Angular velocity

“ 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. ”

“ We come now to the reason for the fourth law, that, with a given actual velocity, the centrifugal force varies inversely as the diameter of the circle. If any of you ever revolved a weight at the end of a cord with some velocity, and let the cord wind up, suppose around your hand, without doing anything to accelerate the motion, then, while the circle of revolution was growing smaller, the actual velocity continuing nearly uniform, you have felt the continually increasing stress, and have observed the increasing angular velocity, the two obviously increasing in the same ratio. ”

“ But although these circumstances be inefficient, a variation in the mean temperature would certainly occasion a corresponding change in the velocity of rotation: for in the science of dynamics, it is a principle in a system of bodies, or of particles revolving about a fixed centre, that the momentum, or sum of the products of the mass of each into its angular velocity and distance from the centre is a constant quantity, if the system be not deranged by an external cause. ”

“ If any diameter of the spherical portion be taken as the axis, and any plane through that axis be considered, the velocity of a particle at right angles to that plane can be at once expressed as the rate at which the velocity-potential varies per unit distance along the circle, symmetrical about the axis, on which the particle lies. The integral of the velocity-potential round this circle vanishes, and so the angular momentum for any thin uniform ring of particles about the axis also vanishes, and as the sphere is made up of such rings, the whole angular momentum is zero. ”

“ The angular velocity measured from the resting system is thus decreasing with increasing distance from the axis. [12] The circumference velocity always remains smaller than , so that no restriction of the dimensions of rigid body, as it occurs at accelerated translation, is there at this place.
Another simple example is the following.
The center should carry out a hyperbolic motion (uniformly accelerated translation) , and the body should rotate about the direction of translation. ”

“ Let α1, α2, α3 be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings. Then, from the principles of § 60 it is evident that at each instant α2/α1 = α2/α3, and consequently that α3 = α1; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant. ”

“ The angular velocity communicated to the body will depend, firstly, upon the length of the perpendicular from the centre of gravity upon the direction of the impelling force, and secondly, upon the law of density of the material or the manner in which the metal is distributed. The direction of rotations will depend upon the position of the centre of figure with regard to that of gravity. ”

“ We have therefore to investigate the motion of a body subject to an attraction varying as the distance, and to a resistance varying as the velocity.
731.] The following application, by Professor Tait [1] , of the principle of the Hodograph, enables us to investigate this kind of motion in a very simple manner by means of the equiangular spiral.
Let it be required to find the acceleration of a particle which describes a logarithmic or equiangular spiral with uniform angular velocity about the pole. ”

“ The components of linear momentum are then Mẋ, Mẏ, and the angular momentum relative to G as base is Iθ., where M is the mass and I the moment of inertia about G. If the extraneous forces be reduced to a force (X, Y) at G and a couple N, we have
Mẍ = X, Mÿ = Y, Iθ.. = N.
(9) If the extraneous forces have zero moment about G the angular velocity θ. is constant. Thus a circular disk projected under gravity in a vertical plane spins with constant angular velocity, whilst its centre describes a parabola. ”

“ If a pair of wheels is required to transmit a certain maximum horse power, the angular velocity of the shaft being ω, the pressure P which the teeth must be designed to sustain at the pitch circle is 550 H.P./ωR, where R is the radius of the pitch circle of the wheel, whose angular velocity is ω. ”

“ In the first place, if we suppose the rings to be solid, it is evident that they must rotate as a whole, the angular velocity of all parts being the same, but the linear or actual velocity being much greater at the outer edge of the ring than the inner, because of the greater circumference of the circle traveled over in rotation. ”

“ The components of the first ones are thus related to one axis for each component, those of the second ones are related to two axes for each component, thus the first has to be written with one and the latter with two indices. In mechanics, force and velocity belong to the first, torque and angular velocity to the second. E. g. the components of an angular velocity were to be written as ; only after passing to Grassmann's "supplement", i.e. the perpendicular upon the rotating plane, the ordinary notation is justified. ”

“ If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate μ about a line parallel to the resultant momentum F at an angle θ, the velocity of the vector of angular momentum, as in the case of a top, is
C1pμ sinθ − C2μ2 sin θ cos θ ”

“ Interpreted dynamically, the left-hand side of this equation represents the velocity of the vector of angular momentum about G, so that the right-hand side represents the moment of the applied force about G, in this case the reaction of the plane, which is parallel to GA, and equal to gM·GA/GL; and so the angle AGL must be less than the angle of friction, or slipping will take place. ”

“ Then if V represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V - R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V - R and the particle of time in which it is generated conjunctly. But the velocity itself is as the contemporaneous increment of the space described directly and the same particle of time inversely. ”

