portional to the rotary velocity of the disk, as well as to the angular velocity of the axis, diminishes with the former, and as it diminishes, the axis must descend, acquiring angular velocity due to the height of fall: hence the rapid gyration and the descending spiral motion which accompanies the loss of rotary velocity. A more curious and puzzling effect of the friction of the axle is presented, when we come to take into consideration, instead of our theoretical solid, the discrepancies of figure presented by the actual gyroscope. If, with a high initial rotation, the common gyroscope be placed on its point of support with its axis somewhat inclined above a horizontal position, it will soon be observed to rise. In my analytical examination (p. 543) I have stated as a deduction from the second equation (4), that "the axis of figure can never rise above its initial angle of elevation." That equation supposes that the rotary velocity n remains unimpaired, and is the expression of a fundamental principle of dynamics-that of "living forces" (so-called), which requires that the living force generated by gravity be directly proportional to the height of fall, and involves as a corollary that through the agency of its own gravity alone, the centre of gravity of a body can never rise above its initial height. The anomaly observed, therefore, either requires the action of some foreign force; or, that the living force lost by the rotating disk, shall, through some hidden agency, be expended in performing this work of lifting the mass. * The discrepancy here exhibited between the motion proper to our theoretical solid of revolution and the experimental gyroscope is due to the division of the latter into two distinct masses, one of which rotates, with friction, upon points or surfaces in the other; and to the fact that at the point of support (in the latter) there is not perfectly free motion in all directions. The friction at the extremities of the axle of the disk, tends to impress on the mass which constitutes the "mounting," a rotation in the same direction. Were the motion of the latter upon its fixed point of support perfectly free, the mounting and disk would soon acquire a common rotatory velocity about the axis of the disk. But the mounting is perfectly free to turn about the vertical axis through the point of support, though not about any other. If we decompose, therefore, the rotation which would be impressed upon the mounting into two components, one about this vertical, and the other about a horizontal axisthe first takes full effect, and the latter is destroyed at the point fo support. If the axis of the instrument is above the horizon tal, this component of rotation is in the same direction as the gyration due to gravity, and adds to it; if the axis is below the horizontal, the component is the reverse of the natural gyration, and diminishes it. *The first of these equations (as I have remarked in a note to p. 547) is the expression of another fundamental principle-more usually called the "principle of areas." But I have shown that the axis soon acquires, independent of this cause, a gyration whose deflecting or sustaining force is just equivalent to the downward component of gravity. The addition to this gyratory velocity caused by friction when the axis is inclined upwards puts the deflecting force in excess, and the axis is raised; it is raised, as in all other cases in which work is done through acquired velocity-viz., by an expenditure of living force; but in this instance, through a most curious and complicated series of agencies. The phenomenon may be best illustrated in the following manner. Let the outer extremity of the common gyroscope, having its axis inclined above the horizontal, be supported by a thread attached to some fixed point vertically above the point of support, so that gyration shall be free. Here gravity is eliminated, and the axis of our theoretical solid of revolution would remain perfectly motionless; but the gyroscope starts off, of itself, to gyrate in the same direction that it would were its extremity free. This gyration increases (if the rotary velocity is great) until the deflecting force due to it, lifts the outer extremity from its support on the thread, and it continues indefinitely to rise. Try the same experiment with the axis below the horizontal. The gyration will commence spontaneously as before, but in the reverse direction: it will increase until the inner extremity is lifted from the point of support, (the action of the deflecting force being here reversed,) the instrument supporting itself on the thread alone. If the experiment is tried with the axis perfectly horizontal, no gyration takes place, for the component of rotation, due to friction, is, in this position, zero. The foregoing reasoning accounts, I believe, for all the observed phenomena of the experimental gyroscope, and shows how, from the theory of our imaginary solid of revolution, a consideration of the effects of the discrepancies of form, and of the actual disturbing forces, leads to their satisfactory explanation. The great similarity between the phenomena of the top and gyroscope, renders it not uninteresting to compare the laws of motion of the two. If we conceive a solid of revolution terminated at its lower extremity by a point (the ordinary form of the top), resting upon a horizontal plane without friction, and having a rotary motion about its axis of figure, such a body will be subject to the action of two forces; its weight, acting at the centre of gravity, and the resistance of the plane, acting at the point vertically upwards. According to the fundamental principles of dynamics, the centre of gravity will move as if the mass and forces were concentrated at that point, while the mass will turn about this centre as if it were fixed. Calling R the resistance of the plane, M the mass, and Mg the weight of the top, and z the height of the centre of gravity above the plane, we shall have for the equation of motion of the centre of gravity* d2z M -R-Mg (1.) As the angular motion of the body is the same as if the centre of gravity was fixed, and as R is the only force which operates