Rotation The basic quantity in otation is the angula displacement simila ole to the linea displacement studied in chapte units: adians, eolutions, degees, θ Poblem soling stategy is the same as fo poblems with linea motion... Phy - Sping 3 Rotation Poblems with constant angula acceleation ae soled just like those with constant linea acceleation. Eample (linea): A ca can acceleate at 5.8 m/s. If it stats fom est, how fa does it tael befoe it eaches a speed of 5 m/s? 5m/s Eample (angula): A potte s wheel can acceleate at 5.8 ad/s. If it stats fom est, how many eolutions does it make befoe it eaches an angula elocity of 5 ad/s? Phy - Sping 3
Analogies between Rotation and Linea Motion (I) Rotational angula position: q (adians) angula elocity: w (ad/sec) angula acceleation: a (ad/s ) Linea position: (metes) elocity: (m/s) acceleation: a (m/s ) θ θ αt t αt α ( θ θ ) at t at a( ) Phy - Sping 3 3 Relation between linea and angula aiables Fundamental elations a θ s s θ distance taeled (adius fom otation cente)(angula displacement [adians]) linea elocity (adius)(angula elocity[ad/s]) a t a t α Tangential acceleation (adius)(angula acceleation [ad/s ]) Phy - Sping 3 4 a Radial acceleation (angula elocity[ad/s]) (adius)
Angula Vaiables Relation between angula fequency and otation fequency: Reminde about fequency and peiod: so ( πad ) f w in ad/s (p)(fequency in Hz) f T π T (adian units sometimes disappea in calculations) Phy - Sping 3 5 Angula and linea aiables Eample: A compute had disk otates at 54 pm (otations pe minute) what is its angula elocity? If the eading head is located 3. cm fom the otation ais, what is the speed of the disk below it? What is the linea acceleation of that point on the disk? If each bit of infomation occupies 5.µm of length on the disk, how many bits pe second can the witing head wite? If the disk takes 4.s to spin up fom est, what is the aeage angula acceleation duing spin up? 3. cm Phy - Sping 3 6 3
Kinetic enegy of otating object Fo now, conside only a igid body otating about a fied ais 3 K K m m ( m m m ) I m m 3 3 m 3 m 3 3 3 I m i i Rotational inetia sum of (mass)(adius fom ais) Phy - Sping 3 7 Units: kg m Rotational inetia plays a ole in otational motion equialent to that of mass in linea motion: esists angula acceleation kinetic enegy is popotional to it constucted so that the equation fo kinetic enegy of a otating object (K½I? ) look like that fo an object in linea motion (K½m ) depends on: the mass of a body the location of the otation ais the shape of the body I m i i sometimes call moment of inetia Phy - Sping 3 8 4
Rotational Inetia Eample: An asymmetical dumbbell is made fom a kg mass and a kg mass connected by a od (of negligible mass) o length.5m.. kg. kg.5m What is the kinetic enegy if the od is otated at 5 e/s about a point in the cente of the od? about the cente of the kg mass? Phy - Sping 3 9 Rotational Inetia Fo etended objects, the sum can be eplace with an integal I i mi I dm Simplest eample is a hoop otated about an ais though its cente point, pependicula to the plane of the hoop. R I R dm R m Phy - Sping 3 5
A few othe impotant shapes: Rotational Inetia Solid disk/cylinde about cente: I½MR R Solid sphee about cente: I(/5)MR Thin od about cente: I(/)ML R See HRW page 49 fo moe etensie table L Phy - Sping 3 Rotational Inetia Composite shapes: Simply add otational inetia, but be caeful about ais location: Eample: Disk of mass M and od of mass M otated about a common cental ais: I M R M L R L Phy - Sping 3 6
Paallel Ais Theoem Useful fo calculating the otational inetia about an ais paallel to an ais though the cente of mass: h I I cm Mh Rotational inetia otational inteia about cm ais (total mass)(distance between ais) Inteesting coelay: of all the possible paallel ais, the one that gies the smallest intetia is the one though the cente of mass Phy - Sping 3 3 Paallel Ais Theoem Eample: A clock pendulum consists of a od of length cm (of negligible mass), with a g disk of adius 5 cm at the end. What it the otational inetia about the piot point? cm 5 cm Phy - Sping 3 4 7
Toque Toque plays the ole in angula motion equialent to that of Foce in linea motion τ F τ F F F sinφ F φ F Toque ecto poduct of displacement fom ais and foce Magnitude of toque (pependicula moment am)(foce) (piot-to-foce distance)(pependicula component of foce) Units: Netwon-metes (not called Joules when you ae talking about toque) Reiew HRW p. 46-47 fo popeties of ecto poduct Phy - Sping 3 5 Toque Analogy between otational and linea: τ Iα F ma toque (otational inetia)(angula acceleation) Eample: A od kg od of length 4.m is esting on a piot which is. m fom one end. What is the angula acceleation when the othe end of the od is eleased? mg Phy - Sping 3 6 8