“ It would appear in the case of a plane possessing in itself no mass, that the motion on the application of a normal impulse borders on the indeterminate, for a tangential component, however small, would result in an indefinite velocity in an edgewise direction being superposed. ”

“ Now if we apply to the perturbation the maximum value which the observations allow, when they have been corrected for the known acceleration due to the variation of the eccentricity of the terrestrial orbit, we find the velocity of the attractive force to amount to fifty millions of times the velocity of light. ”

“ On the other hand, a body thrown upward has uniformly retarded motion, that is, its acceleration is downward. The velocity acquired by a falling body in unit time is called its acceleration, or the acceleration due to gravity, and is equal to 32.16 ft. (980 cm.) per second, downward, each second of time. In one second, therefore, a falling body gains a velocity of 32.16 ft. ”

“ As the inter-vortical molecules acquire different magnitudes before [216] being finally driven toward the center of any system, they will also have differing rates of descent, "their velocities being inversely as their masses, so that if one should become a thousand times larger and slower than the impelling particle of light, it would still have a velocity equal to the diameter of the earth in one minute;" and the average speed would be much greater than this, though always much less than the velocity of light. ”

“ In reality, however, such increase of momentum is only apparent; the momentum of the body and fluid system combined is actually less than that of the body at the same velocity in vacuo by the amount due to its fluid displacement. That is to say, if the body be of the same specific gravity as the fluid the total dynamic system possesses no momentum at all, whatever the velocity. ”

“ The slowness with which powders subside, depends partly on the specific gravity of the substance, and partly on the magnitude of the particles themselves. Bodies, in falling through a resisting medium, after a certain time acquire a uniform velocity, which is called their terminal velocity, with which they continue to descend: when the particles are very small, and the medium dense, as water, this terminal velocity is soon arrived at. ”

“ Now we want to investigate whether this definition is not in sharp contrast to the ordinary illustration of velocities. The distances proportional to the relevant velocities are used in ordinary mechanics as representing the velocities of uniform motions. Formula (2) leads to the same result at the limits of our ordinary experience. Only at velocities nearly comparable to the velocity of light, a notable difference occurs that quickly leads to infinite distortion. ”

“ Admitting the second postulate of the theory of relativity, or if, in any way, the velocity of propagation of light by earthly sources seems to us unchangeable, supposing such an apparatus were realized, a certain number f of fringes would be seen to cross the wire of the eyepiece, when the source passes from rest to velocity v,
where l is the difference of path of the interfering rays, λ the length of the wave, and c the velocity of light. ”

“ But the increment δq of any variable cannot be greater than , where δt is the time during which the increment takes place, and is the greatest value of the velocity during that time. In the case of a system moving from rest under the action of forces always in the same direction, this is evidently the final velocity. ”

“ So for velocities, just as for intervals of time, there is need of a common standard which shall be understood and accepted by everyone, and which shall be the same for all. As has already been stated, the Author considers the velocity of a freely falling body adapted to this purpose, since this velocity increases according to the same law in all parts of the world; thus for instance the speed acquired by a leaden ball of a pound weight starting from rest and falling vertically through the height of, say, a spear’s length is the same in all places ”

“ But now let there be a different rotational velocity imparted to the jointed square round the axis of the two projecting hooked rods, such as to give a resultant moment of momentum round any given line through the center of inertia of the system; and let pairs of the hooked rods in the model thus altered, which is no longer a model of a mere spring balance, be applied as connections between millions of pairs of particles as before, with the lines of resultant moment of momentum all similarly directed. ”

“ But in reference to an observer [Pg 135] in a balloon, or who passes in another train, the velocity has not the same value, and it may cease to be a uniform velocity. We know only relative movements, or, to be quite accurate, movements relative to some material object or other. According to our choice of this object, this standard of comparison, the same velocity may be uniform or accelerated. In the long run, it is clear, we should have to have recourse to Newton’s hypothesis of absolute space to be able to say whether a given velocity is really uniform or accelerated. ”

“ By experiment it has been found that a body falling freely from rest in a vacuum near the earth's surface follows approximately the law , where s = space (height) in feet, t = time in seconds. Find the velocity and acceleration (a) at any instant; (b) at end of the first second; (c) at end of the fifth second.
which tells us that the acceleration of a falling body is constant; in other words, the velocity increases 32.2 ft. per sec. every second it keeps on falling. ”

“ Evidently there exists just as little an obstacle in space to smallness of size as to any given velocity of motion, and there are reasons for supposing that gravity must propagate itself with great velocity. Precisely because the normal velocity of the atoms is great, the material concerned in producing gravity can be very limited in quantity, and notwithstanding that exert a very considerable pressure. ”