Chapter 8 Torque and Angular Momentum
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1 Chaper 8 Torque and Angular Moenu Reiew of Chaper 5 We had a able coparing paraeer fro linear and roaional oion. Today we fill in he able. Here i i Decripion Linear Roaional poiion diplaceen Rae of change of poiion Aerage rae of change of poiion, a a li li Inananeou rae of change of poiion 0 0 a a, a Aerage rae of change of peed a li li Inananeou rae of change of peed 0 0 Ineria I Influence ha caue acceleraion Moenu p L The relaion (ofen phyical law) for roaional oion can be found by a iple ubiuion of roaional ariable for he correponding linear ariable. Roaional Kineic energy A wheel upended a i ai can pin in pace. Since he poin of he wheel are oing, he wheel ha kineic energy. All he piece in a rigid body reain a he ae locaion relaie o all he oher piece. or a roaing objec, he par furher away fro he ai of roaion are oing faer. r The oal kineic energy of all he piece will be K oal i i i 8-
2 K oal i i i i r i i r The quaniy in parenhee i called he roaional ineria (or he oen of ineria) I i i r i inding he Roaional Ineria (page 6). If he objec coni of a all nuber of paricle, calculae he u direcly.. or yerical objec wih iple geoeric hape, calculu can be ued o perfor he u. 3. Since he roaional ineria i a u, you can alway enally decopoe he objec ino eeral par, find he roaional ineria of each par, and hen add he. The roaional ineria depend on he locaion of he roaion ai. The ae objec will hae a differen roaional ineria depending on where i i roaing. Look a he forula for a hin rod below. 8-
3 The roaional kineic energy of a rigid objec roaing wih angular elociy i K I Copare o he ranlaional kineic energy K Torque A quaniy relaed o force, called orque, play he role in roaion ha force ielf play in ranlaion. A orque i no eparae fro a force; i i ipoible o eer a orque wihou eering a force. Torque i a eaure of how effecie a gien force i a wiing or urning oehing. The orque due o a force depend of he agniude of he applied force, he force poin of applicaion, and he force direcion. ir definiion of orque r r Becaue roaion hae direcion, we aign he + ign o orque ha caue counerclockwie roaion, and ign o orque ha caue clockwie roaion. Wha i he ign of he orque in he figure aboe? Torque are eaured in he uni of force ie diance. Thi i he ae dienion a work. Howeer, orque ha a differen effec han work. To keep he wo concep diinc, we eaure work in joule and orque in newon-eer. Second definiion of orque r 8-3
4 r ind he leer ar (or oen ar) by eending he line of he force and drawing a line fro he ai of roaion o ha i croe he line of he force a a righ angle. inding he leer ar i ofen he o difficul par of a orque proble. inding Torque Uing he Leer Ar (p 69). Draw a line parallel o he force hrough he force poin of applicaion; hi line i called he force line of acion.. Draw a line fro he roaion ai o he line of acion. Thi line u be perpendicular o boh he ai and he line of acion. The diance fro he ai o he line of acion along hi perpendicular line i he leer ar (r ). If he line of acion of he force goe hrough he roaion ai, he leer ar and he orque are boh zero. 3. The agniude of he orque i he agniude of he force ie he leer ar: r 4. Deerine he algebraic ign of he orque a before. Cener of Graiy When he graiaional force ac on an objec, all he all piece of he objec eperience he graiaional force. Thi ery large nuber of force aken around an ai will creae a orque. How do we deal wih ha? orunaely, we can grealy iplify he proble. The oal force can be conidered o ac a ingle poin called he cener of graiy. If he graiaional field i unifor in agniude and direcion, he cener of graiy i locaed a he cener of a. becoe 8-4
5 Work done by a orque The epreion for he work done i W r r Power i he rae of doing work W P Roaional Equilibriu We reeber ha for an objec o reain a re, he ne force acing on i u be equal o zero. (ewon fir law.) Howeer, ha condiion i no ufficien for roaional equilibriu. Wha happen o he objec o he righ? The force hae he ae agniude. 8-5
6 Condiion for equilibriu (boh ranlaional and roaional): 0 and 0 The obedien pool. and ake he pool roll o he lef, 4 o he righ, and 3 ake i lide. Proble-Soling Sep in Equilibriu Proble (page 74). Idenify an objec or ye in equilibriu. Draw a diagra howing all he force acing on ha objec, each drawn a i poin of applicaion. Ue he cener of graiy (CM) a he poin of applicaion of any graiaional force.. To apply he force condiion, chooe a conenien coordinae ye and reole each force ino i - and y-coponen. 3. To apply he orque condiion, chooe a conenien roaion ai generally one ha pae hrough he poin of applicaion of an unknown force. Then find he orque due o each force. Ue whicheer ehod i eaier: eiher he leer ar ie he agniude of he force or he diance ie he perpendicular coponen of he force. Deerine he direcion of each orque; hen eiher e he u of all orque (wih heir algebraic ign) equal o zero or e he agniude of he CW orque equal o he agniude of he CCW orque. 4. o all proble require all hree equilibriu equaion (wo force coponen equaion and one orque equaion). Soeie i i eaier o ue ore han one orque equaion, wih a differen ai. Before diing in and wriing down all he equaion, hink abou which approach i he eaie and o direc. There are any good eaple worked ou for you in he e. See page Eaple: Wha i he alle angle a ladder can ake o ha i doe no lide? 8-6
