Chapte 10 and elements of 11, 1 Rotaton of Rgd Bodes What s a Rgd Body? Rotatonal Knematcs Angula Velocty ω and Acceleaton α Rotaton wth Constant Acceleaton Angula vs. Lnea Knematcs Enegy n Rotatonal Moton: Moment of Ineta, I Rotatonal Dynamcs A new opeaton wth vectos: Coss Poduct Toque τ Newton s nd Law fo Rotatonal Moton Wok and Powe n Rotatonal Moton Angula Momentum, L Dynamcs Pecesson, Gyoscopes Consevaton of Angula Momentum
Rgd Body and Rotatonal Moton Objects wth sze that s, mass dstbuted n a volume can be consdeed as a collecton of many pont-lke patcles When these patcles do not tanslate elatve to each othe (.e., the object s not stetched o compessed), the system s called a gd body: t cannot be defomed The geneal moton of a gd body can be splt nto two types: 1. Tanslatonal (Lnea) Moton,, v, a, F, p... Rgd body and ts motons: A patcle of the body. Rotatonal (Angula) Moton Lnea velocty Angula velocty,,,,, L... Axs of otaton Tanslatonal velocty
Angula Knematcs Angula Poston θ In puely otatonal moton, all ponts on the object move n ccles aound the axs of otaton ( O ), wth each pont descbed by a vecto poston Def: The angle θ made by the poston vecto wth espect to an abtay axs (say x) s called angula poston θ O y θ x Ex: see the two ponts on the adjacent bcycle wheel: notce that, as long they ae not on the same adus, the ponts on the gd body wll have dffeent angula postons In ou appoach to otatons, angles wll be measued n adans: 1 adan s the angle at the cente of a ccle subtendng an ac equal to the adus of the ccle When the angle at the cente s expessed n adans, the length of the ac subtended s gven by: l Ex: The ccumfeence π of a complete otaton subtends an angle of π Conventon: angles measued counteclockwse ae postve angle measued clockwse ae negatve + l = θ θ = 1 ad l = θ θ
Angula Knematcs Angula Dsplacement and Velocty How can we use angula postons to descbe otatons? Notce that even though the angula postons of dffeent ponts of a wheel ae n geneal dffeent, when the wheel otates, all ponts otate though the same angle Def: The change n angula poston of all the ponts on a otatng gd body s called angula dsplacement: 1 Δθ θ θ1 Abtay adus x Then, to quantfy how fast the angula poston changes we have to defne fst the aveage angula velocty as the angula dsplacement dvded by tme: 1 t t t 1 Theefoe, lke n the lnea case, the nstantaneous angula velocty s gven by: lm t 0 t d dt Ex: If the bcycle wheel makes two complete otatons evey second, we say that t has a constant angula speed of (π/1 s) = 4π ad/s SI ad s
Angula Knematcs Angula Acceleaton. Angula vecto dectons Hence we can defne the aveage angula acceleaton as the ate at whch the angula velocty changes wth tme: 1 t t t 1 The decton of angula velocty s gven by a ght hand ule SI 0 0 ad s Hence, the nstantaneous angula acceleaton: lm d d t 0 t dt dt Although t s not as ntutve as n the tanslatonal case, the angula velocty and acceleaton ae vectos, pependcula on the ccle of otaton: Ex: If the bcycle wheel spns each second though an angle lage and lage by π, we say that each complete otaton ts aveage angula velocty s π, so ts nstantaneous angula velocty nceases by π ad/s, so t has a constant angula acceleaton π ad/s The decton of angula acceleaton s paallel o antpaallel wth the angula velocty dependng on whethe ω nceases o deceases
Angula Knematcs Relatng angula and lnea veloctes So, the angula dsplacement, velocty and acceleaton chaacteze the ente gd body: all ponts have the same Δθ, ω and α Howeve, the lnea dstance, velocty and acceleaton of ponts at vaous ad fom the axs of otaton ae dffeent: each pont has a dffeent Δl, v and a The lnea knematcs of each pont on a gd body can be elated to the oveall angula chaactestcs based on the elatonshp l = θ Fo nstance, consde a wheel otatng wth constant angula speed ω. A pont at dstance fom the cente of otaton wll otate wth constant lnea speed v tavelng an ac dl n a tme dt: theefoe, we obtan dl d v dt dt So, as long as they ae not at the same dstance fom the cente of otaton, the dffeent ponts on a gd body have dffeent lnea speeds, nceasng fom zeo n the cente of otaton to a maxmum value on the oute m of the otatng gd body v 3 = ω 3 v v = ω Δl = Δθ v 1 = ω 1 Δθ 3 1 v ω ω x
