rot paper 1

1 CHAPTER 4 RIGID BODY ROTATION 4.1 Introduction No real solid body is perfectly rigid. A rotating nonrigid body will be distorted by centrifugal force* or by interactions with other bodi es. Nevertheless most people will allow that in practice some solids are fairly ri gid, are rotating at onl y a modest speed, and any distortion is small compared with the overall si ze of the body. No ex cuses, therefore, are needed or offered for analysing to be gin with the rotation of a rigid body. *I do not in this chapter delve deeply into whether ther e really is “such thing” as “centrifugal force”. Some would try to persuade us that there is no such thing. But is there “such thing” as a “gravitational force”? And is one any more or less “real” than the other? These are deep questions best left to the philosophers. In physics we use the concept of “force” – or indeed any other concept – according as to whether it enables us to supply a description of how physical bodies behave. Many of us would, I think, be challenged if we were faced with an examination questio n: “Explain, without using the term centrifugal force , why Earth bulges at its equator.” We have already discussed some aspects of solid body rotation in Chapter 2 on Moment of Inertia, and indeed the pr esent Chapter 4 should not be plunged into without a good understanding of what is meant by “moment of inertia”. One of the things that we found was that, while the comfortable relation L = Iω which we are familiar with from elementary physics is adequate for problems in two dimensions, in three dimensions the relation becomes L = Iω, where I is the inertia tensor , whose properties were discussed at some length in Chapter 2. We also learned in Chapter 2 ab out the concepts of principal moments of inertia , and we introduced the notion that, unless a body is rotating about one of its principal axes, the equation L = Iω implies that the angular momentum and angular velocity vectors are not in the same direction. We shall discuss this in more detail in this chapter. A full treatment of the rotation of an asymmetric top (whose three principal moments of inertia are unequal and which has as its momental ellipsoid a triaxial ellipsoid) is very lengthy, since there are so many cases to consid er. I shall restrict consideration of the motion of an asymmetric top to a qualitative argument that shows th at rotation about the principal axis of greatest moment of inertia or about the axis of least moment of inertia is stable, whereas rotation about the intermediate axis is unstable. I shall treat in more detail the free rotation of a symmetric top (which has two equal principal moments of inertia) and we shall se e how it is that the angular velocity vector precesses while the angular momentum vector (in the absence of external torques) remains fixed in magnitude and direction. I shall also discuss the situation in which a symmetric top is subjected to an external torque (in which case L is certainly not fixed), such as the motion of a top. A similar situation, in which Earth is subject to ex ternal torques from the Sun and Moon, causes Earth’s axis to precess with a period of 26,000 years, and this will be dealt with in a chapter of the notes on Celestial Mechanics. 2 Before discussing these particular problems, there are a few preparatory topics, namely, angular velocity and Eulerian angles, kinetic energy, Lagran ge’s equations of motion, and Euler’s equations of motion. 4.2 Angular Velocity and Eulerian Angles Let O xyz be a set of space-fixed axis, and let O x 0y0z0 be the body-fixed principal axes of a rigid body. The orientation of the body-fixed principal axes O x0y0z0with respect to the space-fixed axes O xyz can be described by the three Euler angles .,,ψφθ . These are illustrated in Figure IV.1a. Those who are not familiar with Euler angles or who would like a reminder can refer to their detailed description in Chapte r 3 of my notes on Celestial Mechanics. We are going to examine the motion of a body th at is rotating about a nonprincipal axis. If the body is freely rotating in space with no external torq ues acting upon it, its angular momentum L will be constant in magnitude and dire ction. The angular velocity vector ω, however, will not be constant, but will wander with respect to both the space-fixed and body-fixed axes, and we shall be examining this motion. I am going to call the x x0 z0 z θ φ ψ FIGURE IV.1a 3 instantaneous components of ω relative to the body-fixed axes , ,,3 2 1 ωωω and its magnitude .ω As the body tumbles over and over, its Euler angles will be changing continuously. We are going to establish a ge ometrical relation between the instantaneous rates of change of the Euler angles and the instantaneous components of ω. That is, we are going to find how ω1 , ω2 and ω3 are related to . and, ψ φθ &&& I have indicated, in figure VI.2a, the angular velocities ψ θ &&& and,φ as vectors in what I hope will be agreed are th e appropriate directions. x y x0 z0 z θ φ ψ FIGURE IV.2a θ& φ& ψ& 4 It should be clear that ω1 is equal to the x0-component of φ& plus the x0-component of θ& and that ω2 is equal to the y0-component of φ& plus the y0-component of θ& and that ω3 is equal to the z0-component of φ& plus ψ& . Let us look at Figure IV.2b We see that the x 0 and y0 components of θ& are ψ θcos& and ψ θ−sin& respectively. The x0, y0 and z0 components of φ& are, respectively, ψ θ φ sin sin& , ψ θ φ cos sin& and θ φcos& . Hence we arrive at . cos sin sin1 ψ θ+ψ θ φ=ω & & 4.2.1 ψ θ−ψ θ φ=ω sin cos sin2& & 4.2.2 . cos3 ψ+θ φ=ω & & 4.2.3 θ& φ& ψ& y0 z0 x0 θ ψ ψ FIGURE IV.2b 5 4.3 Kinetic Energy Most of us are familiar with the formula 2 21ωI for the rotational kinetic energy of a rotating solid body. This formul a is adequate for simple si tuations in wh ich a body is rotating about a principal ax is, but is not adequate for a body rotating about a nonprincipal axis. I am going to think of a rotati ng solid body as a collection of point masses, fixed relative to each other, but all revolving with the same angular velocity about a common axis – and those who believe in atoms assure me that this is indeed the case. (If you believe that a solid is a continuum, you can still divide it in your imagination into lots of small mass elements.) In figure IV.3, I show just one particle of the rotating body. The position vector of the particle is r. The body is rotating at angular velocity ω. I hope you’ll agree that the linear velocity v of the particle is (now think about this carefully) .rωv ×= . The rotational kinetic en ergy of the solid body is prω vmv m T . . ) (21 21 2 21 rot ∑∑ ∑ × = = = v The triple scalar product is th e volume of a parallelepiped, which justifies the next step: .) (.21∑ × = prω All particles have the same angular velocity, so: .ωω Lω prω I.. .21 21 21) ( = = × = ∑ v ω r FIGURE IV.3 6 Thus we arrive at the following expre ssions for the rotati onal kinetic energy: .ωω Lω I. .21 21 rot = =T 4.3.1 If the body is rotating about a nonprincipal axis, the vectors ω and L are not parallel (we shall discuss this in more detail in later sections). If it is rotating about a principal axis, they are parallel, and the expression reduces to the familiar .2 21ωI In matrix notation, this can be written .~ 21 rot ωω T I = 4.3.2 Here I is the inertia tensor, ω is a column vector containi ng the rectangular components of the angular velocity and ω~is its transpose, namely a row vector. That is: () ,rot    ωωω     − −− −− − ω ω ω= zyx z y x C F GF B HG H A T 4.3.3 or ( ). 2 2 22 2 2 21 rot yx xz zy z y x H G F C B A T ωω −ωω −ωω −ω+ω+ω = 4.3.4 This expression gives the rotational kinetic energy when the components of the inertia tensor and the angular velocity ve ctor are referred to an arbitr ary set of axes. If we refer them to the principal axes, the off-diagonal elements ar e zero. I am going to call the principal moments of inertia I1 , I2 and I3 . (I could call them A, B and C, but I shall often use the convention that A < B < C, and I don’t want to speci fy at the present which of the three moments is the greatest and which is the grea test, so I’ll call them I1 , I2 and I3 , with I1 = ,) (2 2∑ +z ym etc.). I’ll also call the angular velocity components referred to the principal axes ω1 , ω2 , ω3 . Referred, then, to the principal axes, the rotational kinetic energy is ( ).2 332 222 11 21ω+ω+ω = I I I T 4.3.5 I have now dropped the subscript “rot”, because in this chapte r I am dealing entirely with rotational motion, and so T can safely be understood to m ean rotational ki netic energy. We can also now write the kinetic energy in term s of the rates of change of the Eulerian angles, and the expression we obtain will be useful later when we derive Euler’s equations of motion: 7 () ( ) () . cos sin cos sin cos sin sin2 3212 2212 121ψ+θ φ + ψ θ−ψ θ φ +ψ θ+ψ θ φ = & & & & & & I I I T 4 . 3 . 6 You will probably want a concrete example in order to understand this properly, so let us imagine that we have a concrete brick of dimensions 10 cm × 15 cm × 20 cm, and of density 4 g cm −3 , and that it is rotating about a body diameter (the ends of which are fixed) at an angular speed of 6 rad s−1 . I hope you’ll agree that the mass is 12000 g = 12 kg. The principal moment of inertia about the vertical axis is .mkg 0625.0 cmg000,625 )5.7 10( 120002 2 2 2 31 3 = = + × ×=I Similarly the other principal moments are I 1 = 0.0500 kg m2 and I2 = 0.0325 kg m2 . The direction cosines of the vector ω are . 10 20 1510, 10 20 1520, 10 20 1515 2 2 2 2 2 2 2 2 2+ + + + + + Therefore ω1 = 3.34252 rad s−1, ω2 = 4.45669 rad s−1, ω3 = 2.22834 rad s−1 . 