7 W g f Soluion: We will ue he condiion for roaional equilibriu 0 We can chooe any ai abou which o ake orque. The ai I chooe i where he ladder ouche he floor. The leer ar for he noral force and he fricional force will be zero and heir orque will alo be zero. Recall ha he orque i r If he ladder ha lengh L, he leer ar for he weigh i he hor horizonal line below he floor in he diagra. The leer ar i he perpendicular diance fro he line of he force o he poin of roaion. Here i i L r co The leer ar for he force of he wall puhing again he ladder i Uing he condiion for roaional equilibriu The condiion for ranlaional equilibriu are r Lin 0 g 0 L W Lin g co 0 8-7
8 0 and 0 y Referring o he BD, he -coponen gie W f W 0 0 f Again looking a he BD, he y-coponen g 0 g Oh, no! our equaion: Ue he orque relaion y 0 L W Lin g co 0 f W W g f L Lin g co 0 L W Lin g co g W in co g an W Subiue he wo force equaion fro ewon econd law an g f W 8-8
9 f an Since hi i a aic fricion force, f an an an If he coefficien of aic fricion i 0.4, he alle angle i 5 o. Equilibriu in he Huan Body orce ac on he rucure in he body. Eaple 8.0: The deloid ucle eer on he hueru a hown. The force doe wo hing. The erical coponen uppor he weigh of he ar and he horizonal coponen abilize he join by pulling he hueru in again he houlder. There are hree force acing on he ar: i weigh (g), he force due o he deloid ucle () and he force of he houlder join () conraining he oion of he ar. Since he ar i in equilibriu, we ue he equilibriu condiion. To ue he orque equaion we ue a conenien roaion ai. We chooe he houlder join a he roaion ai a ha will eliinae fro conideraion. (Why?) r g g 0 g 0 r 0 8-9
10 r g g in5r 0 r g g r in5 (30 )(0.75) (0. )in5 66 To uppor he 30 ar a 70 force i required. Highly inefficien!! The Iron Cro. Here i an inereing ideo: hp:// Becaue of he yery, half of he gyna weigh i uppored by each ring. Conider he BD aboe. Wr w w r Wrw r W (0.60) (0.045)in W 8-0
11 The force eered by he laiiu dori and he pecorali ajor on one ide of he gyna body i ore han nine ie hi weigh! The rucure of he huan body ake large ucular force neceary. Are here any adanage o he rucure? Due o he all leer ar, he ucle force are uch larger han hey would oherwie be, bu he huan body ha raded hi for a wide range of oeen of he bone. The bicep and ricep ucle can oe he lower ar hrough alo 80º while hey change heir lengh by only a few cenieer. (p. 8) A ideo of equilibriu and how eaily i can be diruped: hp:// Roaional or of ewon econd law Very iilar o he oher econd law. I Moion of Rolling Objec A rolling objec ha roaional kineic energy and ranlaional kineic energy. K K ran K ro CM I CM Why doe he objec roll (and no lide)? ricional force eer a orque on he objec. 8-
12 Eaple 8.3 The acceleraion of a rolling ball. The roaional for of ewon econd law i The orque on he ball i due o fricion So I rf I rf I I f r We can ue ewon econd law o find he linear acceleraion of he ball. A we uually do, ake he +-ai along he incline. gin f a a Ue he epreion for he fricional force o find, gin f a I gin a r 8-
13 Bu he acceleraion of he ball i relaed o i angular acceleraion, a = r. I gin a r Ia gin a r Ia gin a r gin a I r or a unifor, olid phere, I = (/5)MR and for a hin ring, I = MR. Which ha he larger acceleraion? Conider he effec of he roaional ineria on he acceleraion. Eaple: A olid phere roll down a hill ha ha a heigh h. Wha i i peed a he boo? Soluion: Ue coneraion of energy. Since he ball roll wihou lipping, he fricional force doen do any work. The diplaceen i zero in he definiion W = r co. U gy K U K 0 0 I The ranlaional peed of he ball i relaed o i roaional peed, = r. gy gh ( I gh I r I I( / r) r ) or he olid phere, I = (/5)MR gh I r gh ( 5) gh (7 5) 0 7 gh Thi i le han he anwer we found when we ignored rolling, = gh. Angular oenu We inroduced he idea of linear oenu in chaper 7. We had 8-3
14 dp d A iilar epreion ei for roaional oion dl d The ne eernal orque acing on a ye i equal o he rae of change of he angular oenu of he ye. The angular oenu L I i he endency of a roaing objec o coninue roaing wih he ae angular peed abou he ae ai. Angular oenu i eaured in kg /. If he ne orque i zero, we hae he coneraion of angular oenu L 0 L i L f If he roaional ineria of he ye change, i angular peed will change o copenae. Angular oenu i a ecor. (So i he orque!) The direcion i gien by he righhand rule. Our eaon are a conequence of he coneraion of angular oenu. 8-4
15 8-5 We hae copleed our udy of roaional oion. Try o ee how i i analogou o (he ore failiar) linear oion. Here i a uary: Decripion Linear Roaional poiion diplaceen Rae of change of poiion Rae of change of elociy a Aerage rae of change of poiion a, a Inananeou rae of change of poiion 0 li 0 li Aerage rae of change of elociy a a, a Inananeou rae of change of elociy a 0 li 0 li Equaion of unifor acceleraion a a a i f i f i f ) ( ) ( ) ( ) ( i f i f i f Ineria i i r i I Influence ha caue acceleraion r r ewon econd law y y a a I Work co r W W Kineic energy I
16 Moenu p L Iω dp dl ewon econd law d d Condiion for coneraion of oenu 0 0 Coneraion of oenu p i p f Li L f 8-6
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