Angula Knematcs Relatng angula and lnea acceleatons In geneal the vecto lnea acceleaton of a patcle n ccula moton s not tangent to the tajectoy. It can be consdeed as havng two components one tangent to the tajectoy (paallel o ant-paallel wth the velocty) and one pependcula on the velocty: a t : tangent, descbes how the magntude of the lnea velocty vaes a : adal (o centpetal), descbes how the decton of the velocty vaes The component a t of a patcle at dstance fom the axs of otaton can be easly elated to the angula acceleaton α of the gd body: a t dv d dt dt The component a can be also elated to angula velocty: a v Theefoe, the net acceleaton s gven by: a a a t Physcal stuaton: consde a wheel otatng faste and faste ω ω 0 v a a t a v 0 ω ω x x
Angula Knematcs Unfomly acceleated otaton The lnea-angula elatonshp povdes an easy way to descbe ccula moton wth constant angula acceleaton α, by smply notcng that each pont on the otatng gd body acceleates unfomly along the espectve ac of ccle Assume that the moton stats at t 0 = 0 when the otaton s chaactezed by θ 0, ω 0 v 0 ω 0 t 0 = 0 At tme t ω Δl x v Δθ x Then, f the angula acceleaton α s constant, at a late nstant t, The lnea moton of one patcle of the gd body at dstance fom the cente of otaton l v t a t 1 0 v v a t 0 v v a l 0 1 0 l v v t t t t f each lnea quantty s dvded by, we obtan t t t t 0 1 0 0 t 0 1 0 0 t The otatonal moton of the ente gd body (vald fo any of ts pats)
Poblem: 1. Angula knematcs: A bcycle wheel tuns wth tme dependent angula velocty gven by t bt ct whee b, and c ae postve constants such that, fo t n seconds, ω wll be n ad/sec. a) Fnd the tme dependency of the angula poston of the wheel knowng that t stated at θ 0 = π/. b) Fnd the angula acceleaton of the wheel as a functon of tme. c) At what tme s the angula velocty of the wheel not changng?. Unfomly acceleated otaton: An automoble engne slows down fom 4500 pm to 500 pm n.5 s. Calculate a) ts angula acceleaton (assumed constant) b) the total numbe of evolutons the engne makes n ths tme. 3. Rotatonal and lnea moton: You ae to desgn a otatng cylndcal axle to lft buckets of cement fom the gound to a ooftop. The buckets wll be attached to a hook on the fee end of a cable that waps aound the m of the axle; as the axle tuns, the buckets wll se. a) What should the damete of the axle be n ode to ase the buckets at a steady.00 cm/s when t s tunng at 7.5 pm? b) If nstead the axle must gve the buckets an upwad acceleaton of 0.400 m/s, what should be the angula acceleaton of the axle be?
Cente of Mass Unconstaned gd bodes wll tend to otate aound the cente of mass, cm The cente of mass of objects wth unfomly dstbuted mass and symmetc n shape concde wth the cente of symmety Let s look at how we can compute the cm of a dscete dstbuton of mass: Def: The cente of mass of a system of patcles ndexed, each of mass m and coodnates = (x, y, z ) s a pont wth poston: Comments: Notce that the coodnates of the cm ae mx 1 xcm mx ycm m M my m 1 M m y z cm cm m mz m m 1 M In the case of contnuously dstbuted mass, the sums ae to be eplaced by ntegals. Howeve, the dscete elatonshp can be appled to a system of bodes whch can be consdeed as patcles located n the espectve centes of mass 1 1 0 1 3 4 x (m) 5 1 M Ex: Fve patcles of equal masses m ae algned at postons gven on the fgue below. Fnd the cente of mass mx mx mx mx mx xcm CM 5m 0 1 3 4 m 1. m m z m 1 3 4 5
Execse 1: Cente of mass of a system of patcles A dscete system conssts n 5 patcles of equal mass m placed n the cones of a pyamd wth squae bass of sde a, and heght a. a) What ae the coodnates of the patcles n the povded system of coodnates? x M 5m a a z 1 5 a cm 4 a a 3 y b) Calculate the coodnates of the cente of mass n tems of a.
