Week 9 Chapter 10 Section 15


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1 Week 9 Chapter 10 Secton 15 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent, but the rgd object model s very useful n many stuatons where the deformaton s neglgble Ths smplfcaton allows analyss of the moton of an extended object Angular Poston Axs of rotaton s the center of the dsc Choose a fxed reference lne Pont P s at a fxed dstance r from the orgn A small element of the dsc can be modeled as a partcle at P Angular Poston, Pont P wll rotate about the orgn n a crcle of radus r Every partcle on the dsc undergoes crcular moton about the orgn, O Polar coordnates are convenent to use to represent the poston of P (or any other pont) P s located at (r, ) where r s the dstance from the orgn to P and s the measured counterclockwse from the reference lne Angular Poston, 3 As the partcle moves, the only coordnate that changes s As the partcle moves through, t moves though an arc length s. The arc length and r are related: s = r Radan Ths can also be expressed as s r s a pure number, but commonly s gven the artfcal unt, radan One radan s the angle subtended by an arc length equal to the radus of the arc Whenever usng rotatonal equatons, you must use angles expressed n radans 1
2 Conversons Comparng degrees and radans 360 1rad 57.3 Convertng from degrees to radans 180 rad degrees Angular Dsplacement The angular dsplacement s defned as the angle the object rotates through durng some tme nterval f Ths s the angle that the reference lne of length r sweeps out Average Angular Speed The average angular speed, ω avg, of a rotatng rgd object s the rato of the angular dsplacement to the tme nterval avg f t t t f Instantaneous Angular Speed The nstantaneous angular speed s defned as the lmt of the average speed as the tme nterval approaches zero lm t 0 d t dt Angular Speed, fnal Unts of angular speed are radans/sec rad/s or s 1 snce radans have no dmensons Angular speed wll be postve f θ s ncreasng (counterclockwse) Angular speed wll be negatve f θ s decreasng (clockwse) Average Angular Acceleraton The average angular acceleraton,, of an object s defned as the rato of the change n the angular speed to the tme t takes for the object to undergo the change: f avg t t t f
3 Instantaneous Angular Acceleraton The nstantaneous angular acceleraton s defned as the lmt of the average angular acceleraton as the tme goes to 0 lm d t 0 t dt Angular Acceleraton, fnal Unts of angular acceleraton are rad/s² or s  snce radans have no dmensons Angular acceleraton wll be postve f an object rotatng counterclockwse s speedng up Angular acceleraton wll also be postve f an object rotatng clockwse s slowng down Angular Moton, General Notes When a rgd object rotates about a fxed axs n a gven tme nterval, every porton on the object rotates through the same angle n a gven tme nterval and has the same angular speed and the same angular acceleraton So all characterze the moton of the entre rgd object as well as the ndvdual partcles n the object Drectons, detals Strctly speakng, the speed and acceleraton ( are the magntudes of the velocty and acceleraton vectors The drectons are actually gven by the rghthand rule Rotatonal Knematcs Under constant angular acceleraton, we can descrbe the moton of the rgd object usng a set of knematc equatons These are smlar to the knematc equatons for lnear moton The rotatonal equatons have the same mathematcal form as the lnear equatons The new model s a rgd object under constant angular acceleraton Analogous to the partcle under constant acceleraton model Rotatonal Knematc Equatons t f 1 f t t f f 1 f f t all wth consant 3