to produce rotation about that centre, if we call C the moment of inertia of the top about its axis of figure, and A its moment with reference to a perpendicular axis through the centre of gravity, and y the distance, GK (fig. 2) of the point of support from that centre; the equations of rotary motion will become identical with equations (3) (p. 541), substituting R for Mg Cd v2 =0 Advy-(C-A) v2 v1dt=ya Rdt Advx+(C−A) výv2 d t = − ɣ b Rd t (2.) The first of equations (2) gives us v, as for the gyroscope, equal a constant n. Multiplying the 2d and 3d of equations (2) by vy and v, respectively, and adding and making the same reduction as on p. 53, we shall get A(vyd vy+vzdv)=Ryd.cos 0. But z (the height of the centre of gravity above the fixed plane) =-7 cos 0; hence rd. cos 0=-dz; and equation (1) gives R=M(+g). Substituting these values of Rand yd. cos in the preceding equation, and integrating, we have From the 2d and 3d of equations (2) the equation (c) (of the gyroscope, p. 542) is deduced by an identical process. A(bvy+av)+ Cn cos 0=1, and a substitution in the two foregoing equations of the values of the cosines a and b, and of the angular velocities v, and in terms of the angles, and y (see pp. 540, 541), and for z and d Ꮎ Ti dz dt their values, cos 0, and y sin 0 and a determination of the constants, on the supposition of an initial inclination of the axis a, and of initial velocity of axial rotation n, will give us for the equations of motion of the top: *As there are no horizontal forces in action, there can be no horizontal motion of the centre of gravity except from initial impulse, which I here exclude from which the angular motions of the top can be determined. The first is identical with the first equation (4) for the gyroscope. The second differs from the second gyroscopic equation only in containing in its first member the term My2 sin 20 dz2 d 02 d t2' or its equivalent Maz, expressing the living force of vertical transla d t21 tion of the whole mass. The second member (as in the corresponding equation for the gyroscope) expresses the work of gravity, and the first term of the first member expresses the living force due to the angular motion of the axis. Instead therefore of the work of gravity being expended (as in the gyroscope) wholly in producing angular motion, part of it is expended in vertical translation of the centre of gravity. The angular motion takes place not (as in the gyroscope) about the point of support (which in this case is not fixed), but about the centre of gravity (to which the moments of inertia A and B refer); and that centre, motionless horizontally, moves vertically up and down, coincident with the small angular undulations of the axis through a space which will be more and more minute as the rotary velocity n is greater. dy An elimination of between the two equations (4) and a study of the resulting equation, would lead us to the same general results, as the similar process, p. 544, for the gyroscope. The vertical angular motion, expressed by the variation which the angle undergoes, becomes exceedingly minute (the maximum and minimum values of approximating each other) when n is great, and the axis gyrates with slow undulatory motion about a vertical through the centre of gravity. It would be easy, likewise, to show by substituting for 0 another variable, u=a-0, always (in case of high values of n) extremely small, and whose higher powers may therefore be neglected, that the co-ordinates of angular motion, u and y, approximate more and more nearly to the relation expressed by the equation of the cycloid as n increases; though the approximation is not so rapid as in the gyroscope. All the results and conclusions flowing from the similar process for the gyroscope (see pp. 545, 546, 547, 548) would be deduced. As, however, the centre of gravity, to which these angular motions are referred, is not a fixed point, but is itself constantly rising and falling as increases or diminishes, the actual motion of the axis is of a more complicated character. If GK" (see fig. 2) is the initial position of the axis of the top, the motion of the centre of gravity will consist ina vertical falling and rising through the distance G GGK"(cosz, G'G" — cosz, GG')=7(cos, -cos a) (in which, is the minimum value of) while the extremity of the axis K the point of the top describes a circle, more or less perfect, about G". The rationale of the self-sustaining power of the top is identical with that of the gyroscope; the deflecting force due to the angular motion of the axis plays the same part as the sustaining agent, and has the same analytical expression. Owing to friction, the top likewise rises, and soon attains a vertical position; but the agency by which this effect is produced is not exactly the same as for the gyroscope. If the extremity of the top is rounded, or is not a perfect mathematical point, it will roll, by friction, on the supporting surface along the circular track just described. This rolling speedily imparts an angular motion to the axis greater than the horizontal gyration due to gravity, and the deflecting force becomes in excess, (as explained in the case of the gyroscope,) and the axis rises until the top assumes a vertical position. Even though the extremity of the top is a very perfect point, yet if it happens to be slightly out of the axis of figure (and rotation) the same result will, in a less degree, ensue: for the point, instead of resting permanently on the surface, will strike it, at each revolution, and in so doing, propel the extremity along. The conditions of a perfect point, perfectly centered in the axis of figure, are rarely combined, or rather are practically impossible; but it is easy to ascertain by experiment that the more nearly they are fulfilled, and the harder and more highly polished the supporting surface, the less tendency to rise is exhibited; while the great stiffness (or tendency to assume a vertical position) of tops with rounded points, is a fact well known and made use of in the construction of these toys. The references throughout this paper are to my paper on the gyroscope in the June number of the Am. Journal of Education. |