Obsevaton useful n poblems: Rollng Moton (Wthout Slppng) In fgue (a), f the wheel of adus s ollng wthout slppng, the pont P on the m s at est wth espect to the floo when t touches t, whle the cente C moves wth velocty v to the ght In fgue (b), the same wheel s seen fom a efeence fame whee C s at est. Now pont P s movng wth velocty v. Snce P s a pont at dstance fom the cente of otaton, we have: v Theefoe, when a wheel, o a sphee, o a cylnde olls, ts tanslatonal speed (that s, the speed v cm of ts cente of mass) s elated to ts angula speed ω by vcm a) Wheel seen by someone on the gound: ω C P C v cm b) Wheel seen by someone on the bke: ω v cm v cm Cauton: even though ths has the same fom as the elaton between lnea speed of a pont and the angula speed, t s not the same thng v cm P
The otatonal knetc enegy of a gd body fomed of patcles ndexed by (as the wheel on the ght) s the sum of the knetc eneges of all ts pats 1 1 1 Kot m1v 1 mv... mv Enegy n Rotatonal Moton Knetc enegy By substtutng the otatonal quanttes, we fnd that the otatonal knetc enegy can be wtten 1 1 Kot m m Def: moment of neta I of a system of patcles Physcal stuaton: Consde some gdly connected patcles otatng about the cente of mass wth angula speed ω, and n the same tme tanslatng wth speed v cm Then, an abtay patcle of mass m located at dstance fom the cente of otaton, moves wth speed v v Rotatonal velocty, ω Theefoe, a gd body that has both tanslatonal moton (moton of ts cm) and otatonal moton (about ts cm) has both tanslatonal and otatonal knetc eneges: K mv I net 1 Tanslaton cm 1 cm Rotaton Tanslatonal velocty v cm By the adjacent agument, the net knetc enegy s the sum of the knetc eneges assocated wth ts two motons
Moment of Ineta Concept and fomulas So, whle mass gve the tanslatonal neta, the otatonal neta of an object wth espect to an axs of otaton s gven by ts moment of neta, I 1. If the object s a patcle of mass m n ccula moton of adus, the moment of neta s. If the object s made of a fnte numbe of patcles (dscete dstbuton of mass), the net moment of neta s 3. If the object has ts mass dstbuted contnuously, t contans an nfnty of nfntesmally small pats dm each at dstance fom the cente of otaton, so the sum must be eplaced by an ntegal I I I m m dm Notce that the same object can have dffeent moments of neta f a dffeent axs of otaton s consdeed The otatonal neta nceases the futhe the mass s dstbuted fom the axs of otaton Quz: The two cylndes shown have the same mass. Whch has a lage otatonal neta?