4 Comparson Between Rotatonal and Lnear Equatons Relatonshp Between Angular and Lnear Quanttes Dsplacements s r Speeds v r Acceleratons a r Every pont on the rotatng object has the same angular moton Every pont on the rotatng object does not have the same lnear moton Speed Comparson The lnear velocty s always tangent to the crcular path Called the tangental velocty The magntude s defned by the tangental speed Acceleraton Comparson The tangental acceleraton s the dervatve of the tangental velocty dv d at r r dt dt ds d v r r dt dt Clcker Queston Alex and Bran are rdng on a merrygoround. Alex rdes on a horse at the outer rm of the crcular platform, twce as far from the center of the crcular platform as Bran, who rdes on an nner horse. When the merrygoround s rotatng at a constant angular speed, what s Alex s tangental speed? A. twce Bran s B. the same as Bran s C. half of Bran s D. four tmes of Bran s E. mpossble to determne Speed and Acceleraton Note All ponts on the rgd object wll have the same angular speed, but not the same tangental speed All ponts on the rgd object wll have the same angular acceleraton, but not the same tangental acceleraton The tangental quanttes depend on r, and r s not the same for all ponts on the object 4
5 Centrpetal Acceleraton An object travelng n a crcle, even though t moves wth a constant speed, wll have an acceleraton Therefore, each pont on a rotatng rgd object wll experence a centrpetal acceleraton v ac r r Resultant Acceleraton The tangental component of the acceleraton s due to changng speed The centrpetal component of the acceleraton s due to changng drecton Total acceleraton can be found from these components a a a r r r 4 4 t r Rotatonal Moton Example For a compact dsc player to read a CD, the angular speed must vary to keep the tangental speed constant (v t = r) At the nner sectons, the angular speed s faster than at the outer sectons Rotatonal Knetc Energy An object rotatng about some axs wth an angular speed, ω, has rotatonal knetc energy even though t may not have any translatonal knetc energy Each partcle has a knetc energy of K = ½ m v Snce the tangental velocty depends on the dstance, r, from the axs of rotaton, we can substtute v = r Rotatonal Knetc Energy, cont The total rotatonal knetc energy of the rgd object s the sum of the energes of all ts partcles 1 KR K mr 1 1 KR mr I Where I s called the moment of nerta Rotatonal Knetc Energy, fnal There s an analogy between the knetc energes assocated wth lnear moton (K = ½ mv ) and the knetc energy assocated wth rotatonal moton (K R = ½ I ) Rotatonal knetc energy s not a new type of energy, the form s dfferent because t s appled to a rotatng object The unts of rotatonal knetc energy are Joules (J) 5
6 Moment of Inerta The defnton of moment of nerta s I r m The dmensons of moment of nerta are ML and ts SI unts are kg. m We can calculate the moment of nerta of an object more easly by assumng t s dvded nto many small volume elements, each of mass m Moment of Inerta, cont We can rewrte the expresson for I n terms of m I r m r dm lm m 0 Wth the small volume segment assumpton, I r dv If s constant, the ntegral can be evaluated wth known geometry, otherwse ts varaton wth poston must be known Notes on Varous Denstes Volumetrc Mass Densty mass per unt volume: = m / V Surface Mass Densty mass per unt thckness of a sheet of unform thckness, t : t Lnear Mass Densty mass per unt length of a rod of unform crosssectonal area: = m / L = Moment of Inerta of a Unform Rgd Rod The shaded area has a mass dm = dx Then the moment of nerta s L / M Iy r dm x dx L / L 1 I ML 1 Moment of Inerta of a Unform Sold Cylnder Moments of Inerta of Varous Rgd Objects Dvde the cylnder nto concentrc shells wth radus r, thckness dr and length L dm = dv = Lr dr Then for I Iz r dm r Lr dr 1 Iz MR 6
7 Clcker Queston A secton of hollow ppe and a sold cylnder have the same radus, mass and length. They both rotate about ther long central axes wth the same angular speed. Whch object has hgher rotatonal knetc energy? A. Sold cylnder B. The hollow ppe C. Same D. Informaton s not suffcent to determne ParallelAxs Theorem In the prevous examples, the axs of rotaton concded wth the axs of symmetry of the object For an arbtrary axs, the parallelaxs theorem often smplfes calculatons The theorem states I = I CM + MD I s about any axs parallel to the axs through the center of mass of the object I CM s about the axs through the center of mass D s the dstance from the center of mass axs to the arbtrary axs ParallelAxs Theorem Example The axs of rotaton goes through O The axs through the center of mass s shown The moment of nerta about the axs through O would be I O = I CM + MD Moment of Inerta for a Rod Rotatng Around One End The moment of nerta of the rod about ts center s 1 ICM ML 1 D s ½ L Therefore, I ICM MD 1 1 I ML M ML 1 3 L 7
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