20 cm 10 cm 15 cm FIGURE IV.4 ω 8 Hence T = 0.02103 J 4.4 Lagrange’s Equations of Motion In Section 4.5 I want to de rive Euler’s equati ons of motion, which describe how the angular velocity components of a body change when a torq ue acts upon it. In deriving Euler’s equations, I find it convenient to make use of Lagrange’s equations of motion. This will cause no difficulty to anyone who is already familiar with Lagrangian mechanics. Those who are not familiar with Lagrangian mechanics may wish just to understand what it is that Eu ler’s equations are dealing with and may wish to skip over their derivation at this stage. Later in this series, I hope to add a longer chapter on Lagrangian mechanics, when al l will be made clear (maybe). In the meantime, for those who are not content ju st to accept Euler’s equations but must also understand their derivation, this section gi ves a five-minute course in Lagrangian mechanics. To begin with, I have to introduce the idea of generalized coordinates and generalized forces. The geometrical description of a mechanical sy stem at some instant of time can be given by specifying a number of coordinates. For example, if the system consists of just a single particle, you could specify its rectangular coordinates xyz, or its cylindrical coordinates ρφz, or its spherical coordinates rθφ. Certain theorems to be developed will be equally applicable to any of these, so we can think of generalized coordinates q 1q2q3, which could mean any one of the recta ngular, cylindrical of spherical set. In a more complicated system, for example a polyatomic molecule, you might describe the geometry of the molecule at some instan t by a set of interatomic distances plus a set of angles between bonds. A fairly large number of distances and angles may be necessary. These distances and angles can be called the generalized coordinates. Notice that generalized coordinates need not always be of dimension L. Some generalized coordinates, for example, may ha ve the dimensions of angle. [See Appendix of this Chapter for a brief discussion as to whether angle is a dimensioned or a dimensionless quantity.] While the generalized coordinates at an instan t of time describe the geometry of a system at an instant of time, they alone do not pr edict the future behaviour of the system. I now introduce the idea of generalized forces . With each of the generalized coordinates there is associated a generalized force. With the generalized coordinate q i there is associated a corresponding generalized force Pi. It is defined as follows. If, when the generalized coordinate qi increases by δqi, the work done on the system is Piδqi, then Pi is the generalized force associated with the generalized coordinate qi. For example, in our simple example of a single pa rticle, if one of the generali zed coordinates is merely the x- 9 coordinate, the generalized force associated with x is the x-component of the force acting on the particle. Note, however, that often one of the generalized coordinates might be an angle . In that case the generalized force associated with it is a torque rather than a force. In other words, a generalized force need not necessarily have the dimensions MLT −2. Before going on to describe La grange’s equations of motion, let us remind ourselves how we solve problems in mechanics using Newton’ s law of motion. We may have a ladder leaning against a smooth wall and smooth floor, or a cylinde r rolling down a wedge, the hypotenuse of which is rough (so that the cylin der does not slip) and the smooth base of which is free to obey Newton’s third law of motion on a smooth horizontal table, or any of a number of similar problems in mechanic s that are visited upon us by our teachers. The way we solve these problems is as follows. We draw a large diagram using a pencil , ruler and compass . T h e n w e m a r k i n red all the forces , and we mark in green all the accelerations . If the problem is a two-dimensional problem, we write F = ma in any two directions; if it is a three-dimensional problem, we write F = ma in any three directions. Usually this is easy and straightforward. Sometimes it doesn’t seem to be as easy as it sounds, and we may prefer to solve the problem by Lagrangian methods. To do this, as before, we draw a large diagram using a pencil , ruler and compass . But this time we mark in blue all the velocities (including angular velocities). Lagrange, in the Introduction to his book La méchanique analytique (modern French spelling omits the h) pointed out that there were no diagrams at all in his book, since all of mechanics could be done analytically – hence the title of the book. Not all of us, however, are as mathematically gifted as Lagrange, and we cannot bypass the step of drawing a large, neat and clear diagram. Having drawn in the velocitie s (including angular velocitie s), we now calculate the kinetic energy , which in advanced texts is often given the symbol T, presumably because potential energy is traditionally written U or V. There would be no harm done if you prefer to write Ek , Ep and E for kinetic, potential and tota l energy. I shall stick to T , U or V, and E. Now, instead of writing F = ma , we write, for each generalized coordinate, the Lagrangian equation (whose proof awaits a la ter chapter): .i i iPqT qT dtd=∂∂−    ∂∂ & 4.4.1 The only further intellectual e ffort on our part is to determine what is the generalized force associated with that coordinate. Apart from that, the procedure goes quite automatically. We shall use it in use in the next section. That ends our five-minute cour se on Lagrangian mechanics. 10 4.5 Euler’s Equations of Motion In our first introduction to classical mechanics, we learn that when an external torque acts on a body its angular momentum changes (and if no external tor ques act on a body its angular momentum does not change.) We l earn that the rate of change of angular momentum is equal to the appl ied torque. In the first simple examples that we typically meet, a symmetrical body is rotating about an axis of symmetry, and the torque is also applied about this same axis. The angular momentum is just Iω, and so the statement that torque equals rate of change of angular momentum is merely ,ω=τ&I and that’s all there is to it. Later, we learn that L = Iω, where I is a tensor, and L and ω are not parallel. There are three principal moments of inertia, and L, ω and the applied torque τ each have three components, and the statement “torque equals rate of change of angular momentum” somehow becomes much less easy. Euler’s equations sort this out, and give us a relation be tween the components of the τ , I and ω. For figure IV.5, I have just reproduced, with some small modifications, figure III.19 from my notes on this Web site on Celestial Mechanics, where I defined Eulerian angles . Again it is suggested that those who are unfam iliar with Eulerian angles consult Chapter III of Celestial Mechanics. In figure IV.5, O xyz are space-fixed axes, and O x 0y0z0 are the body-fixed principal axes. The axis O y0 is behind the plane of your scr een; you will have to look inside your monitor to find it. z z0 x0 x y O θ φ ψ τ1 τ3 FIGURE IV.5 11 I suppose an external torque τ acts on the body, and I have drawn the components τ1 and τ3. Now let’s suppose that the body rotates in such a manner that the Eulerian angle ψ were to increase by δψ. I think it will be re adily agreed that the work done on the body is τ3δψ. This means, following our definition of generalized force in section 4.4, that τ3 is the generalized force associated with the generalized coordinate ψ. Having established that, we can now appl y the Lagrangian equation 4.4.1: .3τ=ψ∂∂−    ψ∂∂ T T dtd & 4.5.1 Here the kinetic energy is the expression th at we have already es tablished in equation 4.3.6. In spite of the somewhat fearsome aspect of equation 4.3.6, it is quite easy to apply equation 4.5.1 to it. Thus , ) cos(3 3 ω=ψ+θ φ =ψ∂∂I IT& & 4.5.2 where I have made use of equation 4.2.3. Therefore .33ω=    ψ∂∂&&IT dtd 4.5.3 And, if we make use of equations 4.2.1,2,3, it is easy to obtain .) (2 1 21 122 211 I I I IT− ωω=ωω−ωω=ψ∂∂ 4.5.4 Thus equation 4.5.1 becomes: . ) ( 3 21 2 1 33 τ=ωω− −ω II I& 4.5.5 This is one of the Eulerian equations of motion . Now, although we saw that τ3 is the generalized force asso ciated with the coordinate ψ, it will we equally clear that τ1 is not the generalized force associated with θ, nor is τ2 the generalized force associated with φ. However, we do not have to think about what the generalized forces associated with these two co ordinates are; it is much easier than that. To obtain the remaining two Eulerian equations, all that is necessary is to carry out a cyclic permutation of the subs cripts in equation 4.5.5. T hus the three Eulerian equation are: , ) ( 1 32 3 2 11 τ=ωω− −ω II I& 4.5.6 12 , ) (2 13 1 3 22 τ=ωω− −ω II I& 4.5.7 . ) (3 21 2 1 33 τ=ωω− −ω II I& 4.5.8 These take the place of , ω=τ&I which we are more familiar with in elementary problems in which a body is rotating about a principal axis and a torque is applied around that principal axis. If there are no external torque s acting on the body, then we ha ve Euler’s equations of free rotation of a rigid body: , ) ( 32 3 2 11 ωω− =ω II I& 4.5.9 , ) (13 1 3 22 ωω− =ω II I& 4.5.10 . ) (21 2 1 33 ωω− =ω II I& 4.5.11 Example . x0 y0 ω ? 1 2a 2b θ 13 In the above drawing, a rectangular lamina is spinning with constant angular velocity ω between two frictionless bearings . We are going to apply Eu ler’s equations of motion to it. We shall find that the bearings are exerting a torque on the rectangle, and the rectangle is exerting a torque on the bearings. The angular mo mentum of the rectangle is not constant – at least it is not constant in direction . We shall calculate the torque (its magnitude and its direction) and see what is happening to the angular momentum. We note that the principal (s econd) moments of inertia are ) (2 2 31 32 31 22 31 1 b am I ma I mb I + = = = and that the components of angular velocity are .0 sin cos 3 2 1 =ω θ ω=ω θ ω=ω Also, ω& and all of its components are zero. We immediately obtain, from Euler’s equations, that τ1 and τ2 are zero, and that the torque exerted on the rectangle by the bearings is . cos sin) ( ) (2 2 2 31 21 1 2 3 θ θ ω − =ωω− =τ b am II And since , cos and sin 2 2 2 2b aa b ab +=θ +=θ we obtain . ) (3) ( 2 2 22 2 3 ω +−=τ b aabb am Thusτ, the torque exerted on the rectangle by the bearings is directed normal to the plane of the rectangle (out of the plane of the paper in the instantaneous snapshot above). 14 The angular momentum is given by ω LI= . That is to say: ). 90tan( cot cossin/0sincos0sincos 0 00 00 0 o 22 1 23132 22 31 2 31 22 22 31 2 31 12 222 31 321 θ− =θ = θθ=ω ==ω +=θ ω =ω +=θ ω =    θ ωθ ω     +=     baLLmab LLb abam ma Lb aabm mb Lb aab m LLL This tells us that L is in the plane of the rect angle, and makes an angle 90° − θ with the x- axis, or θ with the y-axis, and it rotates around the vector τ. τ is perpendicular to the plane of the rectangle, and of course the change in L takes place in that direction. The torque does no work, and ω and T are constant. The reader might find an analogy in the situation of a planet in orbit around the Sun in a cicular orbit.. The planet experiences a force that is always perpendicular to its ve locity. The force does no work, and the speed and kinetic energy remain constant. 15 The torque on the plate can be represented as a couple of forces exerted by the bearings on the plate, each of magnitude , 22 23 b a+τ or . ) (6) ( 2 2/32 22 2 ω +− b aabb am Forces exerted by the plate on the bearings are, of course, in the opposite direction. ω ? 