Moment of Ineta Calculaton The net moment of neta of a system of objects (patcles o szable bodes) wth known ndvdual moments I s gven by the sum I I I I net 1 3... If the moment of neta wth espect to an axs passng though ts cente of mass s known, the moment wth espect to a famly of paallel axes can be also calculated by usng Ex: 3 helcopte oto blades 1 I 3I 3 ml 3 I 50 kg m blade Paallel-Axs Theoem: If the moment of neta of an object of mass M about an axs though ts cente of mass s I cm, then the moment about any paallel axs at dstance d s: I I Md cm Coollay: The smallest otatonal neta s wth espect to the cm, so the objects wll tend to otate about t. Ex: The cm of a wench thown n a staght lne wll have a 1D tanslaton whle the body otates aound the cm
Enegy n Rotatonal Moton Consevaton of enegy When evaluatng the consevaton of enegy fo otatng gd bodes, the only change fom ou pevous appoach s that, besde the knetc enegy assocated wth the tanslaton of the cm, the total knetc enegy contans a otatonal tem So the expesson fo the mechancal enegy becomes E K K U mv I kx mgy tansl The consevaton of enegy can stll be wtten ot EW nonconsevatve Execse : Rollng moton. Assume that all the objects on the fgue have the same mass m and ae all eleased down the fctonless nclne fom est, fom the same ntal heght. If the adus of each ollng object s the same, whch object wll move faste at the bottom of the nclne? 1 1 1 cm cm
Poblems 4. Paallel-axs Theoem: A thn, ectangula sheet of metal has mass M and sdes of length a and b. Use the theoem to calculate the moment of neta of the sheet fo an axs that s pependcula to the plane of the sheet and that passes though one cone of the sheet. 5. Atwood machne evsted wth non-deal pulley: Two masses m 1, ae connected ove a pulley as n the fgue. Consde the pulley as a dsk of mass M and adus. If the system stats fom est wth the masses at the same level: a) Calculate the speed of the masses when mass m 1 moves up to heght h wth espect to the ntal level b) Use v to calculate the fnal angula velocty ω of the pulley. c) Calculate the fnal angula acceleaton of the pulley. M A B m 1 m 1 m h h m
What causes otatons? To otate an object whch s ntally at est, a foce s needed Besde the magntude of the foce, the otatonal effect of the foce depends on the poston of the foce wth espect to the axs of otaton, and on ts decton So, n ode to study how one can poduce otatons, one must take nto consdeaton all these quanttes, not only the smple foce as n the puely tanslatonal case Let s fst defne some geometcal chaactestcs: Def 1: A lne along the vecto foce s called the lne of acton Def : The pependcula dstance fom the axs of otaton to the lne of acton s called the leve am Ex: Consde a doo acted upon by a foce F. The otatonal effect depends on a) the stength (magntude) of the foce b) the poston whee the foce s appled wth espect to the hnges (axs of otaton) smalle effect F c) the decton of the foce (o lne of acton) zeo effect maxmum effect F lage effect F F
Vectos Coss poduct Results n a vecto Notaton: Defntons: A B o, n detemnant fom, coss poduct AB AB sn A B A ˆ ybz AB z y A ˆ zbx Ax Bz j A ˆ xby Ay Bx k Gven vectos: A Ax, Ay, Az B Bx, By, Bz A A A x y z AB B B B x y z ˆ ˆj kˆ (No poblem f don t know how to handle ths; use the pevous fom!)
Vectos Moe about the Coss Poduct Intepetaton fo the magntude: the magntude of the coss poduct between two vectos s the poduct between one of the vectos and the component of the othe vecto pependcula on the fst one: Asn B Decton: gven by a ght-hand ule: Algn you fnges along the fst vecto such that you can cul them towad the second vecto. The vecto poduct s pependcula onto the plane of the vectos n the decton ndcated by the thumb A AB BAsn A A A B B A n B out B Comment: Notce that, unlke the dot poduct, the vecto poduct s not commutatve A B B A
