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Title: On a Dynamical Top Author: James Clerk Maxwell Release Date: June 1, 2002 [eBook #5192] [Most recently updated: January 21, 2021] Language: English Character set encoding: UTF-8 Produced by: Gordon Keener *** START OF THE PROJECT GUTENBERG EBOOK ON A DYNAMICAL TOP *** On a Dynamical Top, for exhibiting the phenomena of the motion of a system of invariable form about a fixed point, with some suggestions as to the Earth’s motion James Clerk Maxwell [From the _Transactions of the Royal Society of Edinburgh_, Vol. XXI. Part IV.] (Read 20th April, 1857.) To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men who had successfully threaded the mazes of the planetary motions. The mathematicians of the last age, searching through nature for problems worthy of their analysis, found in this toy of their youth, ample occupation for their highest mathematical powers. No illustration of astronomical precession can be devised more perfect than that presented by a properly balanced top, but yet the motion of rotation has intricacies far exceeding those of the theory of precession. Accordingly, we find Euler and D’Alembert devoting their talent and their patience to the establishment of the laws of the rotation of solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligible propositions supersede equations. In the practical department of the subject, we must notice the rotatory machine of Bohnenberger, and the nautical top of Troughton. In the first of these instruments we have the model of the Gyroscope, by which Foucault has been able to render visible the effects of the earth’s rotation. The beautiful experiments by which Mr J. Elliot has made the ideas of precession so familiar to us are performed with a top, similar in some respects to Troughton’s, though not borrowed from his. The top which I have the honour to spin before the Society, differs from that of Mr Elliot in having more adjustments, and in being designed to exhibit far more complicated phenomena. The arrangement of these adjustments, so as to produce the desired effects, depends on the mathematical theory of rotation. The method of exhibiting the motion of the axis of rotation, by means of a coloured disc, is essential to the success of these adjustments. This optical contrivance for rendering visible the nature of the rapid motion of the top, and the practical methods of applying the theory of rotation to such an instrument as the one before us, are the grounds on which I bring my instrument and experiments before the Society as my own. I propose, therefore, in the first place, to give a brief outline of such parts of the theory of rotation as are necessary for the explanation of the phenomena of the top. I shall then describe the instrument with its adjustments, and the effect of each, the mode of observing of the coloured disc when the top is in motion, and the use of the top in illustrating the mathematical theory, with the method of making the different experiments. Lastly, I shall attempt to explain the nature of a possible variation in the earth’s axis due to its figure. This variation, if it exists, must cause a periodic inequality in the latitude of every place on the earth’s surface, going through its period in about eleven months. The amount of variation must be very small, but its character gives it importance, and the necessary observations are already made, and only require reduction. On the Theory of Rotation. The theory of the rotation of a rigid system is strictly deduced from the elementary laws of motion, but the complexity of the motion of the particles of a body freely rotating renders the subject so intricate, that it has never been thoroughly understood by any but the most expert mathematicians. Many who have mastered the lunar theory have come to erroneous conclusions on this subject; and even Newton has chosen to deduce the disturbance of the earth’s axis from his theory of the motion of the nodes of a free orbit, rather than attack the problem of the rotation of a solid body. The method by which M. Poinsôt has rendered the theory more manageable, is by the liberal introduction of “appropriate ideas,” chiefly of a geometrical character, most of which had been rendered familiar to mathematicians by the writings of Monge, but which then first became illustrations of this branch of dynamics. If any further progress is to be made in simplifying and arranging the theory, it must be by the method which Poinsôt has repeatedly pointed out as the only one which can lead to a true knowledge of the subject,--that of proceeding from one distinct idea to another instead of trusting to symbols and