Name SFU e-mail @sfu.ca Date(YY/MM/DD) / / Section Goup UNIT 13: ANGULAR MOMENTUM AND TORQUE AS VECTORS Appoximate Classoom Time: Two 100 minute sessions Help! Pue logical thinking cannot yield us any knowledge of the empiical wold; all knowledge of eality stats fom expeience and ends in it. -- A. Einstein OBJECTIVES 1. To undestand the definitions of toque and angula momentum as vecto quantities. 2. To undestand the mathematical popeties and some applications of the vecto coss poduct. 3. To undestand the elationship between toque and angula momentum. 4. To undestand the Law of Consevation of Angula Momentum. and NSF. Modified fo SFU by N. Albeding, 2005.
Page 13-2 Wokshop Physics II Activity Guide SFU 1057 OVERVIEW 10 min This unit pesents us with a consolidation and extension of the concepts in otational motion that you studied in the last unit. In the last unit you studied the analogy and elationships between otational and linea quantities (i.e. position and angle, linea velocity and angula velocity, linea acceleation and angula acceleation, and foce and toque) without taking into account, in any fomal way, the fact that these quantities actually behave like the mathematical entities we call vectos. We will discuss the vecto natue of otational quantities and, in addition, define a new vecto quantity called angula momentum which is the otational analogue of linea momentum. Angula momentum and toque ae special vectos because they ae the poduct of two othe vectos a position vecto and a foce o linea momentum vecto. In ode to descibe them we need to intoduce a new type of vecto poduct known as the vecto coss poduct. We will exploe the definition and unique natue of the vecto coss poduct used to define toque and angula momentum. We will study the elationship between toque and angula momentum as well as the theoetical basis of the Law of Consevation of Angula Momentum. At the end of this unit you will expeience the effects of angula momentum consevation by holding masses in you hands and pulling in you ams while otating on a platfom. You will be asked to calculate you otational inetia with you ams in and with you ams out by making some simplifying assumptions about the shape of you body. F
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-3 Authos: Picilla Laws & John Luetzelschwab SESSION ONE: 25 min TORQUE, VECTORS, AND ANGULAR MOMENTUM Obsevation of Toque when F and ae not pependicula. In the last unit, you "discoveed" that if we define toque as the poduct of a leve am and pependicula foce, an object does not otate when the sum of the toques acting on it add up to zeo. Howeve, we didn't conside cases whee F and ae not pependicula, and we didn't figue out a way to tell the diection of the otation esulting fom a toque. Let's conside these complications by geneating toques with sping balances and a leve am once moe. Fo this activity you'll need: A hoizontal pivot A clamp stand to hold the pivot Two identical sping scales A ule A potacto!! h app Activity 13-1: Toque as a Function of Angle (a) Suppose you wee to hold one of the scales at an angle of 90 with espect to the leve am, h, and pull on it with a steady foce. Meanwhile you can pull on the othe scale at seveal angles othe than 90 fom its leve am as shown below. Would the magnitude of the second foce be less than, geate than, o equal to the foce needed at 90? What do you pedict? Explain.! F h pivot! F app # Holding toque: " h (b) You should detemine exactly how the foces compae to that needed at a 90 angle. Detemine this foce fo at least fou diffeent angles and figue out a mathematical elationship between F,, and θ. Set up a speadsheet to do the calculations shown in the table below. Hint: Should you multiply the poduct of the measued values of and F
Page 13-4 Wokshop Physics II Activity Guide SFU 1057 by sin θ o by cos θ to get a toque that is equal in magnitude to the holding toque? Holding Toque h (m) Fh (N) τh (N m) Applied Toque app (m) Fapp (N) θ (deg) θ (ad) cos θ sin θ app Fapp cos θ (N m) app Fapp sin θ (N m) (c) Within the limits of uncetainty, what is the most plausible mathematical elationship between τ and, F, and θ? The activity you just completed should give you a sense of what happens to the magnitude of the toque when the pulling foce, F, is not pependicula to the vecto,, fom the axis of otation. But how do we define the diection of the otation that esults when the toque is applied to an object that is initially at est and not balanced by anothe toque? Let's conside the diections we might associate with angula velocity and toque in this situation.
