MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction.............................................. 1 2. Toque and Angula Momentum a. Definitions.............................................. 1 b. Relationship: τ = d L/dt................................. 1 c. Motion Confined to a Plane............................. 2 d. Cicula Motion of a Mass...............................3 3. Systems of Paticles a. Total Angula Momentum............................... 4 b. Total Toque............................................ 4 c. Rigid Body Motion About a Fixed Axis................. 5 d. Example: Flywheel...................................... 5 e. Kinetic Enegy of Rotation.............................. 6 f. Linea vs. Rotational Motion............................ 6 4. Consevation of Angula Momentum a. Statement of the Law................................... 7 b. If the Extenal Toque is not Zeo....................... 7 c. Example: Two Flywheels................................ 7 d. Kinetic Enegy of the Two Flywheels.................... 8 5. Nonplana Rigid Bodies................................. 9 Acknowledgments............................................9 Glossay...................................................... 9 Poject PHYSNET Physics Bldg. Michigan State Univesity East Lansing, MI 1
ID Sheet: MISN-0-34 Title: Toque and Angula Momentum in Cicula Motion Autho: Kiby Mogan, HandiComputing, Chalotte, MI Vesion: 4/16/2002 Evaluation: Stage 0 Length: 1 h; 24 pages Input Skills: 1. Vocabulay: kinetic enegy (MISN-0-20), toque, angula acceleation (MISN-0-33), angula momentum (MISN-0-41). 2. Solve constant angula acceleation poblems involving toque, moment of inetia, angula velocity, otational displacement and time (MISN-0-33). 3. Justify the use of consevation of angula momentum to solve poblems involving toqueless change fom one state of unifom cicula motion to anothe (MISN-0-41). Output Skills (Knowledge): K1. Define the toque and angula momentum vectos fo (a) a single paticle (b) a system of paticles. K2. Stating fom Newton s 2nd law, deive its otational analog and state when it can be witten as a scala equation. K3. Stat fom the equation fo linea kinetic enegy and deive the coesponding one fo otational kinetic enegy. K4. Explain why consevation of angula momentum may not hold in one system but may if the system is expanded. Output Skills (Poblem Solving): S1. Fo masses in cicula motion at fixed adii, solve poblems elating toque, moment of inetia, angula acceleation, otational kinetic enegy, wok, and angula momentum. S2. Given a system in which angula momentum is changing with time due to a specified applied toque, econstuct the minimum expanded system in which total angula momentum is conseved. Descibe the eaction toque which poduces the compensating change in angula momentum. THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of ou poject is to assist a netwok of educatos and scientists in tansfeing physics fom one peson to anothe. We suppot manuscipt pocessing and distibution, along with communication and infomation systems. We also wok with employes to identify basic scientific skills as well as physics topics that ae needed in science and technology. A numbe of ou publications ae aimed at assisting uses in acquiing such skills. Ou publications ae designed: (i) to be updated quickly in esponse to field tests and new scientific developments; (ii) to be used in both classoom and pofessional settings; (iii) to show the peequisite dependencies existing among the vaious chunks of physics knowledge and skill, as a guide both to mental oganization and to use of the mateials; and (iv) to be adapted quickly to specific use needs anging fom single-skill instuction to complete custom textbooks. New authos, eviewes and field testes ae welcome. PROJECT STAFF Andew Schnepp Eugene Kales Pete Signell Webmaste Gaphics Poject Diecto ADVISORY COMMITTEE D. Alan Bomley Yale Univesity E. Leonad Jossem The Ohio State Univesity A. A. Stassenbug S. U. N. Y., Stony Book Views expessed in a module ae those of the module autho(s) and ae not necessaily those of othe poject paticipants. c 2002, Pete Signell fo Poject PHYSNET, Physics-Astonomy Bldg., Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. Fo ou libeal use policies see: http://www.physnet.og/home/modules/license.html. 3 4