equations. An important contribution to our stock of appropriate ideas and methods has lately been made by Mr R. B. Hayward, in a paper, “On a Direct Method of estimating Velocities, Accelerations, and all similar quantities, with respect to axes, moveable in any manner in Space.” (_Trans. Cambridge Phil. Soc_ Vol. x. Part I.) * In this communication I intend to confine myself to that part of the subject which the top is intended io illustrate, namely, the alteration of the position of the axis in a body rotating freely about its centre of gravity. I shall, therefore, deduce the theory as briefly as possible, from two considerations only,--the permanence of the original _angular momentum_ in direction and magnitude, and the permanence of the original _vis viva_. * The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds. It is easy to understand the motion of a body revolving about a fixed axle. Every point in the body describes a circle about the axis, and returns to its original position after each complete revolution. But if the axle itself be in motion, the paths of the different points of the body will no longer be circular or re-entrant. Even the velocity of rotation about the axis requires a careful definition, and the proposition that, in all motion about a fixed point, there is always one line of particles forming an instantaneous axis, is usually given in the form of a very repulsive mass of calculation. Most of these difficulties may be got rid of by devoting a little attention to the mechanics and geometry of the problem before entering on the discussion of the equations. Mr Hayward, in his paper already referred to, has made great use of the mechanical conception of Angular Momentum. Definition 1 The Angular Momentum of a particle about an axis is measured by the product of the mass of the particle, its velocity resolved in the normal plane, and the perpendicular from the axis on the direction of motion. * The angular momentum of any system about an axis is the algebraical sum of the angular momenta of its parts. As the _rate of change_ of the _linear momentum_ of a particle measures the _moving force_ which acts on it, so the _rate of change_ of _angular momentum_ measures the _moment_ of that force about an axis. All actions between the parts of a system, being pairs of equal and opposite forces, produce equal and opposite changes in the angular momentum of those parts. Hence the whole angular momentum of the system is not affected by these actions and re-actions. * When a system of invariable form revolves about an axis, the angular velocity of every part is the same, and the angular momentum about the axis is the product of the _angular velocity_ and the _moment of inertia_ about that axis. * It is only in particular cases, however, that the _whole_ angular momentum can be estimated in this way. In general, the axis of angular momentum differs from the axis of rotation, so that there will be a residual angular momentum about an axis perpendicular to that of rotation, unless that axis has one of three positions, called the principal axes of the body. By referring everything to these three axes, the theory is greatly simplified. The moment of inertia about one of these axes is greater than that about any other axis through the same point, and that about one of the others is a minimum. These two are at right angles, and the third axis is perpendicular to their plane, and is called the mean axis. * Let $A$, $B$, $C$ be the moments of inertia about the principal axes through the centre of gravity, taken in order of magnitude, and let $\omega_1$ $\omega_2$ $\omega_3$ be the angular velocities about them, then the angular momenta will be $A\omega_1$, $B\omega_2$, and $C\omega_3$. Angular momenta may be compounded like forces or velocities, by the law of the “parallelogram,” and since these three are at right angles to each other, their resultant is \begin{displaymath} \sqrt{A^2\omega_1^2 + B^2\omega_2^2 + C^2\omega_3^2} = H \end{displaymath} (1) and this must be constant, both in magnitude and direction in space, since no external forces act on the body. We shall call this axis of angular momentum the _invariable axis_. It is perpendicular to what has been called the invariable plane. Poinsôt calls it the axis of the couple of impulsion. The _direction-cosines_ of this axis in the body are, \begin{displaymath} \begin{array}{c c c} \displaystyle l = \frac{A\omega_1}{H}, ... ...ga_2}{H}, & \displaystyle n = \frac{C\omega_3}{H}. \end{array}\end{displaymath} Since $I$, $m$ and $n$ vary during the motion, we need some additional condition to determine the relation between them. We find this in the property of the _vis viva_ of a