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-5 Authos: Picilla Laws & John Luetzelschwab Obseve #1 Obseve #2 Activity 13-2: Angula Rotation, Toque, and Diection (a) Suppose a paticle is moving aound in a cicle with an angula velocity that has a magnitude of ω associated with it. Accoding to obseve #1, does the paticle appea to be moving clockwise o counte clockwise? How about the diection of the paticle's motion accoding to obseve #2? (b) Is the clockwise vs. counte-clockwise designation a good way to detemine the diection associated with ω in an unambiguous way? Why o why not? (c) Can you devise a bette way to assign a minus o plus sign to an angula velocity? (d) Simila consideation needs to be given to toque as a vecto. Can you devise a ule to assign a minus o plus sign to a toque? Descibe the ule. 30 min Discussion of the Vecto Coss Poduct An altenative to descibing positive and negative changes in angle is to associate a positive o negative vecto with the axis of otation using an abitay but well accepted ule called the ight hand ule. By using vectos we can descibe sepaate otations of many body systems all otating in diffeent planes about diffeent axes. By using this vecto assignment fo diection, toque can be descibed mathematically as a "vecto coss poduct".
Page 13-6 Wokshop Physics II Activity Guide SFU 1057 The vecto coss poduct is a vey stange type of vecto multiplication woked out many yeas ago by mathematicians who had neve even head of angula velocity o toque. The peculia popeties of the vecto coss poduct and its elationship to angula velocity and toque ae explained in most intoductoy physics textbooks. The key popeties of the vecto which is the coss poduct of two vectos and F ae that: (1) the magnitude of the coss poduct is given by Fsinθ whee θ is the angle between the two vectos; F = F sinθ. Note that the tem Fsinθ epesents the component of F along a line pependicula to the vecto. (2) the coss poduct of two vectos and F is a vecto that lies in a diection to both and F and whose diection is given by the ight hand ule. Extend the finges of you ight hand in the diection of the fist vecto and then otate you finges towads the second vecto F and you thumb will then point in the diection of the esultant coss poduct τ. These popeties of the coss poduct ae pictued below.! "! line to F! # extension of Figue 13-1: Diagam of the Vecto Coss Poduct The spatial elationships between, F and τ ae vey difficult to visualize. In the next activity you can connect some thin ods of vaious sizes to each othe at angles of you own choosing and make some "vecto coss poducts". Fo this activity you will need the following items: Rods and connectos Bamboo skewes and modelling clay A potacto
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-7 Authos: Picilla Laws & John Luetzelschwab Activity 13-3: Making Models of Vecto Coss Poducts (a) Pick out ods of two diffeent lengths and connect them at some angle you choose. Conside one of the ods to be the vecto and the othe to be the F vecto. Measue the angle θ and the lengths of and F in metes. Then compute the magnitude of the coss poduct as Fsinθ in newton-metes (N m). Show you units! Note: You should assume that the magnitude of the foce in newtons is epesented by the length of the od in metes. τ = F = F sinθ = (b) Attach a "coss poduct" od pependicula to the plane detemined by and F with a length of Fsinθ Sketch the location of elative to F in the space below. Show the diection and magnitude of the esultant toque τ. Finally, show you coss poduct model to an instucto o teaching assistant fo confimation of its validity. "-skewe connecto -skewe line to! extension of F-skewe (c) In the diagams below the vectos and F lie in the plane of the pape. Calculate the toques fo the following two sets of and F vectos. In each case measue the length of the vecto in metes and assume that the length of the F vecto in cm epesents the foce in newtons. Use a potacto to measue the angle, θ, between the extension of the -vecto and the F-vecto. Calculate the magnitude of the toques. Place the appopiate symbol to indicate the diection of the toque in the cicle as follows: = aow into the page = aow out of the page
Page 13-8 Wokshop Physics II Activity Guide SFU 1057 F F = m F = N θ = τ = N m = m F = N θ = τ = N m 20 min Momentum and its Rotational Analogue Once we have defined the popeties of the vecto coss poduct, anothe impotant otational vecto is easily obtained, that of angula momentum elative to an axis of otation. Activity 13-4: Angula and Linea Momentum (a) Wite the otational analogues of the linea entities shown. Note: Include the fomal definition (which is diffeent than the analogue) in spaces maked with an asteisk (*). Fo example the otational analogue fo velocity is angula velocity and the definition of its magnitude is ω dθ/dt athe than v/.
