / a / )ˆ adal component dt ω( d ˆ / dt) d )( d / dt)
Eample Relaton between angula & lnea vaables (cont d) How ae the angula speeds of the two bccle spockets n Fg. elated to the numbe of teeth on each spocket? The chan does not slp o stetch, so t moves at the same tangental speed v on both spockets: ω v ω ω ω The angula speed s nvesel popotonal to the adus. Let N and N be the numbes of teeth. The condton that tooth spacng s the same on both spockets leads to: π N π N N N Combnng the above two equatons: ω ω N N v
Descpton of geneal otaton Wh t s not alwas ght to defne a otaton b a vecto ognal otaton about as otaton about as ognal otaton about as otaton about as z z z The esult depends on the ode of opeatons
Descpton of geneal otaton Up to ths pont all the otatons have been about the z-as o n - plane. In ths case the otatons ae about a unt vecto nˆ whee nˆ s nomal to the - plane. But n geneal, otatons ae about a geneal decton. Defne a otaton about b as: RH ule nˆ nˆ nˆ In geneal + +, & d d d + but f nfntesmall small, we can defne a vecto b + d d d d ω ω nˆ ; dt dω α α nˆ dt If the aes of otaton ae the same, + +
Knetc eneg & otatonal neta K N (/ ) m v vˆ N (/ ) ω (/ ) m N ( ω ) m A pont n a Rgd bod I as of otaton otatonal neta/ moment of neta
Knetc eneg & otatonal neta (cont d) K ( / ) Iω vˆ m Moe pecse defnton of I : I lm N dm N m ρ( ) dv A pont n a Rgd bod otaton as Compae wth: K ( / ) mv denst volume element And emembe consevaton of eneg: K U K + U + + W othe
Knetc eneg & otatonal neta (cont d) Moment of neta of a thn ng (mass M, adus R) (I) dm λds ;ds lnea mass denst Rd R ds I dm R λds 3 λ π R ( Rd ) λr π MR otaton as (z-as)
Knetc eneg & otatonal neta (cont d) Moment of neta of a thn ng (mass M, adus R) (II) otaton as (-as) dm λds;ds Rd R cos R ds I π ( R cos ) λrd λr 3 π π cos d 3 λr ( / )( + cos ) d 3 ( / ) λr [ + (/ )sn ] π cos cos sn cos π 3 ( / ) λr ( ) (/ ) MR
Knetc eneg & otatonal neta (cont d) Table of moment of neta
Knetc eneg & otatonal neta (cont d) Tables of moment of neta
, measued w..t. COM O COM Paallel as theoem The as of otaton s paallel to the z-as otaton as though P d a P b -a dm -b otaton as though COM COM: cente of mass I b) ] dm dm [( a) + ( ( + ) dm a dm b dm + ( a + b ) I com com dm am bm + d I com + Md com M m m com dm m M M total mass
Paallel as theoem (cont d) COM O otaton as though P d a P b -a dm -b Knowng the moment of neta about an as though COM (cente of mass) of a bod, the otatonal neta fo otaton about an paallel as s : I Icom + Md otaton as though COM
Coespondence between lnea & angula quanttes lnea angula dsplacement veloct acceleaton mass knetc eneg v d / dt ω d / dt a dv / dt α d ω / dt m K ( / ) mv I K ( / ) Iω
Eecses Poblem A mete stck wth a mass of.6 kg s pvoted about one end so that t can otate wthout fcton about a hozontal as. The mete stck s held n a hozontal poston and eleased. As t swngs though the vetcal, calculate (a) the change n gavtatonal potental eneg that has occued; (b) the angula speed of the stck; (c) the lnea speed of the end of the stck opposte the as. (d) Compae the answe n (c) to the speed of a patcle that has fallen.m, statng fom est. cm v Soluton (a) U Mg U cm Mg( cm (.6kg)(9.8m / s.784 J cm ) cm )(.5m.m) v.5 m. m
Poblem (cont d) (b) K + + U K U K ML K U U U (/ ) Iω ; I (/ 3) ω ( U ) / I (6.784 J ) /[(.6kg)(.m) ] 5.4 ad / s (c) v ω (.m)(5.4 ad / s) 5.4m / s (d) v v + a ( ); v,.m; a 9.8m / s v a ( ) 4.43m / s
Poblem The pulle n the fgue has adus R and a moment of neta I. The ope does not slp ove the pulle, and the pulle spns on a fctonless ale. The coeffcent of knetc fcton between block A and the tabletop s µ k. The sstem s eleased fom est, and block B descends. Block A has mass m A and block B has mass m B. Use eneg methods to calculate the speed of Block B as a functon of dstance d that t has descended. Soluton Eneg consevaton: E K + U K + U + W K U m gd K U, (/ ) m, W A v B µ m k + (/ ) m A B gd, ω v v v / R v + (/ ) Iω mb gd (/ )( ma + mb + I / R ) v + µ km v gd( m m + m A B B µ kma) + I / R A gd A ω ω B d I
Poblem 3 You hang a thn hoop wth adus R ove a nal at the m of the hoop. You dsplace t to the sde though an angle β fom ts equlbum poston and let t go. What s ts angula speed when t etuns to ts equlbum poston? Soluton K + U K I I U, K cm K + U + Md Mg cm, ; (/ ) Iω MR + MR MgR( cos β ), U MR cm, R β pvot pont R MgR ω ( cos β ) MR ω β g ( cos ) / R the ogn the ognal locaton of the cente of the hoop