Rotation

“ We want to say, that a rotation in a plane corresponds to a certain direction of the perpendicular, and namely it shall be the direction into that side, at which an observer must be located, so that for him the rotation is counter-clockwise. ”

“ These illustrations will suffice to show how a rapidly rotating body may contain energy in virtue of its rotation, just as a cannon-ball contains energy in virtue of its speed of translation, or as a clock-weight has energy in virtue of the fact that it has some distance to fall before it reaches the earth. The rotating body need not necessarily have the shape of a wheel—it may be globular in form; nor need the axes of rotation be fixed in bearings, like those of the fly-wheel; nor of course is there any limit to the dimensions which the rotating body may assume. ”

“ Those which are not favourably placed move in a right line (as the shortest path) to consort with bodies of their own nature. But this Motion of Rotation admits of nine differences; regarding, 1. the centre round which the bodies move; 2. the poles on which they move; 3. their circumference or orbit, according to their distance from the centre; 4. their velocity, according to the greater or less rapidity of their rotation; 5. the course of their motion, as from east to west, or from west to east; 6. their declination from a perfect circle by spiral lines more or less distant from their centre ”

“ For instance, if the actual rotation of the earth from west to east is taken positive, the direction of the earth s axis from south to north will be taken positive, and if a man walks forward in the positive direction, the positive rotation is in the order, head, right-hand, feet, left-hand.
If we place ourselves on the positive side of a surface, the positive direction along its bounding curve will be opposite to the motion of the hands of a watch with its face towards us. ”

“ If the motion be uniformly continued the sensation of rotation will gradually cease entirely and the observer will imagine himself at rest. But if rr be placed outside the axis of rotation, at once on the rotation beginning, a strikingly apparent, palpable, actually visible inclination of the entire paper box is produced, slight when [Pg 289] the rotation is slow, strong when the rotation is rapid, and continuing as long as the rotation lasts. It is absolutely impossible for the observer to escape perceiving the inclination, although here also all outward points of reference are wanting. ”

“ Let us now consider the movements of rotation, which follow from wounds of portions of the encephalon. They are of two distinct types: the one is a rotary movement round a circle, the other a motion of rolling or spinning. In the first case, the animal remains in his normal attitude, but tends always to go to one side, and describe an orbit more or less extended. In the second case, the animal can progress but little in moving himself. When he attempts to move, he is forced into a revolving motion, and turns on himself like a ball. ”

“ 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. ”

“ A body may be moving with a double rotation, in which these two independent components are not equal; but in such a case we can consider the body to be moving with a composite rotation—a rotation of the A or B kind and, in addition, a rotation about a plane. ”

“ He is only aware that he has been gently turned round a little two or three times. Now let the speed of rotation be slackened. He has a sensation of being gently turned round a little in the opposite direction. Each time the speed is lessened he has this sense of being turned the reverse way. From these experiments we see that what we are conscious of is change of rate of rotation, or, in technical language, acceleration, positive or negative.
Fig. 30.—Diagram of semicircular canals. ”

“ If now the axis of rotation is given a motion of revolution about the axis of figure, then when one half a revolution has been completed the axis of rotation will be in the position P′′P1′′, the equator line will have shifted to E′′E1′′, and the latitude of the place O will have changed from E′O to E′′O, the whole change in the latitude, E′E′′ being 2i, or twice the angle between the axis of figure and the axis of rotation. ”

“ 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. ”

“ But what is meant is elemental rotation, a rotation of the infinitesimal elements as they move. We have an example of such motion in the air when a ring of smoke escapes from the funnel of a locomotive or the lips of a tobacco-smoker, in the motion of part of the liquid when a cup of tea is stirred by drawing the spoon from one side to the other, or when the blade of an oar is moving through the water. In these last two cases the depressions seen in the surface are the ends of a vortex which extends between them and terminates on the surface. ”