Wok and Powe Wok: analogy with linea motion (constant toque, constant foce): W τ θ W F Eample: A flywheel at est is subject to a toque of 5 Nm fom an electic moto. How many tuns will the flywheel make befoe it has a kinetic enegy of J? Powe: analogy with linea motion: P τ P F Eample: If the flywheel in the eample aboe has a otational inetia of kgm, what is the powe supplied by the moto when the flywheel eaches J? Phy - Sping 3 7 Vectos in angula motion Diection of angula elocity is pependicula to plane of otation with diection gien by ight-hand ule: Same with angula acceleation: Speeding up: α Slowing down: α Phy - Sping 3 8 9
Phy - Sping 3 9 Analogies between Rotation and Linea Motion (II) Rotational angula position: q (adians) angula elocity: w (ad/sec) angula acceleation: a (ad/s ) toque: t (N m) Linea position: (metes) elocity: (m/s) acceleation: a (m/s ) Foce: F (N) τ θ τ α τ θ θ α α θ θ α P W I I K t t t ) ( F P F W ma F m K a at t at ) ( Phy - Sping 3 Rolling Motion Rolling is a paticula combination of tanslational (linea) motion and otation motion. Conside one full eolution: Fo olling without slipping: α θ R a R R θ πad,,a all efe to the motion of the cente of the olling object πr
Rolling Motion To get the elocity of any point on a olling object, you must add the ectos of the puely otational elocity and the tanslational elocity of the cente: A B C Eample: find the instantaneous speeds at points A, B, and C fo a hoop olling at m/s. Note that the elocity at point C is zeo. The point of contact is instantaneously at est. This is why you use static fiction to ealuate the fictional foce on a olling object. Phy - Sping 3 EX. A bicycle tie olls acoss the gound as shown below. What point on the tie is moing staight down?. a. b 3. c 4. d 5. all of the aboe. 6. none of the aboe. Phy - Sping 3
Kinetic Enegy fo otation tanslation Simply add the tanslational kinetic enegy (of the whole mass moing linealy with the cente-of-mass elocity) plus the otational kinetic enegy (of the puely otational motion about the cente of mass) cm K M cm I Total kinetic enegy tanslational KE otational KE (tue fo all igid bodies) Fo olling (without slipping) bodies, thee is a special elation between the tanslation and the otation: Rw ( MR I) ( M I R ) K (tue fo olling objects of adius R) / cm Phy - Sping 3 3 Kinetic enegy of olling bodies Eample: A solid disk olls (without slipping) down a amp stating fom est to a table. metes below. What is the speed with which it olls along the table?.m f Note that the answe does not depend on the mass of the disk the angle o shape of the amp the adius of the disk It does depend on the shape (disk, hoop, sphee, etc.) Phy - Sping 3 4
E. A hollow cylinde and a solid cylinde stat fom est at the same position at the top of a amp. Which has the lage cente-of-mass speed when it eaches the bottom of the amp?. The hollow cylinde.. The solid cylinde. 3. The speeds ae the same. Phy - Sping 3 5 E. Two solid cylindes with diffeent masses stat fom est at the same position at the top of a amp. Which has the lage cente-of-mass speed when it eaches the bottom of the amp?. The heay cylinde.. The light cylinde. 3. The speeds ae the same. Phy - Sping 3 6 3
Foces and toques in olling Feely olling object: a f s N mg a Mg sin θ f s Ma cm Net foce (mass)(acceleation of cente of mass) f s R Iα Net toque about cm (otational inetia)(angula acceleation) Eample: If the coefficient of fiction between a ball (sphee) and a amp is., what is the steepest angle whee the ball will oll without slipping? Phy - Sping 3 7 Yo-yo s Rolling object can dop moe slowly if the mass is concentated nea the edge of the body. Slowness chaacteized by I β MR A yo-yo is a way of getting a ey lage β by using a small adius to fi the olling and a lage adius R to gie a lage otational inetial: I MR R This is studied in detail in the IPL ( Mawell s Wheel ) Phy - Sping 3 8 4
E. Two bass knobs ae added at the same distance fom the cente of Mawell's Wheel, which is then aised by winding the sting aound the shaft. When the Wheel eaches the end of the sting, its speed will be. lage if the knobs ae placed close to the shaft.. lage if the knobs ae placed fathe fom the shaft. 3. the same egadless of whee the knobs ae placed. Phy - Sping 3 9 Angula Momentum Angula momentum is defined with espect to some fied point: p m m l p m( ) Angula momentum ecto poduct of (ecto fom fied point to object) (linea momentum of object) elated to toque: d l / dt τ like momentum, this has a lot moe applications when you conside a system of paticles athe than a single paticle. Phy - Sping 3 3 5
Angula Momentum Eample : m/s kg A B m kg m/s m What is the angula momentum of the system about point A? about point B? Phy - Sping 3 3 Angula Momentum Eample : A ball is located at position (3m)i(4m)j and is has linea momentum p(kgm/s)i(kgm/s)k. What is the angula momentum of the ball with espect to the oigin? y p l l l y z y z p p p z y z y p p p y z Answe: l(4kgm /s)i(-3kgm /s)j(-8kgm /s)k Check: l p l Phy - Sping 3 3 6
