Rotation axis

“ Each string is supposed to be rotating in the same direction about its axis, and the angular velocity of rotation is a measure of the magnetic force. In consequence of this rotation there will be differences of pressure in different directions in the medium; the pressure along the axes of the strings will be less than it would be if the medium were at rest, that in the directions at right angles to the axes will be greater, the medium will behave as though it were under tension along the axes of the molecules under pressure at right angles to them. ”

“ The impressed forces are not the only causes which affect the axis of a body during the phenomenon of rotation. This species of motion calls into action other forces depending on the inertia of the mass, which produce effects upon the axis, and which play a prominent part in the theory of rotation. While the body revolves on its axis, the component particles of its mass move in circles, the centres of which are placed in the axis. The radius of the circle in which each particle moves is the line drawn from that particle perpendicular to the axis. ”

“ It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place. Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. ”

“ It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings. ”

“ The composition of finite rotations about parallel axes is a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.
In any continuous motion of a solid about a fixed point O, the limiting position of the axis of the rotation by which the body can be brought from any one of its positions to a consecutive one is called the instantaneous axis. This axis traces out a certain cone in the body, and a certain cone in space, and the continuous motion in question may be represented as consisting in a rolling of the former cone on the latter. ”

“ Therefore, we must make the following supposition: that if the earth does turn round the sun, its axis of rotation remains parallel to itself, having no reference whatever to the sun, whilst the earth is going round the sun; or, to express it in other words—that the earth has the power of preserving its axis of rotation in the same position all the time. Now, it is remarkable, as a mechanical fact, that nothing is so permanent in nature as the axis of rotation of anything which is rapidly whirled. We have examples of this in every-day practice. The first is the motion of a boy's hoop. ”

“ 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. ”

“ 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. ”

“ In real fluids the rotation is not concentrated at the axis as in the case discussed in § 72, but is distributed about the axial region or core. As a matter of convention in the perfect fluid, it is usual to suppose the core to be in a state of uniform rotation—that is, to have constant angular velocity, and the motion of the part external to the core to be cyclic and irrotational, there being no discontinuity at the surface of the core, the velocity being a continuous function of the position . ”

“ 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. ”

“ In mechanism it is usually the central line either of a rotating shaft or axle having journals, gudgeons, or pivots turning in fixed bearings, or of a fixed spindle or dead centre round which a rotating bush turns; but it may sometimes be entirely beyond the limits of the turning body. For example, if a sliding piece moves in circular fixed guides, that piece rotates about an ideal fixed axis traversing the centre of those guides.
Let the angular velocity of the rotation be denoted by α = dθ/dt, then the linear velocity of any point A at the distance r from the axis is αr ”

“ 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. ”

“ 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) . ”

“ 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. ”

“ 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. ”