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-9 Authos: Picilla Laws & John Luetzelschwab Linea Entity Rotational Analogue x (position) v (velocity) * a (acceleation) * F (Foce) * m (mass) * F = ma Definition (b) What do you think will be the otational definition of angula momentum in tems of the vectos and p? Hint: This is simila mathematically to the definition of toque and also involves a vecto coss poduct. Note that toque is to angula momentum as foce is to momentum. (c) What is the otational analogue in tems of the quantities I and ω? Do you expect the angula momentum to be a vecto? Explain. (d) Summaize you guesses in the table below. Linea Equation p = mv (definition in tems of and p ) p = mv (analogue using I and ω ) Rotational Equation L = L =
Page 13-10 Wokshop Physics II Activity Guide SFU 1057 10 min Obseving a Spinning Bicycle Wheel If a bicycle wheel is spinning faily apidly, can it be tuned easily so that its axis of otation points in a diffeent diection? If its axis it pefectly vetical while it is spinning will the wheel fall ove? Altenatively, does it fall ove when the wheel is not spinning? To make these obsevations we will use: An old bicycle wheel mounted on an axle A piece of sting to wap aound the im of the wheel to stat it spinning Activity 13-5: Is Spinning Moe Stable? (a) Do you expect it to take moe toque to change the axis of otation of a wheel that is spinning apidly o one that is spinning slowly? O do you expect the amount of toque to be the same in both cases? Explain. (b) Hold the wheel axis along a vetical line while the wheel is not spinning and change the axis fom a vetical to a hoizontal diection. Descibe the "toque" it takes qualitatively. (c) Have someone help you get the wheel spinning apidly while you hold the axle vetical. While the wheel is spinning change the axis to the hoizontal diection. Descibe the "toque" it takes qualitatively. How does the toque compae to that needed to change the diection of the axis of otation of the wheel when it is not spinning? Did you obseve what you expected to obseve?
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-11 Authos: Picilla Laws & John Luetzelschwab (d) Does the magnitude of the angula velocity vecto change as you change the axis of otation of the wheel? Does its diection change? Does the angula velocity vecto change o emain the same? Explain. (e) Does the angula momentum vecto change as you change the axis of otation of the spinning wheel? Why o why not? (f) If possible use you answe to pat (e) above to "explain" what you obseved in pat (c) above. 15 min Toque and Change of Angula Momentum Ealie in this couse you applied a vey bief foce along a line though the cente of mass of a olling cat. Do you emembe how it moved? What happened when you applied a gentle but steady foce along a line though the cente of mass of the cat? Let's do analogous things to a disk which is fee to otate on a elatively fictionless beaing, with the idea of fomulating laws fo otational motion that ae analogous to Newton's laws fo linea motion. Fo this obsevation, you will need: A otational motion appaatus A clamp stand to mount the system on Sting mass hange and masses
Page 13-12 Wokshop Physics II Activity Guide SFU 1057 Figue 13-2: Rotational Motion Appaatus. Figue out how to use a system like that shown in Figue 13-2 to obseve the motion of the disk unde the influence of a bief toque and a steady toque. In descibing the Laws of Rotational Motion be sue to conside vecto popeties and take both the magnitudes and diections of the elevant quantities into account in you wodings. Activity 13-6: Applied Toques and Resultant Motion a) What happens to the angula velocity and hence the angula momentum of the disk befoe, duing, and afte the application of a bief toque? State a Fist Law of Rotational Motion (named afte youself, of couse) in tems of toques and angula momenta. Hint: Newton's fist law states that the cente-of-mass of a system of paticles o a igid object that expeiences no net extenal foce will continue to move at constant velocity. The Rotational Fist Law in wods: The Rotational Fist Law as a mathematical expession:
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-13 Authos: Picilla Laws & John Luetzelschwab (b) What happens to the magnitude and diection of the angula velocity (and hence the angula momentum) of the disk duing the application of a steady toque? How do they change elative to the magnitude and diection of the toque? If possible, give a pecise statement of a Second Law of Rotational Motion elating the net toque on an object to its change in angula momentum. Note: Take both magnitudes and diections of the elevant vectos into account in you statement. Hint: Newton's second law of motion states that the cente-of-mass of a system of paticles o igid object that expeiences a net extenal foce will undego an acceleation invesely popotional to its mass. The Rotational Second Law in Wods: The Rotational Second Law as a Vecto Equation:
Page 13-14 Wokshop Physics II Activity Guide SFU 1057 SESSION TWO: ANGULAR MOMENTUM CONSERVATION 25 min Poblem Review Fast vs. Slow d L dt Action Recall that τ = d L dt. Suppose that you stat a wheel spinning so that its L vecto is pointing up, and that you then flip the wheel so that its L vecto points down. Which equies moe toque duing the "flipping time" a fast flip o a slow one? To find out, you will need the following: An old bicycle wheel mounted on an axle A piece of sting to wap aound the im of the wheel to stat it spinning Activity 13-7: Fast Flips and Slow Flips (a) Which action do you pedict will equie moe applied toque on a spinning wheel a fast flip o a slow flip? Explain the easons fo you pediction. (b) Stat a wheel spinning faily apidly. Ty flipping it slowly and then as apidly as possible. What do you obseve about the equied toques? (c) Did you pediction match you obsevations? If not, how can you explain what you obseved?