1 θ L θ τ ? F F 16 Example . Figure IV.6 shows a disc of mass m, radius a, spinning at a cons tant angular speed ω about at axle that is inclined at an angle θ to the normal to the disc . I have drawn three body-fixed principal axes. The x- and y- axes are in the plane of the disc; the direction of the x-axis is chosen so that th e axle (and hence the vector ω) is in the zx-plane. The disc is evidently unbalanced and there must be a torque on it to ma intain the motion. Since ω is constant, all components of ω&are zero, so that Eu ler’s equations are , ) (23 2 3 1 ωω − =τ I I , ) (31 3 1 2 ωω − =τ I I . ) (12 1 2 3 ωω − =τ I I Now . , , , cos ,0 , sin2 21 32 41 22 41 1 3 2 1 ma I ma I ma I = = = θ ω=ω =ωθ ω=ω Therefore .2sin cos sin ,022 81 22 41 2 3 1 θ ω −=θ θ ω −=τ =τ=τ ma ma and (Check, as always, that this expression is dime nsionally correct.) Thus the torque acting on the disc is in the negative y-direction. Can you reconcile the fact that th ere is a torque actin g on the disc with th e fact that is it moving with constant angul ar velocity? Yes, most decidedly! What is not constant is the angular momentum L, which is moving around the ax le in a cone such that j L2τ−=& , where j is the unit vector along the y-axis. [ ] θ ω FIGURE IV.6 z x y 17 4.6 Free Rotation of a Rigid Asymmetric Top By “asymmetric top” I mean a body whose three principal moments of inertia are unequal. While we often think of a “top” as a symmetric body spinning on a table, in this section the “top” will not necessarily be symmetr ic, and it will not be in contact with any table, nor indeed subjected to a ny external forces or torques. A complete description of the motion of an asymmetric top is quite complicated, and therefore all that we shall attempt in this chapter is a qualitative description of certain aspects of the motion. That our description is going to be “qualita tive” does not by any means imply that this section is not going to be replete with equations or that we can give our poor brains a rest. The first point that we can make is that, provided that no external torques act on the body , its angular momentum L is constant in magnitude and direction. A second point is that, provided the body is rigid and has no internal degrees of freedom , the rotational kinetic energy T is constant. I deal briefly with nonr igid bodies in section 4.7. Although the angular velocity vector ω is by no means fixed in either magnitude and direction, and the body can tumble over and over, these two conditions impose some constraints of the magnitude and direction of ω . We are going to examine these two conditions to see what constraints are imposed on ω. One of the things we shall find is that rotation of a body about a principal axis of greatest or of least moment of inerti a is stable against small displacements, whereas rotation about the principal axis of intermediate moment of inertia is unstable. Absence of an external torque means th at the angular momentum is constant: , constant 2 32 22 12= + + = L L L L 4.6.1 so that, at all times, . 2 2 32 32 22 22 12 1 L I I I =ω+ω+ω 4.6.2 Thus, for a given L. the angular velocity components always satisfy () () ().12 32 3 2 22 2 2 12 1=ω+ω+ω IL IL IL 4.6.3 That is to say, the angular velocity vector is constrained such that the tip of the vector ω is always on the surface of an ellipsoid of semi axes . /,/,/3 2 1 ILILIL In addition to the constancy of angular mome ntum, the kinetic energy is also constant: 18 .2 3321 2 2221 2 1121T I I I =ω +ω +ω 4.6.4 Thus the tip of the angular velocity vector must also be on the surface of the ellipsoid () () ().1 /2 /2 /22 32 3 2 22 2 2 12 1=ω+ω+ω IT IT IT 4.6.5 This ellipsoid (which is similar in shape to the momental ellips oid) has semi axes ./2,/2,/23 2 1 IT IT IT Thus, how ever the body tumbles over and over, ω is constrained in magnitude and direction so that its tip is on the curv e where these two ellipses intersect. Suppose, for example, that we have a rigid body with ,mkg5.0 ,mkg3.0 ,mkg2.02 32 22 1 = = = I I I and that we set it in motion such that the angular momentum and kinetic energy are L = 4 J s , T = 20 J . (The angular momentum and kinetic energy will be determined by the magnitude and direction of the initial velocity v ector by which it is set in motion.) The tip of ω is constrained to be on the curve of intersection of the two ellipsoids 1 8 3.13 2022 3 22 2 22 1= + +ω ω ω & 4.6.6 and .1 94.8 55.11 14.1422 3 22 2 22 1= + +ω ω ω 4.6.7 It is not easy (or I don’t find it so) to imagine what this curve of in tersection looks like in three-dimensional space, but one of my stude nts, Leif Petersen, prepared the drawing below, and I am grateful to hi m for permission to reproduce it here. You can see that the curve of intersection is not a plane curve. In case it’s of any help, you might want to note that equations 4.6.6 and 4.6.7 can be written 1600 25 9 42 32 22 1 = + + ω ω ω 4.6.8 and ,400 5 3 23 32 22 1 = + + ω ω ω 4.6.9 but I’m going to leave the equations in the fo rm 4.6.6 and 4.6.7, and in figure IV.7, I’ll sketch one octant of the two ellipsoidal surfaces. 19 © copyright Leif Petersen 20 The continuous blue curve shows an octant of the ellipsoid L = constant, and the dashed black curve shows an oc tant of the ellipsoid T = constant. Th e angular momentum vector can end only on the curv e (not drawn) where the two ellipsoids intersect. Two points on the curve are indicated in Figure IV.7. If, for example, ω is oriented so that ω1= 0, the other two components must be ω2 = 8.16 and ω3 = 6.32. If it is oriented so that ω2 = 0, the other two components must be ω3 = 7.30 and ω1 = 8.16. If ω3 = 0, there are no real solutions for ω1 and ω2. This means that, for the given values of L and T, ω3 cannot be zero. Now I’m going to address myself to the stab ility of rotation when a symmetric top is initially set to spin about one of its principal axes, wh ich I’ll take to be the z-axis. We’ll suppose that initially ω 1 = ω2 = 0, and ω3 = Ω. In that case the angular momentum and the kinetic energy are . and2 321 3 Ω = Ω= I T IL In any subsequent motion, the tip of ω is restricted to move along the curve of intersection of the ellipsoids given by equations 4.6.3 and 4.6.5. That is to say, along the curve of intersec tion of the ellipsoids 11.55 13.33 8.94 8 14.14 20 (0, 8.16, 6.32) (8.16, 0, 7.30) FIGURE IV.7 ω1 ω2 ω3 T L 21 122 3 2 232 2 2 132 1=Ωω+    Ωω+    Ωω II II 4.6.10 and .122 3 2 232 2 2 132 1=Ωω+    Ωω+    Ωω II II 4.6.11 For a specific example, I’ll suppose that the mo ments of inertia are in the ratio 2 : 3 : 5, and we’ll consider three cases in turn. Case I. Rotation about the axis of least moment of inertia. That is, we’ll take I 3 = 2, .5 ,32 1 = = I I Since I3 is the smallest moment of inertia, each of the ratios I3/I1 and I3/I2 are less than 1, and . and 23 23 13 13 II II II II> > The two ellipsoids are () ()1 400.0 667.022 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.12 and () ().1 632.0 816.022 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.13 I’ll try and sketch these: 22 Initially, we suppose, the body was set in motion rotating about the z-axis with angular speed Ω, which determines the values of L and T, which will remain constant. The tip of the vector ω is constrained to remain on the surface of the ellipsoid L = 0 and on the ellipsoid T = 0, and hence on the in tersection of these two surfaces. But these two surface touch only at one point, namely ( ω1, ω2 , ω3) = (0 , 0 , Ω). Thus there the vector ω remains, and the rotation is stable. Case II. Rotation about the axis of greatest moment of inertia. That is, we’ll take I 3 = 5, .3 ,22 1 = = I I Since I3 is the greatest moment of inertia, each of the ratios I3/I1 and I3/I2 are greater than 1, and . and 23 23 13 13 II II II II< < The two ellipsoids are () ()1 67.1 50.222 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.14 ω2/Ω ω3/Ω ω1/Ω 1 0.667 0.400 0.632 0.816 L T FIGURE IV.8 23 and () ().1 29.1 58.122 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.15 I’ll try and sketch these: Again, and for the same reason as for Case I, we see that this motion is stable. Case III. Rotation about the intermediate axis. That is, we’ll take I 3 = 3, .2 ,52 1 = = I I This time I3/I1 is less than 1 and I3/I2 is less than 1, and . and 23 23 13 13 II II II II< > The two ellipsoids are ω3/Ω 1 ω1/Ω 1.58 2.50 1.29 1.58 FIGURE IV.9 L T 24 () ()1 50.1 60.022 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.16 and () ().1 22.1 77.022 3 22 2 22 1=Ωω+ Ωω+ Ωω 4.6.17 I’ll try and sketch these: Unlike the situation for Cases I and II, in wh ich the two ellipsoids touch at only a single point, the two ellipses for Case III intersect in the curve shown as a dotted line in figure IV.10. Thus ω is not restricted to ly ing along the z-axis, but it can move anywhere along the dotted line. The motion, therefore, is not stable. You should experiment by throwing a body in th e air in such a manne r as to let it spin around one of its principal axes. A recta ngular block will do, though the effect is particularly noticeable with something like a table-tennis bat or a tennis racket. 1.22 1.50 (0.45, 1.00 , 0.00 )ω2/Ω 0.77 ω3/Ω 1 0.60 ω1/Ω FIGURE IV.10 T L 25 Here is another approach to reach the same result. We imagine an asymmetric top spinning about one of its principa l axes with angular velocity zω ˆω= . It is then given a small perturbation, so that its angular velocity is now .ˆ ˆ ˆ z y xωzω+η+ε= 4.6.18 Here the “hatted” quantities ar e the unit orthogonal vectors; ε and η are supposed small compared with ωz . Euler’s equations are : ( ),3 2 1 I I Iz− ωη=ε& 4.6.19 ( ),1 3 2 I I Iz − εω=η& 4.6.20 ( ).2 1 3 I I Iz − εη=ω& 4.6.21 If ,zω<<εη& then zω is approximately constant. Elimination of η from the first two equations yields .) )( ( 212 3 1 3 2ε  ω − −−=εIII II Iz&& 4.6.22 Elimination of ε instead results in a similar equation in η. If I3 is either the largest or the smallest of the three moments of inertia, the two parentheses in the denominator have the same sign, so the expression in the brackets is positive. Equation 4.6.22 is then the equation for simple harmonic motion, and the motion is stable. If, however, I 3 is intermediate between the other two, the two parentheses have opposite sign, a nd the expression in brackets in negative. In that case ε and η increase exponentially, and the motion is unstable. Mr Neil Honkanen of the University of Vict oria conducted an expe riment to illustrate the stability of rotation about the three principal axes. The body in question was a small “brick” of mild steel (density 7.83 g/cm 3) of dimensions 83 inch × 43 inch × 211 inch, mass 54.1 g. In round figures, this corre sponds to principal moments of inertia A0 = 2 ×10−6 kg m2, B0 = 7 × 10−6 kg m2, C0 = 8 × 10−6 kg m2. He suspended it from an electromagnet, which he set in rotation at a bout 25 revolutions per second, and then let it fall, while photographing it stroboscopically . He did three experiments rotation respectively about the three pr inciple axes. You can see fr om the photographs below that the rotation is stable when th e rotation is about th e axes of greatest or least moment of inertia, but is unstable when the rotation is about the axis of intermediate moment of inertia. 