169. WE come now to the case of the motion of a particle subject not only to given forces, but to undetermined reactions. This occurs when the particle is attached to a fixed, or moving, point by means of a rod or string, and when it is forced to move on a curve or surface. In applying to a problem of this kind the general equations of motion of a free particle, we must assume directions and intensities for the unknown reactions, treating them then as known, and it will always be found that the geometrical circumstances of the motion will furnish the requisite number of additional equations for the determination of all the unknown quantities in terms of the time, and the position of the particle. One case of this kind has been already treated of (§ 84), namely, that of a particle moving on an inclined plane under gravity. There the undetermined reaction is the pressure on the plane, which however is evidently constant, and equal to the resolved part of the particle's weight perpendicular to the plane. The laws of kinetic friction are but imperfectly known, and the few investigations which will be given of motion on a rough curve or surface are of very slight importance. 170. The simplest case is RER A particle is constrained to move on a given smooth plane AR curve, under given forces in the plane of the curve, to determine the motion. Taking rectangular axes in this plane, the forces may be resolved into two, X, Y, parallel respectively to the axes of x and y, the mass of the particle being taken as unity. In See Nolis О addition there will be R, the pressure between the curve and particle, which acts in the normal to the curve, since the curve is smooth and there is therefore no friction. Let P be the position of the particle at the time t ; and let the forces X, Y, R, act on the particle as in the figure, R being estimated positive towards the centre of curvature. Draw TP, a tangent to the constraining curve at P. Then the direction cosines of TP are These two equations, together with the equation of the given curve, are sufficient to determine the motion completely. which might at once have been obtained by resolving along suMSS the tangent. Now, it has been shewn in Chap. II. that if the forces re- 3 solved into X and Y are such as occur in nature, Xd + Ydy is the complete differential of some function - p (x, y). Integrating (3) on this hypothesis, we have supposing v to represent the velocity of the particle at the point xy. Suppose the particle to start at the time t = 0, from a point whose co-ordinates are a, b, with a velocity V. This shews that a particle, constrained to move under the forces X, Y, along any path whatever from the point a, b to the point x, y, has, on arriving at the latter point, the kinetic energy increased by a quantity entirely independent of the path pursued: another simple case of the conservation of energy. 171. To find the reaction of the constraining curve. Resolving along the normal PR, towards the centre of curvature, This might, of course, have been obtained from (2) above, by multiplying them respectively by p 172. To find the point where the particle will leave the constraining curve. For this it is evident that we have only to put R = 0, as then the motion will be free. where is the chord of curvature in the direction PF. Comparing this with the formula v2=fs (§ 82), we see that the particle will leave the curve at a point where its velocity is such as would be produced by the resultant force then acting on it, if continued constant during its fall from rest through a space equal to of the chord of curvature parallel to that resultant. (Compare § 144.) This result is, from the analytical point of view, of little importance; but it is of great interest in connection with Newton's mode of treating such questions. 173. The formulæ just given are much simplified when R we consider gravity only to be acting. Taking in this case the axis of y vertically upwards, our forces become X=0 and Y=-g; and the velocity, and the pressure on the curve, are given by Suppose we change the origin to the point from which the particle's motion is supposed to commence; and take the axis of y vertically downwards; we shall evidently have This shews that the velocity depends merely on the distance beneath a horizontal plane through the original position of rest. Hence, whatever be the nature of the curve on which a particle slides under gravity, its motion will always be in the same direction till it rises to the same level as that to the fall from which its velocity is due. If it cannot do so, its motion will be constantly in the same direction; if it can, its velocity will become zero, and the particle will then either come permanently to rest, or return to the point from which it started. Read B 404 |