MISN-0-34 1 MISN-0-34 2 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan F p 1. Intoduction Just as fo tanslational motion (motion in a staight line), cicula o otational motion can be sepaated into kinematics and dynamics. Since otational kinematics is coveed elsewhee, 1 the discussion hee will cente on otational dynamics. Ou goal is to deive the otational analog of Newton s second law and then apply it to the cicula motion of a single paticle and to systems of paticles. In paticula we wish to develop the elationship between toque and angula momentum and discuss the cicumstances unde which angula momentum is conseved. 2. Toque and Angula Momentum 2a. Definitions. The toque and angula momentum ae defined as vecto poducts of position, foce and momentum. Suppose a foce F acts on a paticle whose position with espect to the oigin O is the displacement vecto. Then the toque about the point 0 and acting on the paticle, is defined as: 2 τ = F. (1) Now suppose the paticle has a linea momentum P elative to the oigin. Then the angula momentum of the paticle is defined as: L = p. (2) The diections of τ and L ae given by the ight-hand ule fo coss poducts (see Fig. 1). 2b. Relationship: τ = d L/dt. Using the definitions of toque and angula momentum, we can deive a useful elationship between them. 1 See Kinematics: Cicula Motion (MISN-0-9) and Toque and Angula Acceleation fo Rigid Plana Objects: Flywheels (MISN-0-33). 2 See Foce and Toque (MISN-0-5). 0 0 (a) Figue 1. Vecto elationships fo: (a) toque (b) angula momentum (both diected out of the page). Stating fom Newton s second law, witten in the fom the toque is: (b) F = d p dt, (3) τ = F = d p dt. (4) This can be ewitten using the expession fo the deivative of a coss poduct: Help: [S-1] τ = d d ( p) dt dt p = d L v p. dt Now p = m v so v p = 0 (because the vecto poduct of paallel vectos is zeo), so the toque is: 3 τ = d L dt. (6) Thus the time ate of change of the angula momentum of a paticle is equal to the toque acting on it. 2c. Motion Confined to a Plane. The expession τ = d L/dt fo a paticle takes on a scala appeaance when the motion of the paticle is confined to a plane. Conside a paticle constained to move only in the x-y plane, as shown in Fig. 2. The toque on the paticle is always pependicula to 3 This equation is valid only if τ and L ae measued with espect to the same oigin. (5) 5 6
MISN-0-34 3 F p Figue 2., F, and p ae all coplana fo motion in a plane. x v m Figue 3. Fo a paticle in cicula motion, and v ae pependicula. this plane as is the angula momentum [wok this out using Eqs. (1) and (2)]. Equivalently, we say that τ and L have only z-components. Since thei diections emain constant, only thei magnitudes change. Then: τ = dl dt (motion in a plane). (7) This equation holds only if F and p ae in the same plane; if not (and they won t be fo non-plana motion), the full Eq. (6) must be used. 4 2d. Cicula Motion of a Mass. The toque and angula momentum fo the special case of a single paticle in cicula motion can be easily elated to the paticle s angula vaiables. Suppose a paticle of mass m moves about a cicle of adius with speed v (not necessaily constant) as shown in Fig. 3. The paticle s angula momentum is: L = m v, (8) but since and v ae pependicula, 5 the magnitude of L is: L = mv (9) and the diection is out of the page. Equation (9) may be ewitten in tems of the angula velocity (since v = ω) as: L = m 2 ω. (10) 4 The component equations ae τ x = dl x/dt, τ y = dl y/dt, τ z = dl z/dt. 5 See Kinematics: Cicula Motion (MISN-0-9). MISN-0-34 4 Similaly, the toque is: τ = dl dt whee α is the paticle s angula acceleation. 