26 27 28 29 4.7 Nonrigid rotator The rotational kinetic energy of a bo dy rotating about a principal axis is2 21ωI, where I is the moment of inertia about that princi pal axis, and the angular momentum is L = Iω. (For rotation about a nonprincipal axis, see se ction 4.3.) Thus the rotational kinetic energy can be written as L2/(2I). When an asymmetric top is rotating about a nonprincipa l axes, the body experiences internal stresses, which, if the body is non rigid, result in pe riodic strains which periodically distort th e shape of the body. As a result of this, rotational kinetic energy becomes degraded into hea t; the rotational kinetic en ergy of the body gradually decreases. In the absence of external torques, however, the angular momentum is constant. The expression L 2/(2I) for the kinetic energy shows that the kinetic energy is least for a given angular momentum when th e moment of inertia is greatest. Thus eventually the body rotates about its principal axis of greatest moment of inertia. After that, it no longer loses kinetic energy to h eat, because, when the body is rotating about a principal axis, it is no longer subject to internal stresses. The time taken (the “relaxation time”) for a bod y to reach its final st ate of rotation about its principal axis of greatest moment of in ertia depends, among othe r things, on how fast the body is rotating. A fast ro tator will reach its final stat e relatively soon, whereas it takes a long time for a slow rotator to reach its fi nal state. Thus it is not surprising to find that, among the asteroids, most of the fast rotators are principal axis rotators, whereas many slow rotators are also nonp rincipal axis rotators. Th ere are, however, a few fast rotators that are still rotating about a nonprincipal axis. It is a ssumed that such asteroids may have suffered a collision in the recent past. 4.8 Force-free Motion of a Symmetric Top Notation: I 1 , I1 , I3 are the principal moments of inertia. I3 is the unique moment. If it is the largest of the three, the body is an oblate symmetric top; if it is the smallest, it is a prolate spherical top. O x 0, Oy0 , Oz0 are the corresponding body- fixed principal axes. ω1 , ω2 , ω3 are the components of the angular velocity vector ω with respect to the principal axes. In the analysis that follows, we are going to have to think about three vectors. There will be the angular momentum vector L, which, in the absence of external torques, is fixed in magnitude and in direction in laboratory space. There will be the direction of the axis of symmetry, the O z axis, which is fixed in the body, but not necessarily in space, unless the body happens to be rotating about its axis of symmetry; we’ll de note a unit vector in this 30 direction by 0zˆ. And there will be the instantaneous angular velocity vector ω which is neither space- nor body-fixed. What we are going to find is the following. We shall find that ω precesses in the body about the body-fixed symmetry axis in a cone called the body cone. The angle between ω and 0zˆis constant (we’ll be calling this angle α), and the magnitude ω of ω is constant. We shall find that the sense of th e precession is the same as the sense of the spin if the body is oblate, but opposite if it is prolate. The di rection of the symmetry axis, however, is not fixed in space, but it precesses about the space-fixed angular momentum vector L in another cone. This cone is narrower than the body cone if the body is oblate, but broader than the body cone if the body is prolate. The net result of these two precessional motions is that ω precesses in space about the space-fixed angular momentum vector in a cone called the space cone . For a prolate top, the semi vertical angle of the space cone can be anything from 0o to 90o; for an oblate top, however, the semi vertical angle of th e space cone cannot exceed 19o 28' . That’s quite a lot to take in in one breath! We can start with Euler’s equations of moti on for force-free rotation of a symmetric top: , ) (1 3 32 11 II I − ωω−=ω& 4.8.1 , ) (1 3 31 21 II I − ωω=ω& 4.8.2 . 033=ω&I 4.8.3 From the first of these we obtain the result ω 3 = constant. 4.8.4 For brevity, I am going to let ,) ( 13 1 3Ω=ω− III 4.8.5 although in a moment Ω will have a physical meaning. Equations 4.8.1 and 4.8.2 become: 2 1 ωΩ−=ω& 4.8.6 and .1 2 ωΩ+=ω& 4.8.7 Eliminate ω2 from these to obtain 31 .12 1 ωΩ−=ω&& 4.8.8 This is the equation for simple harmonic motion and its solution is ) , cos( 0 1 ε+Ω ω=ω t 4.8.9 in which ω0 and ε, the two constants of integration, whose values de pend on the initial conditions in the usual fashion, are the amplit ude and initial phase angle. On combining this with equation 4.8.6, we obtain . ) sin( 0 2 ε+Ω ω=ω t 4.8.10 From these we see that2/12 22 1 ) ( ω+ω , which is the magnitude of the component of ω in the x0y0-plane, is constant, equal to ω0; and since ω3 is also constant, it follows that 2/12 32 22 1 ) ( ω+ω+ω , which is the magnitude of ω, is also constant. The cosine of the angle α between 0zˆand ω is ω3/2/12 32 22 1 ) ( ω+ω+ω , and its sine is ω0/2/12 32 22 1 ) ( ω+ω+ω , so that α is constant. Equations 4.8.9 and 4. 8.10 tell us, then, that the vector ω is precessing around the symmetry ax is at an angular speed Ω. Making use of equation 4.8.5, we find that . ) (cos 1 31 3 ω−Ω=ωω=αIII 4.8.11 If we take the direction of the z0 axis to be the direction of the component of ω along the symmetry axis, then Ω is in the same direction as z0 if I3 > I1 (that is, if the top is oblate) and it is in the opposite direction if the top is prolate. The s ituation for oblate and prolate tops is shown in figure IV.11. FIGURE IV.11 α ω3 ω0 ω α ω3 ω0 Ω Ω Oblate Prolate α ω3 ω0 α ω3 ω0 Ω Ω Oblate Prolate ω 32 We have just dealt with how the instantane ous axis of rotation precesses about the body- fixed symmetry axis, describing the body cone of semi vertical angle α. Now we are going to consider the precession of the body-fixed symmetry axis about the space-fixed angular momentum vector L. I am going to make use of the idea of Eulerian angles for expressing the orientation of one th ree-dimensional set of axes with respect to another. If you are not alrea dy familiar with Eulerian angl es or would like a refresher, you can go to http://astrowww.phys.uvic.ca/~tatum/celmechs/celm3.pdf especially section 3.7, page 29. Recall that we are using O x 0 y0 z0 for body-fixed coordinates, referred to the principal axes. I shall use O xyz for space-fixed coordinates, and there is no loss of generality if I choose the O z axis to coincide with th e angular momentum vector L . Let me try to draw the situation in figure IV.12a. The axes O xyz are the space-fixed axes . The axes O x0y0z0 are the body-fixed principal axes. The angular momentum vector L is directed along the axis O z. The symmetry axis of the body is directed along the axis O z0. The Eulerian angles of the body-fixed axes relati ve to the space fixed axes are ( φ, θ, ψ). Recall, with the aid of figure IV.12b, how these Euler angles are formed: First, a rotation by φ about O z. Second, a rotation by θ about the dashed line 'Ox to form an intermediate set of axes ' ''O zyx . Third, a rotation by ψ about 'Oz to form the body- fixed principal axes O x0y0z0. Spend a little time trying to visualize these th ree sets of axes. Please also convince yourself, from the way the Euler angles were formed through three rotations, that the vector L is in the ''zy plane and has no 'x component. It is also in the 00zyplane and has no 0xcomponent. You will then agree that θ = θ = = cos , sin ,0' ' ' L L L L Lz y x . 4.8.12 Now if , 0'=xL then 'xωis also zero, which means that ω , like L , is in the ' 'zyplane. We have seen that ω makes an angle α with the symmetry axis O z0, where α is given by equation 4.8.11. 33 O z x x0 θ φ ψ L z0 y0 y FIGURE IV.12a O z x x0 θ φ ψ L '0zz y0 y 'y 34 I’ll now add ω to the drawing to make figure IV.13. Like L, it is in the ''zy plane and has no 'x component. I haven’t marked in the angle α. I leave it to your im agination. It is the angle between ω and z0. You should easily agree that . cos , sin ,0' ' ' α ω=ω α ω=ω =ωz y x 4.8.13 From these, together with ' 3 ' ' 1 ' andz z y y I L I L ω= ω= we obtain . tan tan3 1 θ =α I I 4.8.14 For an oblate symmetric top, I3 > I1 , α > θ . For a prolate symmetric top, I3 < I1 , α < θ . Now ω can be written as the vector sum of the ra tes of change of the three Euler angles: .ψφθω &&& ++= 4.8.15 The components of ψθ&&and along O y' are each zero, and therefore the component of ω along O y' is equal to the component of ψ& along O y' . O z x x0 θ φ ψ L z0 y0 y 'y ω 35 ∴ . sin sin θ φ=α ω & 4.8.16 In summary, then: 1. The instantaneous axis of rotation, which makes an angle α with the symmetry axis, precesses around it at angular speed , cos 11 3α ω−=ΩII I 4.8.17 which is in the same sense as ω if the top is oblate and oppos ite if it is prolate. 2. The symmetry axis makes an angle θ with the space-fixed a ngular momentum vector L, where . tan tan 31α =θII 4.8.18 For an oblate top, θ < α. For a prolate top, θ > α. 3. The speed of precession of the symmetry axis about L is ,sinsinωθα=φ& 4 . 8 . 1 9 or, by elimination of θ between 4.8.18 and 4.8.19, . cos 12/1 2 2 12 12 3ω α−+ =φII I& 4.8.20 The net result of this is that ω precesses about L at a rate φ& in the space cone , which has a semi-vertical angle α − θ for an oblate rotator, and θ − α for a prolate rotator. The space cone is fixed in space, while the body co ne rolls around it, always in contact, ω being a mutual generator of both cones. If th e rotator is oblate, the space cone is smaller than the body cone and is inside it. If the rotator is prolate, the body cone is outside the space cone and can be larg er or smaller than it. Write 1 3II c= 4.8.21 for the ratio of the principal moments of inertia. Note that for a pencil, c = 0 ; f o r a sphere, c = 1; for a plane disc or any regular plane lamina, c = 2. (The last of these follows from the perpendicular axes theorem – see Chapter 2.) The range of c, then, is from 0 to 2, 0 to 1 being prolate, 1 to 2 being oblate. 36 Equations 4.8.17 and 4.8.20 can be written α − =ωΩcos)1 (c 4.8.22 and [] . cos)1 ( 12/12 2α − + =ωφc& 4.8.23 Figures IV.15 and IV.16 show, for an oblat e and a prolate rotator respectively, the instantaneous rotation vector ω precessing around the body-fixed symmetry axis at a rate Ω in the body cone of semi vertical angle α; the symmetry axis precessing about the space-fixed angular momentum vector L at a rate φ& in a cone of semi vertical angle θ (which is less than α for an oblate rotator, and greater than α for a prolate rotator; and consequently the instan taneous rotation vector ω precessing around the space-fixed angular momentum vector L at a rate φ& in the space cone of semi vertical angle α − θ (oblate rotator) or θ − α (prolate rotator). 37 Space cone Angular momentum L Body cone ω Ω φ& α θ Symmetry axis FIGURE IV.15 * 38 One can see from figures IV.15 and 16 that the angle between L and ω is limited for an oblate rotator, but it can be as large as 90o for a prolate rotator. The angle between L and ω is θ − α (which is negative for an oblate rotator). We have Symmetry axis Angular momentum L Body cone Space cone φ& θ α Ω ω FIGURE IV.16 39 . tantan) 1( tan tan1tan tan) tan(2α +α −=α θ +α −θ=α−θcc 4.8.24 By calculus this reaches a maximum value of . tanfor 21c cc=α− For a rod or pencil (p rolate), in which c = 0, the angle between L and ω can be as large as 90o. Recalling exactly what are meant by the vectors L and ω, the reader should try now and imagine in his or her mind’ s eye a pencil rotating so that L and ω are at right angles. The spin vector ω is along the length of the pencil and the angular momentum vector L is at right angles to the length of the pencil. For an oblate rotator, the angle between L and ω is limited. The most oblate rotator is a flat disc or any regular flat lamina. The para llel axis theorem shows that for such a body, c = 2. The greatest angle between L and ω for a disc occurs when tan α = √2 (α = 54o 44'), and then tan ( α − θ ) = 1/√8 , α − θ = 19o 28'. In the following figures I illustrate some of these results graphically. The ratio I3/I1 goes from 0 for a pencil through 1 for a sphere to 2 for a disc. