Hence no arrangement is possible, in which a gain of kinetic energy can be obtained when the system is restored to its initial configuration. In other words, "the Perpetual Motion is impossible." 78. The potential energy (§.62) of such a system, in the configuration which it has at any instant, is the amount of work that its mutual forces perform during the passage of the system from any one chosen configuration to the configuration at the time referred to. It is generally convenient so to fix the particular configuration, chosen for the zero of reckoning of potential energy, that the potential energy in every other configuration practically considered shall be positive. To put this in an analytical form, we have merely to notice that by what has just been said, the value of is independent of the paths pursued from the initial to the final positions, and therefore that Σ (Xdx + Ydy + Zdz) is a complete differential. If, in accordance with what has just been said, this be called-dV, V is the potential energy, and Also, by the second law of motion, if m be the mass of a particle of the system whose co-ordinates are x, y, z, we that is, the sum of the kinetic and potential energies is constant. This is called the Conservation of Energy. In abstract dynamics, with which alone this treatise is concerned, there is loss of energy by friction, impact, &c. This we simply leave as loss, to be afterwards accounted for in physics. 78*. [The theory of energy cannot be completed until we are able to examine the physical influences which accompany loss of energy. We then see that in every case in which energy is lost by resistance, heat is generated; and we learn from Joule's investigations that the quantity of heat so generated is a perfectly definite equivalent for the energy lost. Also that in no natural action is there ever a development of energy which cannot be accounted for by the disappearance of an equal amount elsewhere by means of some known physical agency. Thus we conclude that, if any limited portion of the material universe could be perfectly isolated, so as to be prevented from either giving energy to, or taking energy from, matter external to it, the sum of its potential and kinetic energies would be the same at all times. But it is only when the inscrutably minute motions among small parts, possibly the ultimate molecules of matter, which constitute light, heat, and magnetism; and the intermolecular forces of chemical affinity; are taken into account, along with the palpable motions and measurable forces of which we become cognizant by direct observation, that we can recognise the universally conservative character of all natural dynamic action, and perceive the bearing of the principle of reversibility on the whole class of natural actions involving resistance, which seem to violate it. It is not consistent with the object of the present work to enter into details regarding transformations of energy. But it has been considered advisable to introduce the very brief sketch given above, not only in order that the student may be aware, from the beginning of his reading, what an intimate connection exists between Dynamics and the modern theories of Heat, Light, Electricity, &c.; but also that we may be enabled to use such terms as "potential energy," &c. instead of the unnatural "Force-functions," &c. which disfigure some of the modern analytical treatises on our subject.] CHAPTER III. RECTILINEAR MOTION. 79. THE simplest case of motion of a particle which we have to consider is that in a straight line. This may be caused by the applied force acting at every instant in the direction of motion; or the particle may be supposed to be constrained to move in a straight line by being enclosed in a straight tube of indefinitely small bore. As already mentioned, § 69, we shall in every case suppose the mass of the particle to be unity. 80. A particle moves in a straight line, under the action of any forces, whose resultant is in that line; to determine the motion. Let P be the position of the particle at any time t, ƒ the resultant acceleration along OP, O being a fixed point in the line of motion. Let OP, then the equation of motion is (by § 69) d'x In this equation ƒ may be given as a function of x, of dx dt or of t, or of any two or all three combined; but in any case the first and second integrals of the equation (if they can be dx obtained) will give and x in terms of t; that is, the position and velocity of the particle at any instant will be known. The only one of these cases which we will now consider is that in which ƒ is given as a function of x; those in which dx dx and x, being reserved for the Chapter on Motion in a Resisting Medium: while those in which f involves t explicitly possess little interest, as they cannot be procured except by special adaptations; and can even then appear only in an incomplete statement of the circumstances of the particular arrangement. The simplest supposition we can make is that ƒ is constant. 81. A particle, projected from a given point with a given velocity, is acted on by a constant force in the line of its motion; to determine the position and velocity of the particle at any time. Let A be the initial position of the particle, Pits position at any time t, v its velocity at that time, and f the constant acceleration of its velocity. Take any fixed point line of motion as origin, and let OA = a, OP = x. equation of motion is in the The C being a constant to be determined by the initial circumstances of the motion. Suppose the particle projected from A in the positive direction with velocity V, then when t=0, v=V; hence C = V, and But when t = 0, x = a; hence C': = a, and ... (3). 1 x = a + Vt += ft2 Equations (2) and (3) give the velocity and position of the particle in terms of t; and the velocity may be determined in terms of x by eliminating t between them: but the same result will be obtained more directly by multiplying (1) by and integrating. This gives the equation of energy dx dt 82. The most important case of the motion of a particle under the action of a constant acceleration in its line of motion is that of gravity. For the weights of bodies at a given latitude may be considered constant at small distances above the Earth's surface, and therefore if we denote the acceleration due to the Earth's attraction by g, and consider the particle to be projected vertically downwards, equations (2), (3), (4) of § 81 become x being measured as before from a fixed point O in the line of motion. As a particular instance suppose the particle to be dropped from rest at 0. At that instant A coincides with O, and a = 0, V=0. |