The Toque Defnton F Def: The toque of a foce s always defned wth espect to a cetan axs of otaton as the vecto poduct between the poston vecto of the pont whee the foce s appled and the foce vecto: We can descbe the decton usng a sgn conventon: τ > 0 τ < 0 f t tes to otate the f t tes to otate the object counteclockwse object clockwse Ex: A longe leve am s vey helpful n otatng objects: ths s why a wench can be used to loosen a bolt Fomally, ths s due to the fact that a longe nceases the toque fo the same F and θ Also, applyng the foce pependcula on the am wll maxmze the toque fo the same F and, snce snθ s maxmum when θ = 90 + F So, based on the defnton of vecto poduct, the magntude s F sn mn What about the decton of the toque? The vecto toque s pependcula on the plane of the otaton that t attempts to poduce SI θ Axs of otaton
The Toque Calculaton The magntude of the toque can be vewed n two ways: 1. Note that that the foce s pependcula on the leve am l, so Toque = (Foce Magntude) (Leve Am) F sn F θ θ l = snθ Axs of otaton F. Note that that the component of the foce along does not poduce otaton, so we eman only wth the pependcula component: Toque = (Dstance to Axs) (Pependcula Component of Foce) F sn F θ F F Fsn F Axs of otaton
Elements of Statcs Based on ou dscusson about dffeent types moton, we see that the moton s contolled by foces n the case of tanslatons and toques n the case of otatons Theefoe, we can ntoduce the condtons fo an object to be completely at est (o movng wth constant velocty) A sold body that s at est s sad to be n motonal equlbum To be n complete equlbum, a sold body must satsfy both tanslatonal and a otatonal condtons of equlbum: 1. f the net foce s zeo thee s no tanslaton F 0 F F. f the net toque s zeo wth espect wth any abtay cente of otaton (pvot), thee s no otaton 0 Snce the pvot s abtay, we can choose ts such that the equaton won t contan unknown foces x y 0 0 Ex: Rotaton wthout tanslaton: Even when the net foce s zeo, the object can stll otate. So, a zeo balance of foce does not gant complete equlbum
Poblems: 6. Rotatonal equlbum: A unfom, 56 kg beam s suppoted usng a cable connected to the celng, as shown n the fgue. The lowe end of the beam ests on the floo. What s the tenson n the cable? 56 kg 7. Rotatonal and tanslatonal equlbum: A hnged beam suppotng a sgn s held hozontally by a cable, as n the fgue. Calculate the foce F H and the tenson foce F T.
Poblem: 8. A system n equlbum: A od of mass m 1 and length L can otate about a bolt at 1/3 fom one end. A sphee wth unfomly dstbuted mass and adus R s attached at the othe end of the od. If a foce F acts pependcula on the od and keeps the system n equlbum makng an angle α wth the vetcal a) calculate the mass m of the sphee n tems of F, m 1, L, α, g and R. b) fnd the nomal foce on the bolt. F
Rotatonal Dynamcs What f the net toqe actng on a gd body s not zeo? Consde fst a patcle n ccula moton of adus We know that, f ts speed vaes, thee must be a tangental foce acceleatng t, such that, by Newton s nd law F ma m t If we consde the patcle as a gd body, what s the coespondng toque? F ma m I t t What about a gd body seen as an extended object contanng many patcles of mass m at postons fom the cente of otaton? Then, snce the angula acceleaton s the same fo the whole object, we obtan Newton s nd law fo the system: Comments: path m net toque Newton s nd law fo otatons I Ths law woks only fo gd bodes (such that α s the same fo all ponts) The net toque contans only the extenal toques: the ntenal toques cannot modfy the otaton of the system) v F t
Poblems: 9. Roto unfomly acceleated dynamcs: A helcopte oto blade can be consdeed a long thn od, as shown n the fgue. Recall that we calculated the net moment of neta of ths system to be I = 50 kg m. How much toque must be appled to bng the blades up to a speed ω n a tme t? 10. Atwood machne once agan wth pulley dynamcs ncluded: Two masses m 1, ae connected ove a pulley as n the fgue. Consde the pulley as a dsk of mass M and adus. If the system s allowed to move, calculate the acceleaton of the masses, and the angula acceleaton of the pulley. M A v m 1 m v
Consequently, extenal foces F ext have a combned tanslatonal and otatonal effect. If 1. the axs though the cente of mass (say along z- axs) s the axs of symmety. and the axs moves paallel wth tself then a Newton s nd law takes the genec fom Ex: A complex of extenal foces actng on a gd body detemnes a tanslaton and a otaton about the cente of mass Rotatonal Dynamcs Rotaton about a movng axs F ext I Ma z cm z F F 1 cm tanslatonal acceleaton of cm otatonal acceleaton about cm ext 1 z cm dv F F F ma m dt v cm Ex: Axes of symmety: 1 axs axes The toque s ceated by the same foces as the ones detemnng the tanslaton, as long as the foce s not algned wth the cente of mass and the decton of moton I I z F cm z cm 1 cm dz dt