system of invariable form in which there is no friction. The _vis viva_ of such a system must be constant. We express this in the equation \begin{displaymath} A\omega_1^2 + B\omega_2^2 + C\omega_3^2 = V \end{displaymath} (2) Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$ in of $l$, $m$, $n$, \begin{displaymath} \frac{l^2}{A} + \frac{m^2}{B} + \frac{n^2}{C} = \frac{V}{H^2}. \end{displaymath} Let $1/A = a^2$, $1/B = b^2$, $1/c = c^2$, $V/H^2 = e^2$, and this equation becomes \begin{displaymath} a^2l^2 + b^2m^2 + c^2n^2 = e^2 \end{displaymath} (3) and the equation to the cone, described by the invariable axis within the body, is \begin{displaymath} (a^2 - e^2) x^2 + (b^2 - e^2) y^2 + (c^2 - e^2) z^2 = 0 \end{displaymath} (4) The intersections of this cone with planes perpendicular to the principal axes are found by putting $x$, $y$, or $z$, constant in this equation. By giving $e$ various values, all the different paths of the pole of the invariable axis, corresponding to different initial circumstances, may be traced. Figure: Figure 1 * In the figures, I have supposed $a^2 = 100$, $b^2= 107$, and $c^2= 110$. The first figure represents a section of the various cones by a plane perpendicular to the axis of $x$, which is that of greatest moment of inertia. These sections are ellipses having their major axis parallel to the axis of $b$. The value of $e^2$ corresponding to each of these curves is indicated by figures beside the curve. The ellipticity increases with the size of the ellipse, so that the section corresponding to $e^2 = 107$ would be two parallel straight lines (beyond the bounds of the figure), after which the sections would be hyperbolas. Figure: Figure 2 * The second figure represents the sections made by a plane, perpendicular to the _mean_ axis. They are all hyperbolas, except when $e^2 = 107$, when the section is two intersecting straight lines. Figure: Figure 3 The third figure shows the sections perpendicular to the axis of least moment of inertia. From $e^2 = 110$ to $e^2 = 107$ the sections are ellipses, $e^2 = 107$ gives two parallel straight lines, and beyond these the curves are hyperbolas. Figure: Figure 4 * The fourth and fifth figures show the sections of the series of cones made by a cube and a sphere respectively. The use of these figures is to exhibit the connexion between the different curves described about the three principal axes by the invariable axis during the motion of the body. Figure: Figure 5 * We have next to compare the velocity of the invariable axis with respect to the body, with that of the body itself round one of the principal axes. Since the invariable axis is fixed in space, its motion relative to the body must be equal and opposite to that of the portion of the body through which it es. Now the angular velocity of a portion of the body whose direction-cosines are $l$, $m$, $n$, about the axis of $x$ is \begin{displaymath} \frac{\omega_1}{1 - l^2} - \frac{l}{1 - l^2}(l\omega_1 + m\omega_2 + n\omega-3). \end{displaymath} Substituting the values of $\omega_1$, $\omega_2$, $\omega_3$, in of $l$, $m$, $n$, and taking of equation (3), this expression becomes \begin{displaymath} H\frac{(a^2 - e^2)}{1 - l^2}l. \end{displaymath} Changing the sign and putting $\displaystyle l = \frac{\omega_1}{a^2H}$ we have the angular velocity of the invariable axis about that of $x$ \begin{displaymath} = \frac{\omega_1}{1 - l^2} \frac{e^2 - a^2}{a^2}, \end{displaymath} always positive about the axis of greatest moment, negative about that of least moment, and positive or negative about the mean axis according to the value of $e^2$. The direction of the motion in every case is represented by the arrows in the figures. The arrows on the outside of each figure indicate the direction of rotation of the body. * If we attend to the curve described by the pole of the invariable axis on the sphere in fig. 5, we shall see that the areas described by that point, if projected on the plane of $yz$, are swept out at the rate \begin{displaymath} \omega_1 \frac{e^2 - a^2}{a^2}. \end{displaymath} Now the semi-axes of the projection of the spherical ellipse described by the pole are \begin{displaymath} \sqrt{\frac{e^2 - a^2}{b^2 - a^2}} \hspace{1cm}\textrm{and}\hspace{1cm} \sqrt{\frac{e^2 - a^2}{c^2 - a^2}}. \end{displaymath} Dividing the area of this ellipse by the area described during one revolution of the body, we find the number of revolutions of the body during the description of the ellipse-- \begin{displaymath} = \frac{a^2}{\sqrt{b^2 - a^2}\sqrt{c^2 - a^2}}. \end{displaymath} The projections of the spherical ellipses upon the plane of $yz$ are all similar ellipses, and described in the same number of revolutions;