Let the initial angular velocity, h= the initial length, and v=the velocity of decrease of the axis of the cone; then the angular velocity, at the end of a time t, will be equal to (13) An indefinitely great number of indefinitely thin cylindrical shells, just fitting one another, are revolving with different angular velocities, but in the same angular direction, about their common axis; the angular velocity of each shell being proportional to a positive power of its radius. If the system of shells be suddenly united into a solid cylinder, to find the angular velocity of the cylinder about its axis. Let a be the original angular velocity of the outermost shell, and n the said power: then the required angular velocity is equal to 4ω Ferrers and Jackson; Solutions of the Cambridge (14) A series of rough concentric spherical shells fit closely one within another: rotations of given magnitude being impressed upon them about given diameters, no extraneous force acting on the system, to find the ultimate state of the motion. Let Ox, Oy, Oz, be axes of co-ordinates the directions of which are fixed in space, O being the centre of each shell. Let A be the moment of inertia of a shell about a diameter and w its initial angular velocity, the direction-cosines of its initial axis of rotation being l, m, n. Then ultimately the whole system will revolve as a solid sphere about an axis the equations to which are y = Σ (Awl) = Σ (Awm) = Σ (Awn) * Griffin; Solutions of the Examples on the Motion of (15) The line joining the centres of two equal fixed rings is perpendicular to both their planes: two small rings, the masses of which are m, m', which exert on each other at a distance r a mutual attraction mm'f(r), are placed slightly out of a position of stable equilibrium: to find the time of a small oscillation. If c be the distance between the centres of the fixed rings, the required time is equal to πολ {(m + m')ƒ(c)}** (16) A uniform rod can turn freely about one extremity: in its initial position it is horizontal, and is projected horizontally with a given angular velocity: to find the least angle it will make with the vertical during its motion. Let 2a be the length of the rod, and w the initial angular velocity; then 0, the required inclination, may be found from the equation 2aw cos 0=3g sin 0. (17) One extremity of a string is attached to a ring (supposed to have no weight) which slides along a vertical axis, and the other is attached to a particle of equal mass which moves on a horizontal plane: the particle is projected in a direction perpendicular to the plane which passes through the string and axis: to find the position of the string when it has revolved through a horizontal angle of 90°. The string will be horizontal, whatever be the initial velocity of the particle or position of the ring. (18) A uniform rod is moving on a horizontal table about one extremity, and driving before it a particle of mass equal to its own, which starts from rest from a point indefinitely near to the fixed extremity: to find the inclination of the rod to the direction of motion of the particle, when the particle has described any proposed distance along the rod. Let k be the radius of gyration of the rod about its fixed extremity, and r the space described by the particle along the rod at any time. Then the required angle is equal to (19) A screw of Archimedes is capable of turning freely round its axis, which is fixed in a vertical position; a heavy particle is placed at the top of the tube and runs down through it to determine the whole angular velocity communicated to the screw. = Let h = the height of the screw, a = the radius of the cylinder, a = the angle which an indefinitely small element of the screw makes with the vertical, w the required angular velocity; then, m, m', representing the masses of the screw and particle respectively, (20) A square formed of four similar uniform rods, jointed freely at their extremities, is laid upon a smooth horizontal table, one of its angular points being fixed: if given angular velocities in the plane of the table be communicated to the two sides terminating at the fixed point, to determine the greatest value of the angle contained between them during the subsequent motion. If w, w', be the given angular velocities and the required. angle, 5 (w - w')2 Frost; Quarterly Journal of Pure and Applied (21) Four equal particles, exercising no attraction on each other, move in an ellipse about a centre of force at the centre: at the commencement of the motion they were situated at the extremities of the major and minor axes: if at any time they become suddenly connected with each other so as to form a rigid system, to find the angular velocity of the system about the centre of the ellipse. If μ denote the absolute force, and 2a, 2b, the major and minor axes, the system will move about the centre with a constant angular velocity equal to O'Brien and Ellis; Solutions of the Senate-House (22) AB, AC, are two equal rods, capable of motion about a fixed point A: BC is a rod the length of which is at first equal to the sum of the lengths of the two former rods, and it joins loosely their extremities, so as to be close to A: in this state an angular velocity is given to the rods about a vertical axis through A: the rod BC then contracts, its centre rising vertically until ABC becomes an equilateral triangle: to find the work done in the contraction of BC. The required work done is equal to three times the work done in giving the original rotation together with the work which would be done in raising BC and one of the rods AB, AC, to the height of the triangle. CHAPTER XI. COEXISTENCE OF SMALL OSCILLATIONS. CONCEIVE that a particle or a system of particles, subject to certain fixed laws of geometrical connection or constraint, be slightly but generally deranged from a position of stable equilibrium, the invariable elements of the geometry being supposed to be free from particular relations. Then, if in the geometrical equations there be n independent variables, the motion of each member of the system may be represented by the composition of n primary oscillations of different periods, the periods of the n oscillations of any two members of the system being coexistent, while their amplitudes will generally be different. When the periods of the n elementary oscillations are commensurable, the whole system will return to its original state after an interval equal to the least common multiple of these periods; as in the case of vibrating cords and vibrating surfaces. This general property of sympathetic vibrations has been entitled the Principle of the Coexistence of small Oscillations or Vibrations. Should the original derangement of the system from its position of equilibrium, instead of being perfectly general, be effected by peculiar adaptation, we may reduce the n elementary oscillations to any smaller number we may please. If the fixed geometrical elements of the system be not, as we have supposed, free from particular relations, and if it receive a perfectly general derangement, there will as before arise in the system altogether n classes of oscillations; under these circumstances however a peculiarity occasionally presents itself, viz. that, although as we have supposed the original derangement be quite general, yet into the motion of no single member of the system will all the elementary oscillations enter; this case |