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1-0.8-0.6-0.4-0.200.20.40.60.81 cφdot/ωPENCIL SPHERE DISC PROLATE OBLATEα°015 30 45 60 75 90 Ω/ω 40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.20.40.60.811.21.41.61.82 cφdot/ωPENCIL SPHERE DISC PROLATE OBLATEα°0 15 30 45 60 75 90 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 200.050.10.150.20.250.30.350.40.450.5 cΩ/φdotSPHERE DISC OBLATEα° 015 30 45 60 75 90 41 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4-3.5-3-2.5-2-1.5-1-0.50 cΩ/φdotPENCIL SPHERE PROLATEα° 0153045607590 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5-4.5-4-3.5-3-2.5-2-1.5-1-0.50 cφdot/ΩPENCIL PROLATEα° 0 15 30 45 60 75 42 0 10 20 30 40 50 60 70 80 90-1-0.8-0.6-0.4-0.200.20.40.60.81 α, degreesΩ/ω PROLATEOBLATEDISC SPHERE PENCILc 0.000.250.500.751.001.251.501.752.00 0 10 20 30 40 50 60 70 80 9000.20.40.60.811.21.41.61.82 α, degreesφdot/ω PROLATEOBLATEDISC SPHERE PENCILc 0.000.250.500.751.001.251.501.752.00 43 0 10 20 30 40 50 60 70 80 9000.050.10.150.20.250.30.350.40.450.5 α, degreesΩ/φdotOBLATE DISC SPHEREc 1.001.251.501.752.00 44 0 10 20 30 40 50 60 70 80 90-4-3.5-3-2.5-2-1.5-1-0.50 α, degreesΩ/φdotPROLATE PENCILSPHERE c 0.000.250.500.75 0 2 4 6 8 10 12 14-3.5-3-2.5-2-1.5-1-0.50 α, degreesφdot/Ω PROLATEPENCILc 0.00 0.25 0.50 0.75 45 0 10 20 30 40 50 60 70 80 900102030405060708090 α, degreesθ - α, degreesPROLATEc 0 0.05 0.1 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60 70 80 9002468101214161820 α, degreesα - θ, degreesOBLATEc 2.0 1.8 1.6 1.4 1.2 1.0 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10102030405060708090 cθ - α, degreesPROLATE1 1530 45 60 75 89α° 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 202468101214161820 cα - θ, degreesOBLATE 115304555α° 47 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 202468101214161820 cα - θ, degreesOBLATE55 65 75 85 89α° 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20-100102030405060708090 I3/I1Greatest angle between L and ωPROLATE OBLATEPENCIL SPHERE DISC FIGURE XV.18PROLATE OBLATEPENCIL SPHERE DISC FIGURE XV.18 IV.18 48 Our planet Earth is approxima tely an oblate spheroid, its dynamical ellipticity 1 1 3 /) ( III− being about 3.285 × 10−3. It is not rotating exactly abut its symmetry axis; the angle α between ω and the symmetry axis being about one fifth of an arcsecond, which is about six metres on the surface. The rotation pe riod is one sidereal day (which is a few minutes shorter than 24 solar hours.) Equation 4.8.17 tells us that the spin axis precesses about the symmetry axis in a pe riod of about 304 days, all within the area of a tennis court. The actual motion is a little more comp licated than this. The period is closer to 432 days because of the nonrigidity of Earth, and superimposed on this is an annual component caused by the annual movement of ai r masses. This precessional motion of a symmetric body spinning freely about an axis in clined to the symmetry axis gives rise to variations of latitude of amplitude about a fifth of an arcs econd. It is not to be confused with the 26,000 year period of the precession of the equinoxes , which is caused by external torques from the Moon and the Sun. 4.9 Centrifugal and Coriolis Forces We are usually told in el ementary books that there is “no such thing” as centrifugal force . When a satellite orbits around Earth, it is no t held in equilibrium between two equal and opposite forces, namely gravity acting to wards Earth and centrifugal force acting outwards. In reality, we are told, the satellite is accelerating (the centripetal acceleration ); there is only one force, namely the gravitational force, which is equal to the mass times the centripetal acceleration. Yet when we drive round a corner too fast and we feel ourselves flung away from the centre of curvature of our path, the “centrif ugal force” certainly feels real enough, and indeed we can solve problems referred to rota ting coordinate systems as if there “really” were such a thing as “centrifugal force”. Let’s look at an even simpler example, not ev en involving rotation. A car is accelerating at a rate a towards the right. See figur e IV.20 – but forgive my limited artistic abilities. Drawing a motor car is somewhat beyond my skills. * a a FIGURE IV.20 49 There is a plumb-bob hanging from the roof of the car, but, because of the acceleration of the car, it is not hanging verti cally. Some would say that th ere are but two forces on the plumb-bob – its weight and the tension in the st ring – and, as a result of these, the bob is accelerating towards the right . By application of F = ma it is easily possible to find the tension in the string and the angle that the string makes with the vertical. The passenger in the car, however, sees th ings rather differently (figure IV.21.) To the passenger in the car, nothing is accele rating. The plumb-bob is merely in static equilibrium under the action of three forces, one of which is a force ma towards the left. To the question “But what is the agent that is causing this so-called force?” I counter with the question “What is the agent that is cau sing the downward force that you attribute to some mysterious ‘gravity’ ”? It seems that, when referred to the referen ce frame of figure IV.20, there are only two forces, but when referred to the accelerating reference frame of figure IV.21, the system can be described perfectly well by po stulating the existence of a force ma pulling towards the left. This is in fact a principle in classical mechanics, known as d’Alembert’s principle, whereby, if one refers the descri ption of a system to an accelerating reference frame, one can replace an acceleration with a for ce in the opposite direction. It results in a perfectly valid descri ption of the behavior of a system , and will accurately predict how the system will behave. So who’s to say which forces are “real” and which are “fictitious”, and which reference frame is better than another? The situation is similar with respect to centr ifugal force. If you consider a satellite in orbit around Earth, some would say that th ere is only one force acting on the satellite, namely the gravitational force towards Earth. The satellite, being in a circular orbit, is accelerating towards the centr e of the circle, and all is as expected - F = ma. The acceleration is the centripetal acceleration (peto – I desire). An astronaut on board the satellite may have a different po int of view. He is at a cons tant distance from Earth, not * FIGURE IV.21 ma 50 accelerating; he is in static equilibrium, and he feels no net force on him whatever – he feels weightless. If he has b een taught that Earth exerts a gravitational force, then this must be balanced by a force away from Earth. This force, which becomes apparent when referred to a corotating reference frame, is the centrifugal force (fugo – I flee, like a fugitive ). It would need a good lawyer to argue that the invisible gravitational force towards Earth is a real force, while the equa lly invisible force acti ng away from Earth is imaginary. In truth, all forces are “imaginary” – in that they are only devices or concepts used in physics to describe and predict the behaviour of matter. I mentioned earlier a possible awful examinatio n question: Explain why Earth bulges at the equator, without using the term “centrifugal force”. Just thank yourself lucky if you are not asked such a question! People who ha ve tried to answer it have come up with some interesting ideas. I have heard (I don’ t know whether it is true) that someone once offered a prize of $1000 to anyone who could pr ove that Earth is rotating, and that the prize has never been claimed! Some have tried to imagine how you would determine whether Earth is rotating if it were the onl y body in the universe. There would be no external reference points against which one coul d measure the orientation of Earth. It has been concluded (by some) that even to thi nk of Earth rotating in the absence of any external reference points is meaningless, so that one certainly could not determine how fast, or even whether and about what axis, Earth was rotating. Si nce “rotation” would then be meaningless, there would be no cen trifugal force, Earth would not bulge, nor would the Foucault pendulum rotate, nor would naval shells deviate fr om their paths, nor would cyclones and anticyclones exist in the atmosphere.. Centrifugal force comes into existence only when there is an external universe. It is the external universe, then, revolving around the stationary Earth, that causes centrifugal force and all the other effects that we have mentioned. These are deep waters indeed, a nd I do not pursue this aspect further here. We shall merely take the pragmatic view that problems in mechanics can often be solved by referring motions to a corotating reference frame, and that the behaviour of mechanical systems can succe ssfully and accurately be described and predicted by postulating the “existence” of “inertial” forces su ch as centrifugal and Coriolis forces, which make themselves apparent only when referred to a rotating frame. Thus, rather than involving ourselves in diffi cult questions about whether such forces are real, we shall take things easy with just a few simple equations. 51 Σ = O xyz is an inertial reference frame (i.e. not accelerating or rotating). Σ' = O x'y'z' is a frame that is rotating about the z-axis with angular velocity .ˆzω ω= Three questions: 1. If P is a point that is fixed with respect to Σ', what is its velocity v with respect to Σ? 2. If P is a point that is moving with velocity v' with respect to Σ', what is its velocity v with respect to Σ? 3. If P is a point that has an acceleration a' with respect to Σ', what is its acceleration a with respect to Σ? 1. The answer to the first question is, I thi nk, fairly easy. Just by inspection of figure XV.22, I hope you will agree that it is .rωv ×= 4.9.1 In case this is not clear, try the following ar gument. At some instant the position vector of P with respect to Σ is r. At a time δt later its position vector is r + δr , where t rr δωθ =δ sin and δr is at right angles to r, and is directed along the small circle z x y P y ω r FIGURE IV.22 x' O 52 whose zenith angle is θ, in the direction of moti on of P with respect to Σ. Expressed alternatively, . δ φ r ˆ sin t r δωθ = Draw the vector δr on the figure if it helps. Divide both sides by δt and take the limit as δt → 0 to get r&. The magnitude and direction of r& are then expressed by the single vector equation ,rωr ×=& or .rωv ×= The only thing to look out for is th is. In two-dimensional problems we are often used to expressing the relation between linear and angular speed by v = r ω, and it doesn't matter which way round we write r and ω. When we are doing a three-dimensional problem using vector notation, it is important to remember that it is . rωv ×= and not the other way round. 