6 = m2 dω dt = m2 α, (11) 3. Systems of Paticles 3a. Total Angula Momentum. The total angula momentum of a system of paticles is simply the sum of the angula momenta of the individual paticles, added vectoially. Let L 1, L 2, L 3,..., L N, be the espective angula momenta, about a given point, of the paticles in the system. The total angula momentum about the point is: 7 L = L 1 + L 2 +... = N L i. (12) As time passes, the total angula momentum may change. Its ate of change, dl/dt, will be the sum of the ates dl i /dt fo the paticles in the system. Thus dl/dt will equal the sum of the toques acting on the paticles. 3b. Total Toque. The total toque on a system of paticles is just the sum of the extenal toques acting on the system. The toque due to intenal foces is zeo because by Newton s thid law the foces between any two paticles ae equal and opposite and diected along the line connecting them. The net toque due to each such action-eaction foce pai is zeo so the total intenal toque must also be zeo. Then the total toque on the system is just equal to the sum of the extenal toques: N τ = τ i,ext (system of paticles). (13) i=1 Fo the system, then: τ = d L dt. (14) In wods, the time ate of change of the total angula momentum about a given point, fo a system of paticles, is equal to the sum of the extenal toques about that point and acting on the system. 6 See Toque and Angula Acceleation fo Rigid Plane Objects: Flywheels (MISN-0-33). 7 Fo continuous mass distibutions the summation becomes an integation. i=1 7 8
MISN-0-34 5 MISN-0-34 6 3c. Rigid Body Motion About a Fixed Axis. A igid body is a system of paticles whose positions ae all fixed elative to each othe. Since L = mv = m 2 ω fo each paticle in the body, we may wite fo the total angula momentum: ( ) L = m i i 2 ω, (15) i whee we have assumed that the body is otating about a fixed axis with angula velocity ω. The quantity in paentheses, I i m i 2 i, (16) dm Figue 4. Mass on the im of a flywheel (heavy line). m 4 m 3 4 3 1 2 m 2 m 1 Figue 5. Rotational E k equals the sum of paticles E k. is called the moment of inetia of the body. 8 Thus we wite L = Iω (17) and, since the axis of otation is fixed, equations with a scala appeaance hold fo the toque: τ = dl dt = I dω = Iα. (18) dt 3d. Example: Flywheel. As a visual example, we calculate the toque and angula momentum fo the special case of a flywheel whee the entie mass is unifomly distibuted aound the im. Let dm be the mass of an infinitesimal segment of the im as shown in Fig. 4. The angula momentum dl of the mass dm is: dl = dm 2 ω. (19) Since and ω ae the same fo all points on the im, the total angula momentum fo the flywheel is: L = dl = 2 ω dm = M 2 ω = Iω, (20) whee M is the total mass of the flywheel. This is identical to Eq. (17) fo a single mass M in cicula motion. The total toque is also the same, i.e., τ = dl dt = M2 α = Iα. (21) 8 See Unifom Cicula Motion: Moment of Inetia, Consevation of Angula Momentum, Kinetic Enegy, Powe (MISN-0-41). 3e. Kinetic Enegy of Rotation. The total kinetic enegy of a system can be witten in tems of the system s moment of inetia and angula velocity. We stat with the statement that the total kinetic enegy of the system of paticles, each of which is in cicula motion about a fixed axis of otation, is equal to the sum of the kinetic enegies of the individual paticles. Each individual paticle of mass m i moves in a cicle of adius i about the axis of otation. If the positions of the paticles ae all fixed elative to each othe (as in a igid body), then the angula velocity ω is the same fo all paticles. The kinetic enegy of each paticle is thus: E ik = 1 2 m iv 2 i = 1 2 m i 2 i ω 2 The total kinetic enegy of the otating body is theefoe [ E k = 1 2 m 11ω 2 2 + 1 2 m 22ω 2 2 +... = 1 N ] m i i 2 2 By Eq. (16) the sum, m i i 2, is just the moment of inetia I of the body, so the otational kinetic enegy fo a igid body can finally be witten: i=1 ω 2 E k = 1 2 Iω2. (22) See if you can detemine the moment of inetia and kinetic enegy of ou flywheel in Sect. 3d. Help: [S-2] 3f. Linea vs. Rotational Motion. Hee is a compaison of the equations of dynamics in nomal and otational fom: 9 10
