Axis of rotation

“ There is an instantaneous axis of rotation, i.e. by a suitable Lorentz transformation one can achieve that all points of a straight line are at rest.
6) All points of the body have a constant "rest distance" with respect to a certain point located upon the instantaneous axis, the center of rotation.
Except the identification of these general properties of the definition, I have computed some special motions, among them the case that a point of the body carries out a uniformly-accelerated motion (hyperbolic motion) , and that the body is arbitrarily rotating about the direction of translation. ”

“ Now, it is manifest that this rotation cannot be that of the medium as a whole about an axis, for the magnetic field may be of any breadth, and there is no evidence of any motion the velocity of which increases with the distance from a single fixed line in the field. If there is any motion of rotation, it must be a rotation of very small portions of the medium each about its own axis, so that the medium must be broken up into a number of molecular vortices. ”

“ The two forces (other than the one exerted by the fulcrum) acting on a lever always oppose each other in the matter of changing rotation. They may be considered as a pair of parallel forces acting on a body, each tending to produce rotation.
Fig. 86.—The three classes of levers.
122. Moment of Force.—The effectiveness of each force may therefore be determined by computing its moment about the fixed axis (see Art. 84) , that is, by multiplying each force by its distance to the fulcrum or axis of rotation. Let a meter stick have a small hole bored through it at the 50 cm. ”

“ Retardation is continually balanced by acceleration, and the motion of the sphere would, therefore, be approximately uniform about the central axis of rotation. It will be clear that the movements thus described of the masses will be very similar in nature to those of the pendulum masses in the experimental system previously discussed. The fact that the motion of the auxiliary masses over the surface of the sphere is assumed to be completely circular and not vibratory, as in the pendulum case, has no bearing on the general energy phenomena. ”

“ Since the sum total of rotary motion now seen in the planetary system must have been there from the beginning, he conceives the immense vaporous mass forming the sun and his atmosphere to have had a slow rotation on its axis. As the intensely hot mass gradually cooled, it would contract towards the centre. As it contracted, its velocity of rotation would, by the laws of mechanics, constantly increase; so that a time would arrive, when, at the outer boundary of the mass, the centrifugal force due to the rotation would counterbalance the attractive force of the central mass. ”

“ A motion of translation along an axis cannot produce a rotation about that axis unless it meets with some special mechanism, like that of a screw, which connects a motion in a given direction along the axis with a rotation in a given direction round it; and a motion of rotation, though it may produce tension along the axis, cannot of itself produce a current in one direction along the axis rather than the other. ”

“ It therefore appears that the increase or diminution of the distance of the direction of a force from the axis has the same effect upon its power to give rotation as a similar increase or diminution of the force itself. The power of a force to produce rotation is, therefore, accurately estimated, not by the force alone, but by the product found by multiplying the force by the distance of its direction from the axis. It is frequently necessary in mechanical science to refer to this power of a force, and, accordingly, the product just mentioned has received a particular denomination. ”

“ Conservation of Rotation.—Let us suppose a hollow circular cylindrical vessel filled with fluid to be set in rotation, about its axis: then if the fluid possessed viscosity it would, in course of time, acquire sensibly the same speed of rotation as the vessel, so that the whole system would revolve en bloc. With an ideal fluid, however, the rotation of the vessel might be continued indefinitely without imparting any motion to its contents. ”

“ When the rod or string is equally flexible in every direction, the plane of vibration given to it from any original impulse is constantly maintained whatever may be the velocity of rotation communicated to its point of support, provided the axis of vibration remains in the same position, or move only parallel to itself. ”

“ But if ω > p the amplitude of that circular vibration which agrees in sense with the rotation ω will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. If the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (see Mechanics, § 13) . ”

“ Now that which occurs in this example of a body tending to two movements at the same time, is also produced when it is a question of movements of rotation, so that if a force acts upon a body in rotation in such a manner as to give it a movement of the same kind round another axis, a third movement will be originated round a third axis, the direction of which will be nearest to that in which the rotation is most rapid. ”

“ Its revolution with the earth will tend to make it change its angular position, and therefore it places itself parallel to the earth's axis; when in this position the daily rotation no longer causes any change in its direction in space, so it continues to point to the pole star (Fig. 48) . It would be an interesting experiment to measure with a delicate chemical balance the force with which the axis raises itself, and in this way weigh the rotational motion of the earth. ”

“ It is accordingly clear that the acceleration or retardation of rotation is felt. The acceleration operates as a stimulus. The sensation, however, like almost all sensations, though it gradually decreases, lasts perceptibly longer than the stimulus. Hence the long continued apparent rotation after the stopping of the apparatus. The organ, however, which causes the persistence of this sensation must have its seat in the head, since otherwise the axis of apparent rotation could not assume the same motion as the head. ”

“ If we have a spinning ball and we give to it a new kind of rotation, what will happen? Suppose, for example, that the earth were a homogeneous sphere, and that there were suddenly impressed upon it a new rotatory motion tending to send Africa southwards; the axis of this new spin would have its pole at Java, and this spin combined with the old one would cause the earth to have its true pole somewhere between the present pole and Java. ”

“ Despite of this, no rotation arises, since this would be identical to a rotation in , while we know that (with respect to ) the lever is in statical equilibrium. Exactly this torque is necessary to move in a translatory way.
Also equation (3) can be easily confirmed here, by calculating the component of momentum perpendicular to the velocity, and from that we calculate the increase of angular momentum according to (6) . Namely, force performs the work per unit time in ; thus an energy current enters at into the lever, end leaves it at where the rotation axis exerts the force upon it. ”

“ A movement of rotation therefore originates round a third axis placed between the two first, but whilst the real axis of the top, obeying this resultant movement, takes up its new position, the law of gravity continuing to act, displaces and moves it a little further, so that in endeavouring to reach its centre of gravity, it turns round its fulcrum (fig. 871) . ”