Poblem: 11. Rollng fcton: A sold bowlng ball ollng (that s, wthout slppng) down a etun amp nclned at an angle β wth the hozontal. Let s see how we can analyze ths moton as a combnaton between a tanslaton and a otaton. a) What s the ball s tanslatonal and angula acceleatons b) What s the ollng fcton? I 5 MR
Angula Momentum Defnton The otatonal analog to the lnea momentum s the angula momentum L Def: Fo a patcle of mass m movng n the xy-plane wth a momentum p, the angula momentum wth espect to the ogn of the system of coodnates whee the poston of the patcle s, s L p Ex: If the patcle s movng n a ccle L I L L m I L L SI kg m s mv Theefoe, f we consde a symmetc gd body otatng about an axs of symmety wth angula speed ω,, wth moment of neta I elatve to the axs, the net angula momentum wth espect to axs wll be the sum of angula momenta of the patcles n the gd body: z z y ω y L L v p m mv L has the same decton as ω: pependcula on the ccle of otaton, as gven by the ght hand ule Cauton! Ths expesson doesn t wok f the symmety condtons ae not fulflled: n geneal the decton of L s not along the axs and ts decton changes x x
Angula Momentum Dynamcs, pecesson, and gyoscopes Newton s nd law fo otatonal moton can be efomulated n tems of angula momentum: f a net toque acts on a gd body, the angula momentum wll change at a ate equal to the net toque Consequently, f a foce acts on a otatng object such that t detemnes a toque pependcula on the angula momentum, the esult wll be a change n decton of the angula momentum n a moton called pecesson Ex: As long a top spns about ts axs, t s vey stable wth espect to the pull of ts weght even when slanted because ts toque s pependcula on the ts angula momentum such that t detemnes a pecesson. The same phenomenon explans the pecesson of the Eath as t s gavtatonally pulled by the Sun Moeove, f a gd body otates eally fast, small foces applyng pependcula toques wll hadly be able to change the axs of otaton Ths explans the functonalty of gyoscopes as navgatonal nstuments: espectve of the moton of a shp, a spnnng gyoscope wll keep n pontng n the same decton dl dt
Angula Momentum Consevaton Newton s nd law leads us to a condton fo the consevaton of angula momentum: f the net extenal toque actng on an object s zeo, the total angula momentum s constant: dl 0 dl 0 L L I const. dt Thus, f the pats of an solated otatng system edstbute due to some ntenal toques, the net angula momentum stays constant, even though ts stuctue changes: L L I I afte befoe afte Ex 1: No net otatons but otatng body pats A fallng cat twstng ts body An astonaut n space tyng to otate hs body A helcopte must have two otos Ex : Keple s nd law: planets move faste when they ae close to the sun Ex 3: A ballena edstbutng the mass of he body The total angula momentum stays the same, but the angula velocty changes when the mass s edstbuted changng the otatonal neta: befoe I I lage small small lage
Poblems: 1. Angula momentum of a patcle: A small block on a fctonless, hozontal suface has a mass of 0.050 kg. It s attached to a massless cod passng though a hole n the suface. The block s ognally evolvng a dstance of 0.300 m fom the hole, wth an angula speed of 1.75 ad/s. The cod s then pulled fom below, shotenng the adus of the ccle to 0.150 m. What s the new angula speed? 13. Angula momentum of gd bodes: Consde a tuntable to be a ccula dsk of moment of neta I t otatng feely at a constant angula velocty ω. The axs of the dsk s vetcal and the dsk s suppoted by fctonless beangs. A ecod of moment of neta I s dopped onto the tuntable. Thee s fcton between the two dsks. Afte ths "otatonal collson," the dsks wll eventually otate wth the same angula velocty. a) What s the fnal angula velocty, ω f, of the two dsks? b) What s the fnal otatonal knetc enegy, K f, of the two spnnng dsks n tems of the ntal knetc enegy K f and gven quanttes? c) Assume that the tuntable decceleates fo a tme Δt befoe eachng the angula velocty ω f. What was the aveage toque actng on the bottom dsk due to fcton wth the ecod?