MISN-0-34 7 MISN-0-34 8 Table 1. Geneal foms. Nomal Rotational F = dp /dt τ = dl/dt E k = MV 2 /2 E k = Iω 2 /2 Fo a igid body otating about a fixed axis, the following compaisons hold. Table 2. Rigid-body foms. Nomal P = Mv F = dp/dt = Ma Rotational L = Iω τ = dl/dt = Iα Some of the equations shown in the Table 2 can be ewitten in vecto/tenso fom so they have validity beyond linea and cicula motion. 4. Consevation of Angula Momentum 4a. Statement of the Law. The total angula momentum of a system of paticles is conseved if thee ae no extenal toques acting on the system. Thus: τ = 0 = d L (23) dt so L is a constant. Fo a igid body otating about a fixed axis, L = Iω, so that if I changes, thee must be a compensating change in ω in ode fo L to emain constant. 4b. If the Extenal Toque is not Zeo. If the extenal toque on a system is not zeo then angula momentum is not conseved fo the system. Howeve, all is not lost, fo if the system is expanded to include whateve is causing the extenal toque on it (theefoe changing it into an intenal toque), angula momentum will be conseved fo the expanded system. 4c. Example: Two Flywheels. Conside what happens when two identical flywheels, one spinning and one at est, ae suddenly bought togethe. The spinning flywheel has total angula momentum L 0 = M 2 ω 0 befoe it is allowed to come in contact with the second flywheel. When the two flywheels come togethe, the fiction between them causes a toque Figue 6. Two flywheels, one spinning, one at est, ae bought togethe. to be exeted on the spinning one and, since this toque comes fom an extenal souce, the angula momentum of the spinning flywheel is not conseved. If, howeve, ou system consists of both flywheels, the toques each exet on the othe ae intenal to the system and so thee is no longe an extenal toque. Thus fo the expanded system, containing both flywheels, angula momentum is conseved. This tells us that: M 2 ω 0 = M 2 ω f + M 2 ω f = 2M 2 ω f, (24) whee ω f is the final angula velocity at which the two flywheels otate. Once they ae togethe the spinning one slowed down (ω 0 ω f ), the othe having gone fom zeo to ω f the two act as a single flywheel, with mass 2 M, otating with angula velocity ω f = ω 0 /2. Help: [S-3]. 4d. Kinetic Enegy of the Two Flywheels. When flywheels otating at diffeent speeds ae bought into contact, it can be expected that kinetic enegy is not conseved. This is because the nonconsevative fictional foce acts on the two flywheels. The kinetic enegy is initially and aftewad it is: Help: [S-4] E k0 = 1 2 M2 ω 2 0, (25) E kf = M 2 ω 2 f = 1 4 M2 ω 2 0. (26) Theefoe the atio of the final kinetic enegy to the initial kinetic enegy is: E kf E k0 = 1 2. (27) This means that half of the initial kinetic enegy has been dissipated due to the fictional foces between the sufaces of the two flywheels. Thus the total angula momentum of the system is conseved even though the total kinetic enegy is not. 11 12
MISN-0-34 9 5. Nonplana Rigid Bodies Although the flywheel has been used as a specific example of otational motion, the concepts descibed can be applied to all igid bodies whethe they ae plana o not. The otational motion of nonplana igid bodies is discussed elsewhee. 9 MISN-0-34 PROBLEM SUPPLEMENT Poblem 5 also occus on this module s Model Exam. PS-1 Acknowledgments This module is based on one by J. Boysowicz and P. Signell. Pepaation of this module was suppoted in pat by the National Science Foundation, Division of Science Education Development and Reseach, though Gant #SED 74-20088 to Michigan State Univesity. Glossay angula momentum (vecto): L = p, whee p is the linea momentum of a paticle at a position with espect to the oigin. consevation of angula momentum: the total angula momentum of a system is conseved if the extenal toque on the system is zeo. moment of inetia (of a system of paticles): mi i 2. the sum I igid body: a system of paticles whose positions ae fixed elative to each othe. the application of dynamics to otating bod- otational dynamics: ies. otational kinetic enegy: the kinetic enegy of a paticle o system of paticles otating about a fixed point. toque (vecto): located at. τ = F, whee F is the foce acting on a paticle 9 See Rotational Motion of a Rigid Body (MISN-0-36). 