704 MECHANICS 39. Indeter minate- ness of solution. the equilibrium position the laps of the string, from the ring outwards, are parallel to the respective sides of a closed equilateral polygon, taken all in the same direction. That the solution is unique will be seen at once by considering a displace ment of the ring, for the resultant of the forces obviously tends to diminish the displacement. When there are but three forces, their directions must be inclined at angles of 120 to one another (fig. 39). Thus we have immediately the solution of the celebrated geometrical pro blem, " Find the point the sum of whose distances from three given points is the least possible." 128. If, in the first problem of 124 above, the particle were -supported by three strings, instead of two, each attached to a fixed point, we should first have to assure ourselves that all three are brought into play. For, if not, the problem is reduced to the former case. The obvious condition is that, when the three strings are simultaneously tight, and the points of suspension are not in one vertical plane, the particle supported shall be situated within the triangular prism formed by vertical planes passing through each pair of points. If this condition be satisfied, the pro cess for determining the tensions of the strings is merely to construct a parallelepiped, three of whose edges lie along the strings, while the conterminous diagonal is vertical. This leads to an obvious geometrical construction ; and, when it is carried out, the lengths of the various edges are to the diagonal as the corresponding tensions to the weight of the particle. When the three points are in one vertical plane, nothing short of infinitely perfect fitting will, in general, bring all three strings simultaneously tight ; and in this case the problem, mathematically considered, is indeterminate. 1 When the strings are sufficiently exten sible, all will be brought into play ; and, with sufficient data, the problem is determinate. Kinetics of a Particle with One Degree of Freedom. 129. Here the motion is rectilinear, or at least takes place in some assigned curve. The simplest case is that of a falling stone, when the effect of the resistance of the air is set aside and the acceleration due to gravity is reckoned the same at all elevations. This has already been treated with sufficient detail as a matter of pure kinematics, 28, 29. Sliding 130. When the particle, instead of falling freely, is con- on in-, strained by a smooth inclined plane on which it slides, we see that (so long as it moves on the line of greatest slope) its weight My has components, M^sina tangential to the plane and M/^cosa perpendicular to it, a being the inclination of the plane to the horizon. The latter com ponent produces the normal pressure on the plane, and is the only contributor to it, since there is no curvature. The former produces the acceleration of the motion. Thus the acceleration is now p sina only; but, with this change, the results of 29 still hold. Hane 131. If the plane be rough, with coefficient of statical rough, friction //., it can furnish ( 120) a force of friction tending to prevent motion, of any amount up to juMgrcosa . If this be less than M^sina, motion will commence, and the force acceleratin it will be clined plane. where /u. isthe coefficient of kinetic friction (121). Thusthe 1 Of course, physically, there is no indeterminateness, even with perfectly inextensible strings. same results as in 29 still hold, but with g (since - //cosa) instead of g. As we have seen that fj. <p., accelerated motion can take place down an inclined plane in certain cases where the mass, if once at rest, would not start. 132. As a slightly more complex case, let us now take again the problem of free motion in a vertical line, but allow for the diminution of gravity as the distance from the earth increases. The weight of a particle of mass m at the earth s surface is mg. At a distance se from the centre, it is, therefore, E 2 m ^ where R is the radius of the earth, supposed spherical. This acts downwards, or in the direction opposite to that in which x increases. Hence, equating it, with its proper sign, to the rate of acceleration of momentum, we have for the equation of motion Body fallim earth from ; great dista:i Here the right hand member is a function of x only. Multiply by xdt, and integrate, and we have, leaving out the ex traneous factor m (the possibility of doing this showing that the motion is the same for all masses), If V be the speed at the earth s surface (where x = R), Also if the particle turns, to come down again, at the height h above the surface, Hence /~i ^ y "E+Ji I.B/I - -5-U R + A; .,, E + h This shows the amount of error in the approximate formula ( 28) for projectiles,
- V = ^.
If the particle be supposed to have been originally at rest, at a Meteo practically infinite distance from the earth (a case which may occur with a meteorite for instance), we have x = when x=<x>, and our formula becomes The speed with which the mass reaches the surface (where
- = R) is therefore /2</R, i.e., that which it would acquire by
falling, under constant acceleration g, through a height equal to the earth s radius. In this special case, the second integral is The second integral, in its general form, is a little complex ; but we may avoid it by means of a geometrical construction, founded on the results of the investigation of planetary motion soon to be given. 133. Let us next take a case in which the accelera- Hail- tion depends upon the speed of the moving body. A stone sufficiently simple one is furnished by a falling raindrop, ram ~ or hailstone, when the resistance of the air is taken into r lx account. For the moderate speeds with which such bodies move, the resistance varies, at least approximately, as the square of the speed. To avoid needless complexity, we neglect here the variation of gravity due to changes of vertical height. Suppose the particle to have been projected vertically upwards from the origin with the speed V, and let v be its speed at any time t, and a; its distance from the origin at that time. If we assume k to be the speed with which the particle must move so that the retardation due to the resistance may be equal to g, the v 2 retardation when the speed is v will be represented by g-^ . /C Let the axis of x be drawn vertically upwards ; then the _resist- ance acts with gravity, and the equation of motion upwards is
dv