2. If P is moving with velocity v' with respect to Σ', then its velocity v with respect to Σ must be v = v' + ω × r . 4.9.2 This shows that × +  ≡   Σ Σω 'dtd dtd 4.9.3 What this equation is intended to convey is that the operation of differentiating with respect to time when referred to the inertial frame Σ has the same result as differentiating with respect to time when referred to the rotating frame Σ', plus the operation ω × . We shall understand this a little better in the next paragraph. 3. If P is accelerating with respect to Σ', we can apply the operation 4.9.3 to the equation 4.9.2: ) '( ) '( 'rωvω rωv a ×+ × + ×+   = Σdtd .) ( ' ' ' rωωvω vω a ××+×+×+ = ∴ .' 2 ) ( ' vω rωω a a × + ×× + = 4.9.4 This, then, answers the third question we posed . All we have to do now is to understand what it means. To start with, let us return to the case where P is neither moving nor accelerating with respect to Σ'. In that case, equation 4.9.4 is just 53 , ) ( rωω a × × = 4.9.5 which we could easily have obt ained by applying the operator 4.9.3 to equation 4.9.1. Let us try and understand what this m eans. In what follows, a “hat” ( ^ ) denotes a unit vector. We have φ rω ˆ sinθ ω=× r and hence φz φ z rωω ˆˆ sin ˆ sin ˆ ) (2×θ ω=θ ω×ω= ×× r r and φzˆˆ×is a unit vector directed towards the z-axis. Notice that th e point P is moving at angular speed ω in a small circle of radius r sin θ . The expression φz rωωa ˆˆ sin ) (2×θ ω= ××= r , then, is just the familiar centripetal acceleration , of magnitude r ω2 sin θ, directed towards the axis of rotation. We could also think of ) (rωω ×× as a triple vector product . We recall that B)C(A B)C(A C) (BA • • − = × × so that . ) ( ) (2 . rωrω rωω ω− = ×× That is . ˆ ˆ cos ) (2 2r z rωω ω−θ ω= ×× r r This can be illustrated by th e vector diagram shown in fi gure IV.23. The vectors are drawn in green, in accordance with my conve ntion of red, blue and green for force, velocity and acceleration respectively. FIGURE IV.23 ) ( rωω × × rˆ2ωr zˆ cos2θ ωr 54 However, equation 4.9.4 also tells us that, if a particle is moving with velocity v' with respect to Σ', it has an additional acceleration with respect to Σ of v'ω×2 , which is at right angles to v' and to ω. This is the Coriolis acceleration. The converse of equation 4.9.4 is . '2 ) ( ωvωrω a a' × + ×× + = 4.9.6 If a particle has a force F = ma acting on it with respect to Σ, when referred to Σ' it will appear that the particle is moving under the influence of three forces: the “true force” F, the centrifugal force acting away from the axis of rotation, and the Coriolis force , which acts only when the particle is in motion with respect to Σ', and which is at right angles to both v' and ω: .'2 ) ( ωv ωrω F F' × + ×× + = m m 4.9.7 It is worth now spending a few moments thinking about the dire ction of the Coriolis force . 2ωv'×m Earth is spinning on its axis with a period of 24 si dereal hours (23h and 56m of solar time.). The vector ω is directed upwards through the north pole. Now go to somewhere on Earth at latitude 45o N. Fire a naval shell to th e north. To the east. To the south. To the west. Now go to the equator a nd repeat the experiment. Go to the north pole. There you can fire only due south. Repeat the experiment at 45o south, and at the south pole. Each time, think a bout the direction of the vector . 2ωv'×m If your thoughts are to my thoughts, your mind to my mind, you s hould conclude that the shell veers to the right in the northern hemisphere and to the le ft in the southern hemisphere, and that the Coriolis force is zero at the equator. As ai r rushes out of a high pressure system in the northern hemisphere, it will swirl clockwise around the pressure cen tre. As it rushes in to a low pressure system, it will swirl countercloc kwise. The opposite situation will happen in the southern hemisphere. You can think of the Coriolis force on a naval shell as being a consequence of conservation of angular momentum. Go to 45 o N and point your naval gun to the north. Your shell, while waitin g in the breech, is moving around Ea rth’s axis at a linear speed of ωR sin 45o, where R is the radius of Earth. Now fire the shell to the north. By the time it reaches latitude 50o N, it is being carried around Earth’ s axis in a small circle of radius only R sin 40o. In order for angular momentum to be conserved, its angular speed around the axis must speed up – it will be deviated towards the east. Now try another thought experiment ( Gedanken Prüfung.). Go to the equator and build a tall tower. Drop a stone from the top of the tower. Think now about the direction of the vector . 2ωv'×m I really mean it – think hard. Or again, think about conservation of angular momentum. The stone drops closer to Ea rth’s axis of rotation. It must conserve angular momentum. It falls to the east of the tower ( not to the west!). 55 The two forces on the stone are its weight mg and the Coriolis force. Earth’s spin vector ω is to the north. The Coriolis force is at right angl es to the stone’s velocity. If we resolve the stone’s velocity into a vertically down component y & and a horizontal east component x&, the corresponding components of the Coriolis force will be ym&ω2t o t h e east and xm&ω2 upward. However, I’m going to assume that y x&&<< and the only significant Coriolis force is the eastward component ym&ω2 , which I have drawn. Another way of stating the approximation is to say that the upward component of the Coriolis force is negligible compared with the weight mg of the stone. After dropping for a time t, the y-coordinate of the stone is found in the usual way from , , ,2 21gt y gt y g y = = = & && and the x- coordinate is found from . , , 2 23 31 2gt x gt x gt y x ω= ω= ω=ω= & &&& Thus you can find out how far to the east it has falle n after two seconds, or how far to the east it has fallen if the height of the tower is 100 metres. The equation to the trajectory would be the t-eliminant, which is .9832 2ygxω= 4.9.8 * FIGURE IV.24 East x y ym&ω2 mg North 9 ω 56 For Earth, ω = 7.292 × 10−5 rad s−1, and at the equator g = 9.780 m s−2, so that .m 10 788.4981 102 − −× =ω g The path is graphed in figure IV.25 for a 100-metre tower. The horizontal scale is exaggerated by a factor of about 6000. 0 2 4 6 8 10 12 14 16 18 20 220102030405060708090100 Displacement to east, millimetresHeight above ground, metres I once asked myself the questi on whether a migrating bird could navigate by using the Coriolis force. After all, if it were flyi ng north in the northern hemisphere, it would experience a Coriolis force to its right; might this give it navigational information? I published an article on this in The Auk 97, 99 (1980). Let me know what you think! You may have noticed the similarity betw een the equation for the Coriolis force ωv' F × =m2 and the equation for the Lorentz force on an electric charge moving in a magnetic field: .Βv F × =q The analogy can be pursued a bi t further. If you rotate a coil in an electric field, a current will flow in the coil. That’s electromagnetic induction, and it is the principle of an electric generato r. Sometime early in the twentieth century, the American physicist Arthur Compton (that’ s not Denis Compton, of whom only a few of my readers will have hear d, and very few indeed in No rth America) successfully tried an interesting experiment. He made some toroidal glass tubes, filled with water coloured FIGURE IV.25 57 with KMnO 4, so that he could see the water, and he rotated these tubes about a horizontal or vertical diameter, and, lo, the water flowed around the tubes, just as a current flows in a coil when it is rotated in a magnetic field. Imagine a toroidal tube set up in an east-west vertical plane at the equator. The top part of the tube is slightly further from Earth’s rotation axis than the bottom part, and conse quently the water near the top of tube has more angular momentum per unit mass, aroun d Earth’s axis, than the water near the bottom. Now rotate the tune through 180o about its east-west horizontal diameter. The high angular momentum fluid moves closer to Earth’s rotation axis, and the low angular momentum fluid moves further from Earth’s axis. Therefor e, in order to conserve angular momentum, the fluid must flow around th e tube. By carrying out a series of such experiments, Compton was able, at least in principle, to measure the speed of Earth’s rotation, and even his latitude, without l ooking out of the window, and indeed without even being aware that there wa s an external universe out ther e. You may think that this was a very difficult experiment to do, but you do it yourself every da y. There are three mutually orthogonal semicircular canals in side your ear, and, every time you move your head, fluid inside these semicircular canals fl ows in response to the Coriolis force, and this fluid flow is detected by little nerves, which send a message to your brain to tell you of your movements and to help you to keep your balance. You have a wonderful brain, which is why understanding physics is so easy. Going back to the Lorentz force, we reca ll that a moving charge in a magnetic field experiences a force at right angles to its velo city. But what is the origin of a magnetic field? Well, a magnetic field exists, for example, in the interior of a solenoid in which there is a current of moving electrons in th e coil windings, and it is these circulating electrons that ultimately cause the Lorentz force on a charge in the interior of the solenoid – just as it is the galaxies in th e universe revolving around our stationary Earth which are the ultimate cause of the Coriolis force on a particle movi ng with respect to our Earth. But there I seem to be getting into deep waters again, so perhaps it is time to move on to something easier. 4.10 The Top We have classified solid bodi es technically as symmetric, asymmetric, spherical and linear tops, according to the relative sizes of th eir principal moments of inertia. In this section, or at least in the titl e of this section, I mean “top” in the nontechnical sense of the child’s toy – that is to say, a symmetric body, pointed at one end, spinning around its axis of symmetry, with the pointed end on the gr ound or on a table. Technically, it is a “heavy symmetric top with one point fixed.” 