702 MECHANICS the forces. Hence the equations of motion of a particle are changed into those of equilibrium by putting Statical 1 20. We have now all that is necessary for the friction, dynamics of a single particle, with exception of the experi mental laws of friction. These, very nearly as they were established by Coulomb, we will now give. To produce sliding of one flat-faced solid on another requires a tangential force which is directly proportional to the normal pressure between the surfaces, and whose actual magnitude is found from this pressure by means of a factor called the " coefficient of statical friction." This coefficient depends upon the nature of the solids, the roughness or smoothness of the surfaces in contact, and the amount of tallow, oil, &c., with which they have been smeared. It also depends upon the time during which they have been left in contact. It is only in extreme cases dependent on the area of the surfaces in contact. 121. When the forces applied are insufficient to pro duce sliding, the whole amount of friction is not called into play ; it is called out to an amount just sufficient to balance the other forces. Thus there are two quite distinct problems connected with the statics of friction : the first, to determine the amount of friction called into play under given circumstances ; the second, to find the limiting circumstances under which, with friction, equili brium is possible. When motion is produced, there is Kinetic still friction (now called " kinetic"). It follows the same friction. l aws as does statical friction, only that the coefficient, which is approximately independent of the velocity, is usually considerably less than the statical coefficient. Equili brium Statics of a Particle. 122. By 117, forces acting at the same point, or on the same material particle, are to be compounded by the . i same laws as velocities. Therefore the sum of their re- jJdl LiL-it . _ , solved parts m any direction must vanish if there is equi librium; whence the necessary and sufficient conditions are found by resolving in three directions at right angles to one another. They follow also directly from Newton s statement with regard to work, if we suppose the particle to have any velocity, constant in direction and magnitude (and by 6 this is the only general supposition we can make, since absolute rest has for us no meaning). For the work done in any time is the product of the displacement during that time into the algebraic sum of the effective com ponents of the applied forces, and there is no change of kinetic energy. Hence this sum must vanish for every direction. Practically, as any displacement may be resolved into three, in any three directions not coplanar, the vanishing of the work for any one such set of three suffices for the criterion. But, in general, it is convenient to assume them in directions at right angles to each other. Hence, for the equilibrium of a material particle, it is necessary, and sufficient, that the (algebraic) sums of the applied forces, resolved in any one set of three rectangular directions, should vanish. ^This statement gives at once the result that, if X 1( Y 1( Z 1; X 2> Y s> z a> &c -> be the components (parallel to the three axes) of the forces P 1( P 2 , &c., acting on the particle, we must have ) = 0, 2(Y) = 0, 2(Z) = 0. When these conditions are not satisfied, there is a resultant force P, with direction cosines A, /*, v, such that Attrac- 123. By far the most extensive series of examples of the composition of forces acting on a single particle is furnished by the theory of " attraction," where each particle of the attracting mass exerts upon the attracted particle a Fig. 34. Then /3y represents force in the direction of the line joining them, and of magnitude depending on their masses and their mutual distance only. See POTENTIAL. 124. When there are but three forces acting on the Exan , particle, their directions to give equilibrium must obviously of pa; ., be in one plane. For, if the third were not in the plane of the other two, it would have an uncompensated com- ponent perpendicular to that plane. Hence this case is always at once reducible to the triangle or the parallelogram of forces ; and the magnitudes of each of the three forces are respectively proportional to the sines of the angles between the directions of the other two. Thus, when a pellet is supported by two strings, as in fig. 34, we may proceed as follows to determine their tensions. Let P be the pellet, of weight W, and let AP, BP be the strings attached to points A and B respectively. Let their tensions be T and T . The remark above shows that the strings must hang in a vertical plane, since the force W acts in a vertical line. Since A, B, and the length of the strings are given, the figure is perfectly definite. Draw Py vertically upwardSj and make its length represent, on any assumed scale, the value of W, Draw y/2 parallel to AP, and let it meet BP in T, and P/3 represents T , in direction and also in magni tude, on the same scale in which yP represents W. This case leads to nothing but the determination of the tensions, since the form of the figure is definite. Next, let one of the tensions be given in magnitude. To effect this, we may suppose the end of PB not to be fastened at B, but to pass over a smooth pulley and support a weight Q. Let fig. 35 represent the state of equilibrium, and let the same construc tion as before be made. Then we must have yP : P/3 : : W : Q ; or, writing it in terms of angles, sinAPB:sinAPy::W:Q. A and B and the direction of yP being given, this datum suffices for the drawing of the figure; i.e., for the calculation of the angles. A little con sideration will show that, however small Q be, provided the string supporting it be long enough, there is always one definite position of equilibrium. The actual calculations in such a case as this are troublesome. It was chosen mainly on that account, so as to show, in a R simple case, how pure geometrical pro cesses may occasionally save the neces sity of a tedious trigonometrical in vestigation. But a still simpler method will be afterwards explained, viz., that, for a position of stable equilibrium, the potential energy must be a minimum. Now, to apply this to our example, we see that any downward displace ment of Q produces an upward motion of P. But when AP is nearly vertical the vertical displacement of P is indefinitely smaller than that of Q, so that Q must go down. On the other hand if APB be nearly a straight line, a displacement of P produces an indefinitely smaller displacement of Q. Hence P must go down. And these results are in character independent of the relative magni tudes of P and Q, provided both be finite. Finally, let both tensions be constant. Here we must imagine pulleys both at A and at B (fig. 36), with weights Fig. 35.
Fig. 36.