“ Beginning then with the earth's velocity of rotation at its lowest, we may suppose it gradually and continually increased, and, as we have already seen, the change in the energy of the system is to be expressed by multiplying the relative rotation into the change of the earth's angular velocity. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ We all know that the train’s motion is retarded, and this loss of motion represents the requisite expenditure of force. Now, is there any process in nature resembling, in however remote a degree, the application of a brake to check the earth’s rotation? There is. The tidal wave, which sweeps, twice a day, round the earth, travels in a direction contrary to the earth’s motion of rotation. ”

“ 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. ”

“ 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) . ”

“ In order to illustrate the theory of rotation experimentally, we must have a body balanced on its centre of gravity, and capable of having its principal axes and moments of inertia altered in form and position within certain limits. ”

“ Contact between w and f is maintained by means of a weight or spring, the rotation of f being sufficiently slow to insure its being continuous, while the necessary rapidity of cutting speed for the tools is attained by rotating h at the required speed of rotation. ”

“ It is known, from the principles of mechanics, that this endeavor can never strictly be executed, because the axial rotation of the globe incessantly so acts as to throw every body, while in motion, in our hemisphere, to the right of the line on which it is moving, no matter whether that line be from east to west, north to south, or at any conceivable angle with the meridians or the equator. ”

“ Similarly, in mechanics, the arc of a vibrating pendulum may be gradually enlarged by successive impacts until it becomes a circle. The part of a rotation differs generically from a complete one, yet it may approach infinitely near to it, and with only such a difference as exists between one arc and another slightly shorter. ”

“ I fasten a horizontal board in the frame RR of my rotatory apparatus, lie down upon the same with my right ear upon the board, and cause the apparatus to be uniformly rotated. As soon as I no longer perceive the rotation, I turn around upon my left ear and immediately the sensation of rotation again starts up with marked vividness. The experiment can be repeated as often as one wishes. A slight turn of the head even is sufficient for reviving the sensation of rotation which in the perfectly quiescent state at once disappears altogether. ”

“ The velocity of rotation which a body receives from a given impulse is great in exactly the same proportion as the moment of inertia is small. Thus the moment of inertia may be considered in rotatory motion analogous to the mass of the body in rectilinear motion. ”

“ 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. ”

“ We may obtain a more concrete representation of the new ideas by imagining a crisscross of threads extending between each individual star and all the others. Rotation, which develops dynamical effects, is, then, the rotation of an object with respect to this network of threads; and it is obviously a matter of indifference to suppose that the object is rotating among the threads and stars, or that threads and stars are rotating around the object. ”

“ 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. ”

“ A body in rapid rotation tends to remain in its plane; that is, its axis rests parallel with itself, and instead of obeying any force tending to divert its direction, it produces in consequence of the combination of two simultaneous movements, a displacement of the axis, generally much feebler and of a different kind from that which this force exercises on the same body in repose. ”

“ This axis intersects the surface at the north and south poles. The time which the earth occupies in making a complete rotation around this axis is called a sidereal day. The sidereal day is a little shorter than the ordinary day, being only 23 hours, 56 minutes, and 4 seconds. The rotation is performed just as if a rigid axis passed through the centre of the earth; or, to use the old and homely illustration, the earth rotates just as a ball of worsted may be made to rotate around a knitting-needle thrust through its centre. ”

“ 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 movement, as has often been pointed out, is like that of a peg-top which rotates rapidly on its axis while at the same time the axis itself has a slow revolving motion. Thus the phenomena which are presented in the rotation of the earth demonstrate that the axis about which the earth rotates occupies what is, at all events, approximately a fixed position in the earth, though not a 6xed position in space. ”

“ Rotation. Irregular Distribution. Irrotation, —Definition.—The propositions connecting boundary circulation and rotation include all cases of rotation, so that we know that however much the rotation differs in different parts of the fluid, the algebraic sum of the rotation taken over the whole of the region is equal to the integration of the circulation along the boundary (reckoned plus or minus, according to the law laid down) . Thus, the total rotation in a region containing fluid is zero when the sum of the circulation taken over a complete circuit of the boundary is zero ”