58 I have drawn it is figure IV.26, spinning about its symmetry axis, which makes an angle θ with the vertical. The distance between the ce ntre of mass and the point of contact with the table is l. It has a couple of forces acting on it – its weight and the equal, opposite reaction of the table. In figure IV.27, I replace these two forces by a torque, τ, which is of magnitude Mgl sin θ. Note that, since there is an external torque acting on the system, the angular momentum vector is not fixed. FIGURE IV.26 ω θ l ω θ l FIGURE IV.27 9τ 59 Before getting too involved with numerous equa tions, let’s spend a little while describing qualitatively the motion of a top, and also describing the various coordinate systems and angles we shall be discu ssing. First, we shall be making use of a set of space-fixed coordinates. We’ll let the origin O of the coordinates be at the (fixed) point where the tip of the top touches the table. The axis O z points vertically up to the zenith. The O x and Oy axes are in the (horizontal) plane of the ta ble. Their exact orie ntation is not very important, but let’s suppose that O x points due south, and O y points due east. O xyz then constitutes a right-handed set. We’ll also make use of a set of body-fixed axes, which I’ll just refer to for the moment as 1, 2 and 3. The 3-axis is the symmetry axis of the top. The 1- and 2-axes are perpendicular to this. Their exact positions are not very important, but let’s suppose that the 31-plane passes thr ough a small ink-dot which you have marked on the side of the top, and that the 123 system constitutes a right-handed set. We are going to describe the orientation of the top at some instant by means of the three Eulerian angles θ, φ and ψ (see figure IV.28). The symmetry axis of the top is represented by the heavy arrow, and it is tipped at an angle θ to the z-axis. I’ll refer to a plane normal to the axis of symmetry as the “equator” of the top, and it is inclined at θ to the xy-plane. The ascending node of the equator on the xy-plane has an azimuth φ, and ψ is the angular distance of the 1-axis from th e node. The azimuth of the symmetry axis of the top is φ − 90 o = φ + 270º. FIGURE IV.28 z y x O 1 θ φ θ ψ 60 Now let me anticipate a bit and describe the motion of the top while it is spinning and subject to the torque described above. The symmetry axis of the top is going to precess around the z-axis, at a rate that will be described as . φ & Except under some conditions (which I shall eventually describe) this precessional motion is secular. That means that φ increases all the time – it does not oscillate to and fro. However, the symmetry axis does not remain at a constant angle with the z-axis. It oscillates, or nods, up and do wn between two limits. This motion is called nutation (Latin: nutare , to nod). One of our aims will be to try to find the rate of nutation φ& and to find the period and amplitude of the nutation. It may look as though the top is spinning about its axis of symmetry, but this isn’t quite so. If the angular velocity vector were ex actly along the axis of symmetry, it would stay there, and there would be no precession or nuta tion, and this cannot be while there is a torque acting on the top. An exception woul d be if the top were spinning vertically ( θ = 0), when there would be no torque acting on it . The top can in fact do that, except that, unless the top is spinning quite fast, this situ ation is unstable, and the top will tip away from its vertical position at th e slightest perturbation. At high spin speeds, however, such motion is stable, and indeed one of our aims must be to determine the least angular speed about the symmetry axis such motion is stable. However, as mentioned, unless the top is spinning vertically, the vector ω is not directed along the symmetry axis. We’ll call the three components of ω along the three body- fixed axes ω1, ω2 and ω3, the last of these being the component of ω along the symmetry axis. One of the things we shall discover wh en we proceed with the analysis is that ω3 remains constant throughout the motion. Also , you should be able to distinguish between ω3 and .ψ& These are not the same, because of the motion of the node. In fact you will probably understand that . cos3 θ φ−ω=ψ & & Indeed, we have already derived the relations between the component of the angular velocity vector and the rate of change of the Eulerian angles – see equations 4.2.1,2 and 3. We shall be making use of these relations in what follows. To analyse the motion of the top, I am going to make use of Lagrange’s equations of motion for a conservative system. If you are fam iliar with Lagrange’s equations, this will be straightforward. If you ar e not, you might prefer to skip this section until you have become more familiar with Lagrangian mechanics in Chapter 13. However, I introduced Lagrange’s equation briefly in section 4.4, in which Lagr ange’s equation of motion was given as .j j jPqT qT dtd=∂∂−    ∂∂ & 4.10.1 61 Here T is the kinetic energy of the system. Pj is the generalized force associated with the generalized coordinate qj. If the force is a conservative force, then Pj can be expressed as the negative of the derivative of a potential energy function: .    ∂∂−= jjqVP 4.10.2 Thus we have Lagrange’s equati on of motion for a system of conservative forces . j j j qV qT qT dtd ∂∂−=∂∂−    ∂∂ & 4.10.3 Thus, in solving problems in Lagrangian dynami cs, the first line in our calculation is to write down an expression for the kinetic energy. The first line begins: “ T = ...”. In the present problem, the kinetic energy is . ) (2 3321 2 22 1 121ω + ω+ω = I I T 4.10.4 Here the subscripts refer to the principal ax es, 3 being the symmetry axis. The Eulerian angles θ and φ are zenith distance and azimuth respec tively of the symmetry axis with respect to laboratory fixed (space fixed) axes. The Eulerian angle ψ is measured around the symmetry axis. The components of the an gular velocity are related to the rates of change of the Eulerian angles by previously derived formulas (equations 4.2.1,2,3), so the kinetic energy can be expr essed in terms of : and, ψ φθ &&& .) cos ( ) sin (2 321 2 2 2 121θ φ+ψ +θ φ+θ = && && I I T 4.10.5 The potential energy is . constant cos +θ =Mgl V 4.10.6 Having written down the expressions for the kinetic and potential energies in terms of the Eulerian angles, we are now in a position to apply the Lagrangian equations of motion 4.10.3 for each of the three coordinates. We’ll start with the coordinate φ. The Lagrangian equation is .φ∂∂−=φ∂∂−    φ∂∂ V T T dtd & 4.10.7 62 We see that φ∂∂ φ∂∂ V Tand are each zero, so that 0=    φ∂∂ &T dtd, or . constant=φ∂∂ &T This has the dimensions of angular mome ntum, so I’ll call the constant L1. On evaluating the derivative , φ∂∂ &T we obtain for the Lagrangian equation in φ: . cos cos sin1 32 32 1 L I I I =θ ψ+θ φ+θ φ & & & 4.10.8 I’ll leave the reader to carry out exactly the same pro cedure with the coordinate ψ. You’ll quickly conclude that , constant=ψ∂∂ &T which has the dimensions of angular momentum, so call it L3, and you will then arrive at the following for the Lagrangian equation in ψ: . ) cos (3 3 L I =θ φ+ψ&& 4.10.9 But the expression in parentheses is equal to ω3 (see equation 4.2.3, although we have already used it in equation 4.10.5) , so we obtain the result that ω3, the component of the angular velocity about the symmetry axis, is constant during the motion of the top. I t would probably be worth the reader’s time at th is point to distinguish again carefully in his or her mind the difference between ω3 and .ψ& Eliminate ψ& from equations 4.10.8 and 4.10.9: . sincos 2 13 1 θθ −=φIL L& 4.10.10 This equation tells us how the rate of precession varies with θ as the top nods or nutates up and down. We could also eliminate φ& from equations 4.10.8 and 4.10.9: . sincos) cos ( 2 13 1 33 θθ θ −− =ψIL L IL& 4.10.11 The Lagrangian equation in θ is a little more complicated, but we can obtain a third equation of motion from the c onstancy of the total energy: . cos ) cos ( ) sin (2 321 2 2 2 121E Mgl I I =θ +θ φ+ψ +θ φ+θ && && 4.10.12 We can eliminate ψ φ&&and from this, using equations 4.10.10 and 11, to obtain an equation in θ and the time only. After a little algebra, we obtain 63 , sincoscos2 13 1 2     θθ −−θ − =θIL LB A& 4.10.13 where    − = 32 3 121 ILEIA 4.10.14 and .2 1IMglB= 4.10.15 The turning points in the θ-motion (i.e. the nutation) occur where . 0=θ& This results (after some algebra! – but quite straightforward all the same) in a cubic equation in : cosθ =c 03 2 2 1 0 = + + + Bc ca ca a 4.10.16 where ,2 2 12 1 312 3 12 11 0IL IIL IE ILA a − − =   − = 4.10.17 12 131 2 131 12 2 2 IMgl ILLBILLa − = − = 4.10.18 and .22 13 312 3 12 13 2       + − −=   − −=IL IIL IE ILA a 4.10.19 Now equation 4.10.16 is a cubic equation in cos θ and it has either one real root or three real roots, and in the latter case two of them or all three mi ght be equal. We must also bear in mind that θ is real only if cos θ is in the range −1 to +1. We are trying to find the nutation limits, so we are hoping that we w ill find two and only two real values of θ . (If the tip of the top were poised on top of a point – e.g. if it were poised on top of the Eiffel Tower, rather than on a horizontal table − you could have θ > 90o.) To try and understand this bette r, I constructed in my mind a top somewhat similar in shape to the one depicted in figures IV.25 a nd 26, about 4 cm diameter, 7 cm high, made of brass. For th e particular shape and dimensions that I imagined, it worked out to have the following parameters, rounded off to two significant figures: M = 0.53 kg l = 0.044 m I1 = 1.7 × 10−4 kg m2 I3 = 9.8 × 10−5 kg m2 I thought I’d spin the top so that ω3 (which, as we have seen, remains constant throughout the motion) is 250 rad s−1, and I’d start the top at rest ( 0 =θ=φ&& ) at θ = 30o and then let 64 go. Presumably it would then im mediately start to fall, and 30o would then be the upper bound to the nutation. We want to see how fa r it will fall before nodding upwards again. With ω3 = 250 rad s−1 we find, from equation 4.10.9 that L3 = 2.45 ×10−2 J s. Also, I am assuming that 0 =φ& when θ = 30o, and equation 4.10.10 tells us that L1 = 2.121 762 × 10−2 J s. Then with g = 9.8 m s−2, we have, from equation 4.10.15, B = 2.688 659 × 103 s−2. My initial conditions are that φ& and θ&are each zero when θ = 30o, and equations 4.10.10 and 4.10.13 between them tell us that A = B cos 30o, so that A = 2.328 447 × 103 s−2. From equations 4.12.17, 18 and 19 we now have a 0 = −1.324 898 × 104 s−2 a1 = +3.328 586 × 104 s−2 a2 = −2.309 834 × 104 s−2 and we already have B = 2.688 659 × 10 3 s−2. The “sign rule” for polynomial equations, if you are familiar with it, tells us that there are no negative real roots (i .e. no solutions with θ > 90o), and indeed if we solve the cubic equation 4.10.16 we obtain c = 0.824 596, 0.866 025, 6.900 406. The second of these corresponds to θ = 30 ο, which we already knew must be a solution. Indeed we could have di vided equation 4.10.16 by c – cos 30o to obtain a quadratic equation for the remaining two roots, but it is perhaps best to solve the cubic equation as it is, in order to verify that cos 30 o is indeed a solution, th us providing a check on the arithmetic. The third solution does not give us a real θ (we were rather hoping this would happen). The second solution is the lower limit of the nutation, corresponding to θ = 34o 27'. 