1. A flywheel of adius 0.50 m and mass 500.0 kg has a toque of 30.0 N m acting on it. a. Calculate its moment of inetia. b. Find its angula acceleation. c. Assuming the flywheel is initially at est, calculate the time equied to complete the fist two evolutions. d. What is its angula velocity and tangential velocity afte two evolutions? e. Find its kinetic enegy afte two evolutions. 2. A vey lightweight cicula platfom has a weight of 300.0 N placed on it at a distance 25 cm fom its cente. The platfom is placed hoizontally on a pedestal such that a fictional dag foce acts on the platfom at the point whee the weight is. The coefficient of fiction is µ = 0.050. a. Sketch the foces acting on the platfom (assume it has zeo mass) and find thei numeical values. b. Calculate and sketch all toques acting on the system. c. If the platfom initially otates at 8π adians/s, find how long it takes fo it to slow down and stop. d. What else must be included in the system in ode fo angula momentum to be conseved? 3. Suppose the stabilizing gyoscope of a ship has a oto of mass 5.0 10 4 kg, all located on the im at a adius of 0.20 m. The oto is stated fom est by a constant foce of 1.00 10 3 N applied though a belt on the im by a moto. a. Daw a diagam showing all foces acting on the oto. b. Show all toques acting on the oto. c. Compute the length of time needed to bing the oto up to its nomal speed of 9.00 10 2 ev/min. 13 14
MISN-0-34 PS-2 MISN-0-34 PS-3 d. State what else you must include in the system so that the total angula momentum will be conseved. That is, name the othe object whose otational speed about the oto shaft changes oppositely as the oto picks up speed. Descibe the eaction toque which poduces the angula acceleation of the othe object. 4. A system consists of two massless stuts, igidly connected to a sleeve as shown in the diagam. The foce F is always at ight angles to its stut and the axle is vetical, enabling the system to otate feely in a hoizontal plane (the sketch is a view looking down ). Neglect gavity. axle a. State why gavity can be neglected in this poblem. b. Sketch all foces acting on the system. F sleeve c. Wite down and justify the value of the (net) total foce acting on the system. d. Wite down o sketch all toques acting on the system. At the end of the system s fist complete evolution, deive the mass m s: e. moment of inetia about the axle f. angula acceleation g. time h. angula velocity i. tangential velocity j. kinetic enegy k. wok done on it ( F d x) l. total enegy m. change in angula momentum about the axle. Descibe a plausible expanded system within which angula momentum is conseved in the above case. Specifically: n. Descibe the eaction toque which poduces the compensating angula momentum. R m 5. A vetical flywheel of adius R contains (vitually) all of its mass M on its im and has a handle on one spoke at a adius which is less than the adius of the im. You stand next to the flywheel, gasp the handle, and apply a constant tangential foce F to the handle. a. Sketch all foces acting on the flywheel including that due to gavity. b. Wite down and justify the value of the net (total) foce on the flywheel. c. Wite down o sketch all toques acting on the system. At the end of one-half of a evolution, find the im s: d. moment of inetia about the axle e. angula acceleation f. time g. angula velocity h. tangential velocity i. kinetic enegy j. wok done on it k. total enegy l. angula momentum m. Descibe the mechanism by which angula momentum is conseved in this case. Bief Answes: 1. a. I = m i 2 i = M2 = (500 kg)(0.50 m) 2 = 125 kg m 2 b. τ = Iα α = τ I = c. θ = 1 2 αt2 : θ = 4π t = 30 N m 2 = 0.24 ad/s2 125 kg m ( ) 1/2 2θ = α [ ] 1/2 (2)(4π) = 10.2 s 0.24 s d. ω = ω 0 + αt = 0 + (0.24/ s 2 )(10.2 s) = 2.4 ad/s 15 16