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-15 Authos: Picilla Laws & John Luetzelschwab Angula Momentum Consevation Now you can use the vecto expession fo Newton's second law of Rotational Motion to show that, in theoy, we expect angula momentum on a system to be conseved if the net toque on that system is zeo. Only thee things in this wold ae cetain death, taxes and consevation of momentum. Activity 13-8: Angula Momentum Consevation Using mathematical aguments show that, in theoy, wheneve thee is no net toque on an object o system of paticles, angula momentum is conseved. 15 min Flipping a Rotating Wheel What Changes? In Activities 13-5 and 13-7 you should have discoveed that it takes a healthy toque to change the diection of the angula momentum associated with a spinning bicycle wheel. Let's obseve a moe complicated situation involving a simila change of angula momentum. Conside a peson sitting on a platfom that is fee to move while holding a spinning bicycle wheel. What happens if the peson applies a toque to the bicycle wheel and flips the axis of the wheel by 180? This state of affais is shown in the diagam below.
Page 13-16 Wokshop Physics II Activity Guide SFU 1057 Figue 13-3: Spinning wheel being flipped on otating platfom Fo this obsevation you will need the following equipment: A otating platfom A peson A bicycle wheel mounted on an axle A piece of sting to wap aound the im of the wheel to stat it spinning Activity 13-9: What Happens When the Wheel is Flipped? (a) What do you pedict will happen if a stationay peson, sitting on a platfom that is fee to otate, flips a spinning bicycle wheel ove? Why? (b) What actually happens? Does the esult agee with you pediction?
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-17 Authos: Picilla Laws & John Luetzelschwab (c) Use the Law of Consevation of Angula Momentum to explain you obsevation in wods. Hints: Remembe that angula momentum is a vecto quantity. Does the angula momentum of the wheel change as it is flipped? If so, how does the angula momentum of the peson and stool have to change to compensate fo this? 50 min Changing You Rotational Inetia In this activity you will veify the Law of Consevation of Angula Momentum qualitatively by otating on a easonably fictionless platfom with you ams extended. You can then educe you otational inetia by pulling in you ams. This should cause you to otate at a diffeent ate. This phenomenon is populaly known as the ice skate effect. Since people can econfigue themselves they ae not eally igid bodies. Howeve, in this obsevation we will assume that you can behave tempoaily like two igid bodies one with you ams extended with masses and the othe with you ams pulled in with the masses.
Page 13-18 Wokshop Physics II Activity Guide SFU 1057 You can obseve this effect qualitatively by using the following appaatus: A otating platfom You body Two 2-kg masses Figue 13-4: Simplified model of a human body as a combination of cylindical shapes. Activity 13-10: The Effect of Reducing Rotational Inetia (a) Accoding to the Law of Consevation of Angula Momentum, what will happen to the angula speed of a peson on a platfom if his o he otational inetia is deceased? Back up you pediction with equations.
Wokshop Physics II: Unit 13 Angula Momentum & Toque... Page 13-19 Authos: Picilla Laws & John Luetzelschwab (b) Ty spinning on the otating platfom. What happens to you angula speed as you pull you ams in? If you wee asked to veify the Law of Consevation of Angula Momentum quantitatively, you would need to calculate you appoximate otational inetia fo two configuations. This pocess is a eal tou de foce, but it does seve as an excellent eview of techniques fo calculating the otational inetia of an extended set of objects. Estimate the otational inetia of the platfom you used. Assume that each of you ams with the attached hand has a mass that is equal to a fixed % of you total mass as shown in the table below. Idealize youself as a cylinde (athe than a squae) with long thin ods as ams. You may have to look up some data in the text book to do the otational inetia calculations. Am/Hand Women 4.8% Men 5.8% Figue 13-5: Pecentage of the mass of a typical peson's am and hand elative to that pesons total body mass. Ref: Plagenhoef, Stanley. Pattens of Human Motion (Englewood Cliffs, NJ: Pentice-Hall, 1971), Ch. 3
Page 13-20 Wokshop Physics II Activity Guide SFU 1057 axis of otation Activity 13-11: You Rotational Inetia (a) Find the total otational inetia of the otating system consisting of you, a pai of masses, and a otating platfom. Assume that you can hold the 2.0 kg masses at a distance of 5.0 cm fom the axis of otation when you elbows ae in. Hint: Don't foget to account fo the mass and otational inetia of the platfom. Show all you wok caefully. I = 1 2 MR2 axis of otation l I = 1 Ml 2 12 (b) Find the total otational inetia of the otating system if you ae holding a 2.0 kg mass in each hand at am's length fom you axis of otation. I = 1 Ml 2 3 l axis of otation (c) Which pat of system has the lagest otational inetia when you ams ae extended (i.e. the tunk, ams, 2.0 kg masses, o platfom)? Is the esult supising? Explain.