(11) Two equal weights P, P, are tied to the ends of a fine string which passes over two pullies without mass in a horizontal line: supposing a weight W, less than 2P, to be fixed to the middle point of the horizontal portion of the string, to determine how far it will descend. If a the distance between the two pullies, W will fall through a space equal to (12) A solid cylinder is freely moveable about its axis, which is fixed horizontally, and weights W, W', are hung at the ends of a string wound round it and attached to it at some point so as to prevent slipping: after W' has descended from rest for t seconds, it is suddenly cut off, and the system comes to rest in t seconds more to find the weight of the cylinder. The weight of the cylinder is equal to (13) A thin uniform smooth tube is balancing horizontally about its middle point, which is fixed: a uniform rod, such as just to fit the bore of the tube, is placed end to end in a line with the tube, and then shot into it with such a horizontal velocity that its middle point shall only just reach that of the tube: supposing the velocity of projection to be known, to find the angular velocity of the tube and rod at the moment of the coincidence of their middle points. If v be the velocity of the rod's projection, m the mass of the rod, m' that of the tube, 2a, 2a', their respective lengths, and w the required angular velocity; then (14) A circular wire ring, carrying a small bead, lies on a smooth horizontal table: one end of an elastic thread, the natural length of which is less than the diameter of the ring, is attached to the bead and the other to a point in the wire: the bead is placed initially so that the thread coincides very nearly with a diameter of the ring: to find the vis viva of the system when the string has contracted to its natural length. If c be the diameter of the ring, a the natural length of the thread, and μ the modulus of elasticity, the required vis viva is equal to SECT. 2. Vis Viva and the Conservation of the Motion of the Centre of Gravity. The Principle of the Conservation of the Motion of the Centre of Gravity, under its most general form, asserts that, the motion of the centre of gravity of a free system of bodies, disposed relatively to each other in any conceivable manner, is the same as if the bodies were all united at their centre of gravity, and each of them were animated by the same accelerating forces as in its actual state. The discovery of the Principle is due to Newton', by whom it received a demonstration in the particular case where the system is subject to no external force, when the centre of gravity will either remain at rest or move in a straight line with a uniform velocity. D'Alembert afterwards extended the Principle to the case where each body is supposed to be solicited by a constant accelerating force, acting in parallel lines, or directed towards a fixed point and varying as the distance. Finally, Lagrange expressed the Principle under its most general form for every law of force to which the bodies can be subject. 3 2 (1) A smooth groove KAL (fig. 226) is carved in a vertical plane in the body KBCL, which is placed upon a smooth horizontal plane, along which it is able to slide freely to find the form of the groove in order that a particle, placed within it, may oscillate in it tautochronously, the time of an oscillation being given. 1 Principia; Axiomata sive Leges Motus, Cor. 4. 3 Mécanique Analytique, Tom. 1. p. 257, &c. Let P be the place of the particle in the groove at any time; draw PN vertically to meet the horizontal plane at N, which will lie in the line OE formed by the intersection of a vertical plane through the groove with the horizontal plane. Let A be the lowest point of the groove; draw AM horizontally, AA vertically. Let O be a fixed point in OE; let OA' = x', ON = x1, PN=y1, AM=x, PM=y; let k,, k, be the initial values of y1, y; let m=the mass of the particle, m' the mass of the body. = Then, by the Principle of the Conservation of the Motion of the Centre of Gravity, since no forces act upon the particle and body parallel to OE, Also, by the Principle of the Conservation of Vis Viva, (1). But, from the geometry, it is evident that and therefore, if 7 denote the time of a semi-oscillation, T = 2g (k − y) ; This value of 7 must be independent of k in order that the particle may oscillate tautochronously, and therefore we must have, it being necessary that the coefficient of dy be of -1 dimensions in y and k, and therefore from (8) we get, for the equation to the groove, Clairaut; Mémoires de l'Académie des Sciences de Paris, 1742, p. 41. Euler; Opuscula, de motu corporum tubis mobilibus inclusorum, p. 48. (2) In a smooth circular tube are placed two equal particles, which are connected together by an elastic string, the natural length of which is two-thirds of the length of the circumference: the string is stretched until the particles are in contact and the tube is placed upon a smooth horizontal table and left to itself: to determine the ratio of the kinetic energy of the two particles to the work done in stretching the string, when the string resumes its natural length. Let m be the mass of either particle, m' that of the tube. Let Ox be the line of motion of the centre of the tube, O being a fixed point: let C be the position of the centre of the tube at the end of any time t, P being the corresponding position of either particle. Let OC=x, PCx=0, CP=a. Then, by the Principle of the Conservation of the Motion of the Centre Again, by the Principle of the Conservation of Vis Viva, The kinetic energy of the two particles, that is, half their vis viva, is equal to Hence the ratio of the kinetic energy of the two particles to the work done in stretching the string, viz. 2a [Tde, is d 2m d 2 dt a sin e + m' dt dť Ө {(x+a cos 0) + 2m which, by (1), is equal to 2m. (am' sin 6)2 + 2m. {a cos 0 (2m +m')}3 2m (am' sin 0)2 + 2m . {a cos 0 (2m+m')} + m' . (2ma sin )1' 2π and therefore, since = is equal to 2 (m2 + mm' +m") (2m + m') (2m' + m) (3) A rigid quiescent wire, in the form of a semicircle, is suspended from its ends by little rings, moveable along a hori |