MISN-0-34 PS-4 MISN-0-34 PS-5 v = ω = (2.4 ad/s)(0.50 m) = 1.2 m/s e. E k = 1 2 Iω2 = 1 2 (125 kg m2 )(2.4 s) 2 = 360 J 2. a. F ( gavity) = 300 N F ( eaction) = 300 N 3. a. F ( fiction) = µn(nomal) = (0.05)(300 N) = 15 N b. τ( fiction) = F ( fiction) = (0.25 m)(15 N) = 3.75 N m c. τ = Iα; I = M 2 α = τ I = ω = αt fictional foce 15N gavity foce eaction foce of pedestal 0.25m τ M 2 = 3.75 N m (300 N/9.8 m/s 2 = 1.96 /s2 )(0.25 m) 2 fictional toque t = ω α = 8π/ s 1.96/ s 2 = 12.8 s d. If the pedestal and eath ae included then the eath will acquie the compensating angula momentum so angula momentum is conseved fo the combined system. mounting bolt belt moto oto oto b. only one toque (fom belt): out of pape. c. 1.57 10 1 min. gavity d. The moto exets a toque on the ship though the moto s mounting bolts and mounting bed, while the ship tansmits it to the wate and the wate tansmits some of it to the eath. axle 4. a. The foce and toque poduced by gavity ae exactly cancelled by the foce and toque exeted on the system to keep it in a hoizontal plane. b. F axle foce c. net foce is zeo because the system goes nowhee (does not acceleate away fom its pesent location). d. τ = F, into the pape e. I = mr 2 f. α = τ/i = F /(mr 2 ) g. θ = θ 0 + ω 0 t + 1 2 αt2 ( ) ( ) 1 F 2π = 0 + 0 + 2 mr 2 t 2 ( ) 4πmR 2 1/2 t = F h. ω = ω 0 + αt = 0 + F ( ) 4πmR 2 1/2 mr 2 = F ( ) 1/2 F 4π i. v = ωr = m j. E k = 1 2 mv2 = F 2π k. W = F d x = F dθ = F 2π l. E tot = E k = F 2π m. L = mvr = (mf 4π) 1/2 R (into pape) ( ) 1/2 F 4π mr 2 n. If the axle is attached to the eath and if F s eaction foce is against the eath, then it is the eath which acquies the compensating angula momentum about the axle. 17 18
MISN-0-34 PS-6 MISN-0-34 AS-1 5. a. axle foce F axle gavity foce b. zeo; no acceleation of the (cente of the) flywheel. c. F, into page d. MR 2 e. F /(MR 2 ) ( 2πMR 2 f. F ( F 2π g. MR 2 ( F 2π h. M i. F π j. F π k. F π ) 1/2 ) 1/2 ) 1/2 l. (MF 2π) 1/2 R m. The expanded system consists of the flywheel, you, and the eath. The (fictional) foce of you feet and the eaction foce of the mounting bolts against the eath causes the eath s angula momentum about the axle to change in a compensating fashion. SPECIAL ASSISTANCE SUPPLEMENT S-1 (fom TX-2b) so: S-2 (fom TX-3e) S-3 (fom TX-4c) S-4 (fom TX-4d) d( p) dt = d d p p + dt dt d p dt = d d ( p) dt dt p I = 2 dm = M 2 E k = 1 2 Iω2 = 1 2 M2 ω 2 M 2 ω 0 = M 2 ω f + M 2 ω f = 2M 2 ω f = ω 0 = 2ω f ω f = 1 2 ω 0 E kf = 1 2 M2 ω 2 f + 1 2 M2 ω 2 f = M 2 ω 2 f = M 2 ( 1 2 ω 0) 2 = 1 4 M2 ω 2 0 = E kf E k0 = 1 4 M2 ω 2 0 1 2 M2 ω 2 0 = 1 2 19 20
MISN-0-34 ME-1 MODEL EXAM 1. See Output Skills K1-K4 in this module s ID Sheet. The actual exam may contain one o moe, o none, of these skills. 2. A vetical flywheel of adius R contains (vitually) all of its mass M on its im and has a handle on one spoke at a adius which is less than the adius of the im. You stand next to the flywheel, gasp the handle, and apply a constant tangential foce F to the handle. a. Sketch all foces acting on the flywheel including that due to gavity. b. Wite down and justify the value of the net (total) foce on the flywheel. c. Wite down o sketch all toques acting on the system. At the end of one-half of a evolution, find the im s: d. moment of inetia about the axle e. angula acceleation f. time g. angula velocity h. tangential velocity i. kinetic enegy j. wok done on it k. total enegy l. angula momentum m. Descibe the mechanism by which angula momentum is conseved in this case. Bief Answes: 1. See this module s text. 2. See this module s Poblem Supplement, poblem 5. 21 22
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