65 Generally, however, the top will nutate between two values of θ. Let us call these two values α and β, α being the smaller (more vertical) of the two. I’ll refer to θ = α as the “upper bound” of the motion, even though α < β , since this corresponds to the more vertical position of the top. We have looked a little at the motion in θ; now let’s look at the motion in φ, starting with equation 4.10.10: . sincos 2 13 1 θθ −=φIL L& 4.10.10 If the initial conditi ons are such that L1 > L3 cos α (and therefore always greater than φθ&) cos3L is always positive. The motion is then something like I try to illustrate in figure IV.29. This motion corresponds to an initial condition in which you gi ve the top an initial push in the forward direction as indi cated by the little blue arrow. If the initial conditi ons are such that , cos / cos 3 1 β > >α LL the sign of φ& is different at the upper and lower bounds. Thus is illustrated in figure IV.30 FIGURE IV.29 FIGURE IV.30 66 This motion would arise if you were initially to give a little backward push before letting go of the top, as indicated by the little blue arrow. If the initial conditi ons are such that L 1 = L3 cos α, then θ& and φ& are each zero at the upper bound of the nutation, and this was the s ituation in our numerical example. It corresponds to just letting th e top drop when you let it go , without giving it either a forward or a backward push. This is illustrated in figure IV.31. As we discovered while doing our numer ical example, the initial conditions 0=φ=θ && when θ = α lead, in this third type of motion, to α = cos3 1L L 4.10.20 and . cosα =B A 4.10.21 In that case equation 4.10.13 becomes ().sincos cos) cos (cos2 13 2  θθ −α−θ −α =θILB& 4.10.22 The lower bound to the nutat ion(i.e. how far the top falls) is found by putting θ = β when .0=θ& This gives the following quadratic equation for β: .0 1coscos cos2 12 3 2 12 3 2=−α+β −βBIL BIL 4.10.23 In our numerical example, this is FIGURE IV.31 67 c o s2 β − 7.725 002 cos β + 5.690 048 = 0, 4.10.24 which, naturally, has the same two solutions as we obtained when we solved the cubic equation, namely 0.824 596 and 6.900 406. Recalling the definition of B (equation 4.10.15), we see that equation 4.10.23 can be written , sin2cos cos2 2 31β =β −αLMglI 4.10.25 from which we see that the greater L3, the smaller the difference between α and β − i.e. the smaller the amplitude of the nutation. Equation 4.10.12, with the help of equati ons 4.10.10 and 11, can be written: . cos ) cot csc(21 22 3 1 12 121 32 3θ +θ −θ =θ − − Mgl L LIIILE & 4.10.26 The left hand side is the total energy minus the spin and nutation ki netic energies. Thus the right hand side represents the effective potential energy Ve(θ) referred to a reference frame that is co-rotating with the precession. The term Mgl cosθ needs no explanation. The negative of the derivative of the firs t term on the right hand side would be the “fictitious” force that “exists” in the corota ting reference frame. The effective potential energy Ve(θ) is given by .cos 2] cot)/( [csc)2/()( 2 11 2 1 3 12 1e LMglILLI LV θ+θ −θ =θ 4.10.27 I draw Ve(θ) in figures IV.32 and 33 using the valu es that we used in our numerical example – that is: θ +θ −θ =θ cos536228.0 ) cot701154.1 (csc081324.1)(2 eV joules. 4.10.28 Figure 32 is plotted up to 90o (although as mentioned earlie r one could go further than this if the top were not spinning on a horizont al table), and Figure 33 is a close look close to the minimum. One can see that if 1979 .0)2/(22 3 = − I L E the effective potential energy (which cannot go higher than this , and reaches this value only when ) 0 =θ& , the nutation limits are between 30o and 34o 24' . For a given L3, for a larger total energy, the nutation limits are correspondingly wider. But for a given total energy, the larger the component L3 of the angular momentum is, the lo wer will be the horiz ontal line and the narrower the nutation limits. If the top loses energy (e.g. because of air resistance, or 68 friction at the point of contact with the table), the E = constant line will become lower and lower, and the amplitude of the nut ation will become less and less. If )2/(32 3I L E− is equal to the minimum value of Ve(θ) there is only one solution for θ, and there is no nutation. For energy less than this , there is no stationary value of θ and the top falls over. 0 10 20 30 40 50 60 70 80 9000.511.522.533.544.5 θ, degreesVe, joulesFIGURE IV.31 32 69 25 30 35 400.1950.20.2050.210.2150.220.225 θ, degreesVe, joulesFIGURE IV.32 We can find the rate of true regular precession quite simply as follows – and this is often done in introductory books. δL L FIGURE IV.34 33 70 In figure IV.34, the vector L represents the angular momentum at some time, and in a time interval δt later the change in th e angular momentum is δL. The angular momentum is changing because of the external torque , which is a horizontal vector of magnitude θsinMgl (remind yourself from figure XV.26 and 27). The rate of change of angular momentum is given by . τ=L& In time δt the tip of the vector L moves through a “distance” τ δt. Denote by Ω precessional angular velocity (the magnitude of which we have hitherto called φ&). The tip of the angular moment um vector is moving in a small circle of radius L sin θ. We therefore see that τ = ΩL sin θ . Further, τ is perpendicular to both Ω and L. Therefore, in vector notation, . L× =Ω τ 4.10.29 Note that the magnitude of τ is Mgl sin θ and the magnitude of L×Ω is ΩL sin θ, so that the rate of precession is LMgl=Ω 4.10.30 and is independent of θ . One can continue to analyse the motion of a top almost indefinite ly, but there are two special cases that are perhaps wort h noting and which I shall describe. Special Case I. L 1 = L3 . In this case, equation 4.10.27 becomes , cos ) cot (csc)(2 eθ +θ −θ =θCMglV 4.10.31 where . 212 1 MglILC= 4.10.32 It may be rather unlikely that L1 = L3 exactly, but this case is of interest partly because it is exceptional in that Ve(0) does not go to infinity; in fact Ve(0) = Mgl whatever the value of C. Try substituting θ = 0 in equation 4.10.31 and see what you get! The right hand side is indeed 1, but you may have to wo rk a little to get there. The other reason why this case is of interest is that it makes a useful introduction to case II, which is not impossibly unlikely, namely that L1 is approximately equal to L3, which leads to motion of some interest. In figure IV.35, I draw Ve(0)/Mgl for several different C. 71 0 10 20 30 40 50 60 70 80 900.811.21.41.61.82Ve(θ)/(Mgl θ, degreesC = 10 2 1.5 1FIGURE IV.34 From the graphs, it looks as though, if 2≥C , there is one equilibrium position, it is at θ = 0o (i.e. the top is vertical), and the equilibrium is stable, If C < 2, there are two equilibrium positions: the vertical position is unstable, and the other equilibrium position is stable. Thus if the top is spinning fast (large C) it can spin in the vertical position only (a “sleeping top”), but, as the top slows dow n owing to friction and air resistance, the vertical position will become unstable, and the top will fall down to a positive value of θ . These deductions are correct, for 0e=θddV results in θ =θ −4 2sin ) cos1(2C 4.10.33 One solution is θ = 0, and a second differentiation w ill show that this is stable or unstable according to whether C is greater than or less than 2, although the second differentiation is slightly tedious, and it ca n be avoided. We can also note that θ −cos1 is a common factor of the two sides of equation 4.10.33, and it can be divided out to yield a cubic equation in cos θ: ,0 cos cos cos)1 2( 1 23 2=θ −θ −θ + −− C C 4.10.34 35 72 which could be solved to find the second equi librium point – but that again is slightly tedious. A less tedious way might be to take the square root of each side of equation 4.10.33: θ −=θ −2cos1) cos1(2C 4.10.35 and then divide by the common factor (1 − cos θ) to obtain ,1 2 cos − =θ C 4.10.36 which gives a real θ only if .2≥C Note also, if .90 ,o 21=θ =C Special Case II. .3 1L L≈ In other words, L1 and L3 are not very different. In figure IX.36 I draw Ve(0)/Mgl for several different C, for L3 = 1.01 L1. 0 10 20 30 40 50 60 70 80 900.811.21.41.61.82Ve(θ)/(Mgl θ, degreesC = 10 2 1FIGURE IV.35 We see that for quite a large range of C greater than 2 the stable equilibrium position is close to vertical. Even though the curve for C = 2 has a very broad minimum, the actual minimum is a little less than 17o. (I haven’t worked out the exact position – I’ll leave that to the reader.) 36 73 Appendix In Section 4.4 we raised the question as to whether angle is a dimensioned or a dimensionless quantity, and in Section 4.8 we raised the question as to whether angle is a vector quantity. I can present two arguments. One of them will prove in controvertibly that angle is dimensionless. The other will prove, equally incontrovertibly, and equally co nvincingly that angle has dimensions. Angle, as you know, is defined as the ratio of arc length to radius. It is the ratio of two lengths, and is therefore incontrovertibly dimensionless. Q.E.D. On the other hand, it is necessary to state the units in which angle is expressed. You cannot merely talk about an angle of 1. You must state whether that is 1 degree or 1 degree. Angle therefore has dimensions. Q.E.D. So – you may take your pick. In many contexts, I like to think of angle as a dimensioned quantity, having dimensions Θ. That is to say, not a combination of mass, length and time, but having its own dimensions in its own right. I find I can carry on with dimensional analysis successfully like this. Now for the question: Is angle a vector? An angle certainly has both magnitude and a direction associated with it. Thus the direction associated with the angle is at right angles to the plane of the screen, or the paper. However, this evidently isn’t enough for it to be a vector in the sense that we know it. For example, if you tu rn through an angle a, and then through an angle b, you cannot say that the net resultant of these is to turn through an angle c, where . cos22 2 2C ab b a c − + = a b c C 74 Thus, although angle has both magnitude and direction, and could be thought of thus far as a vector, angles do not obey the ordinary triangle law of vector add ition. For this reason, angles are sometimes called “pseudo-vectors”. In fact, as any astronomy student will tell you , the correct relation between the angles is . cos sin sin cos cos cos C b a b a c + = If the angles a, b, c (not C) are very small, then the triangle become s almost plane. The angles add more and more like the usual plane triangle rule for vector addition. This is probably obvious when thinking about the geometry, but you can also convince yourself of it by expanding the sines and cosines (except for cos C) as series, and, to the second order of small quantities )1 sin, 1 (cos2 21≈θ θ−≈θ , you’ll find that the equation C b a b a c cos sin sin cos cos cos + = reduces to . cos22 2 2C ab b a c − + = For this reason it is sometimes said that an “infinitesimal rotation” can be regarded as a true vector. Also for this reason, the time rate of change of an angle, dtdθ, that is to say an angular velocity, can quite safely be treated as a tr ue vector, since the numerator and denominator of the derivatives are both infinitesimals. Thus, although angle has direction associated with it, angle is not a true vector in that angles do not follow the usual rules for vector addition. However, very small angles do approximately follow the addition rules, so that, in the infinitesimal limit, angles can be treated as vectors. And hence angular velocity, being a ratio of infinitesimals ( dθ and dt ), can correctly be treated as vectors.