Angula momentum Rigid body otation about fied ais: L I Angula momentum (otational inetia)(angula elocity) L analogous to pm fo linea momentum Eample: What is the angula momentum of the Eath due to its otation? 4 6 M E 6 I E 5 MR kg R E 9.7 6.4 37 kgm E (πad)/(864s) 7.3 33 7. kgm /s L E m Phy - Sping 3 33 5 ad/s Conseation Angula Momentum Angula momentum is conseed: Fo a system with no net etenal toque acting on it, the total angula momentum is constant angula momentum is still conseed een if thee ae intenal toques etenal foces (as long as they don t poduce toques) Eample: A (non-otating) sack of ice with otational inetia of kgm is dopped onto a disk otating on at ad/s. The disk has a otational inetia of 3. kgm. What is the otational speed of the system afte the ice and disk couple? Phy - Sping 3 34 7
E. A dumbbell is otating about its cente as shown. Compaed to the dumbbell's angula momentum about its cente, its angula momentum about point B is. bigge.. the same. 3. smalle. Phy - Sping 3 35 E. A student, standing on a platfom that otates without fiction, holds a bicycle wheel that otates counteclockwise (as seen fom aboe). Afte tuning the ale of the bicycle wheel upside down, the student and platfom. emain stationay.. begin to otate in a clockwise diection. 3. begin to otate in a counteclockwise diection. Phy - Sping 3 36 8
E. A figue skate stands on one spot on the ice (assumed fictionless) and spins aound with he ams etended. When she pulls in he ams, he angula elocity. emains the same.. inceases. 3. deceases. Phy - Sping 3 37 E. A figue skate stands on one spot on the ice (assumed fictionless) and spins aound with he ams etended. When she pulls in he ams, he angula momentum. emains the same.. inceases. 3. deceases. Phy - Sping 3 38 9
E. A figue skate stands on one spot on the ice (assumed fictionless) and spins aound with he ams etended. When she pulls in he ams, she educes he otational inetia and he angula speed inceases so that he angula momentum is conseed. Compaed to he initial otational kinetic enegy, he otational kinetic enegy afte she has pulled in he ams must be. the same.. lage because she's otating faste. 3. smalle because he otational inetia is smalle. Phy - Sping 3 39 Angula Momentum Eample: A meteo of mass 5 kg cashes into the Eath at the equato, moing in an eastwad diection fom an angle of 3 degees aboe the hoizon, at a speed of 7m/s. What is the esulting change in the angula elocity of the Eath? o φ 7m/s I E 5 L E MR 7. 9.7 33 kgm 37 kgm /s L m Rmsin φ R N 6 ( 6.37 m)( 5 kg)( 7m/s )(.866). kgm /s I Li L f E i Lm I E m f Phy - Sping 3 4
Conseation of Angula Momentum Special case: If two bodies hae zeo net angula momentum, then the otation angles ae elated by I θ I θ This is simila to a system of two bodies at est, whee m m Eample: If you stand on a tuntable and otate a hoop of otational inetia.kgm aboe you head, and you tun by 3 degees when it makes a full tun, what is you otational inetia? Phy - Sping 3 4 Angula Momentum and Toque The fundamental elation is like that between momentum and foce: aeage toque: dl τ dt Sum of toques ate of change of angula momentum τ L t Eample: The otational inetia of the oto system is 35 kgm, and it go fom est to its final speed of 3 e/min in 6.7 second. What is the aeage toque fom the engine? Phy - Sping 3 4
m Conseation of L in collisions Eample: ball hits babell and sticks (totally inelastic) P m befoe just afte late i / i m i c.m. d i /3 d i /3 i m L ( d / 3)( m ) m d / 3 i i I cm 3 m( d /3) m(d / 3) md P (3m)( /3) m L I md 3 md / 3 i i i d i Phy - Sping 3 43 Analogies between Rotation and Linea Motion (III) Rotational angula position: q (adians) angula elocity: w (ad/sec) angula acceleation: a (ad/s ) otational inetia: I (kgm ) toque: t (N m) angula momentum: L (kgm /s) Linea position: (metes) elocity: (m/s) acceleation: a (m/s ) mass: m (kg) Foce: F (N) linea momentum: p (kgm/s) Phy - Sping 3 44
3 Phy - Sping 3 45 Analogies between Rotation and Linea Motion (III) Rotational Linea dt dl L L I L P W I I K t t t f i / τ τ θ τ α τ α θ θ α dt dp F p p m p F P F W ma F m K at t at f i / Phy - Sping 3 46 E. When a ca acceleates fowad, it tends to otate about its cente of mass. The ca will nose upwad. when the diing foce is imposed by the ea wheels (fo font-wheel die the ca would nose downwad).. whethe the diing foce is imposed by the ea o the font wheels.
E. Suppose you had a ca that was mostly wheels and anothe ca that had tiny wheels. If the cas had the same total mass and thei centes of mass ae equal distances fom the gound and each goes fom zeo to 4mph in ten seconds, which one will nose up the most?. Muscle Head.. Pip Squeak 3. Both the same. Phy - Sping 3 47 4