COMPOSITION Of FORCES.] MECHANICAL PHILOSOPHY. STATICS. 



693 



PRESSURE AND TENSION. When a material 

 body is in a state of equilibrium under the influence 

 of forces, the forces applied to the body may either 

 have a tendency to press the particles of the body or to 

 crush it, in which case the forces are called pressures, 

 or they may have a tendency to separate the particles of 

 the body or tear it, in which case they are denominated 

 tensions. A weight placed on a body exerts a pressure 

 on it. Two men pulling a rope, or weights suspended 

 from a rope, exert a tension on the rope throughout its 

 substance ; and if one part of the rope be weaker than 

 another, and the weight or force be sufficiently great, 

 the rope will break or be torn asunder at that part. 

 The tie-beams of a roof, which prevent the weight of 

 the roof from thrusting the walls of a building out of 

 the perpendicular, are under tension, while the walls 

 support the pressure of the roof. It is of great im- 

 portance, practically, to distinguish between pressure 

 and tension ; for some substances will bear a large 

 amount of pressure without injury, but will be torn 

 asunder by a far less amount of tension. 



FLEXIBLE CORDS. In theoretical statics, as we 

 conceive our solid bodies to be perfectly rigid, so we 

 conceive the strings which support our weights to be 

 perfectly flexible, and at the same time perfectly in- 

 extensible. It is needless to remark that no such cords 

 or strings are to be found in nature ; but these hypo- 

 thetical bodies enable us to divest our problems of many 

 difficulties, and to arrive at conclusions which may 

 afterwards be used in practice with great accuracy, 

 when experiment has enabled us to determine the want 

 of flexibility in the material we use. For the same 

 reason, friction, or the resistance which surfaces not 

 perfectly smooth oppose to the motion of a body over 

 them, is at first neglected in our problems. 



ACTION AND REACTION. It is an axiom of 

 statics tliat is, it is a self-evident truth, or one which 

 admits of no other proof than universal experience 

 that whatever force one rigid body exerts upon another 

 rigid body, the latter opposes that force by an equal 

 force, which is called its reaction. Thus if * beam of 

 wood, B (Fig. 7) standing r >g . /, 



upright on a floor, C, 

 support on one of its 

 extremities a 501b. weight, 

 A, the weight A will 

 exert a pressure of 50 Ibs , 

 acting downwards, on the 

 beam, and the beam will 

 convey this pressure to 

 the floor. But the rigi- 

 dity of the beam opposes a force to the weight, which 

 prevents the weight from falling, or crushing the beam ; 

 and thus a reaction equal to 60 Ibs. is exerted upwards 

 by the beam upon the weight. Again, the beam 

 presses on the floor with its own weight, in addition to 

 that of the 50 Ib. weight ; and if the floor be strong 

 enough, and of a material sufficiently rigid to prevent B 

 sinking into it, the floor will sustain the pressure of A 

 and B, which acts downwards ; but it will react up- 

 wards on B with a force equal to the pressure B 

 exerts upon it. 



EQUILIBRIUM OF A MATERIAL PARTICLE. 

 If a material particle be acted upon by two forces which 

 are equal to one another, but acting in opposite direc- 

 tions in the same straight line, they will neutralise one 

 another, and the particle will be at rest. This is self- 

 evident, and depends upon our fundamental idea of the 

 equality of forces. Supposing, however, that the two 

 forces do not act on the particle in the same straight 

 line, but in the direction of straight lines inclined to 

 one another at some angle, in what direction must a 

 force be applied to the particle, and of what magnitude 

 must it be, to neutralise the effect of these two forces, 

 both in the case where the two forces are equal, and 

 also where they are unequal 1 This is one of the most 

 important, and, indeed, the fundamental proposition of 

 statics. Before we can discover it we must adopt some 

 means of representing forces and their directions. 



GEOMETRICAL REPRESENTATION OF 

 Fig. s. FORCES. If we represent the 



material point by a geometrical 

 point A (Fig. 8), we may draw 

 a line, A P, to represent the 

 direction in which a force, say 

 of P pounds, is acting on the 

 particle, , and A Q to represent 

 the direction in which a force of 

 Q pounds is acting on A ; then if we take AP,P inches 

 in length, and A Q,Q, inches in Fig 9 



length, the lines AP and AQ will 

 represent the forces P and Q acting *, 



on A, both in magnitude and direc- 

 tion. Thus if two forces, one of 

 4 Ibs. and another of 3 Ibs., are sup- / 



posed to act on a particle at right - . - - 

 angles, or perpendicular to one 



another, we should represent them by the lines A B, 3 

 inches in length, and AC, 4 inches in length, drawn 

 perpendicularly to one another. (Fig. 8). 



It is not necessary to use an inch as the representative 

 of a pound, or the unit of weight. Any other convenient 

 measure, such as the eighth or tenth of an inch, may be 

 used, if we keep to the same scale throughout. It is usual 

 to indicate the direction in which the force acts by the 

 addition of an arrow-head to the line which represents it 

 RESULTANT. If two forces, P and Q (Fig. 10), 

 Fi(?. 10. represented in magnitude and 



direction by the lines A P and 

 A Q, act upon the point A, so 

 as to cause it to move, it must 

 begin to move in some direc- 

 tion. Let A R represent that 

 direction. Now we can con- 

 ceive that some force, S, could 

 be applied to A in the opposite 

 direction to A R, in which A 

 would begin to move, which 

 would stop its motion and 

 keep it at rest. Let A S represent this force in magni- 

 tude and direction. Then the particle A would be kept 

 at rest by the three forces, P, Q, and S, acting in the 

 directions A P, A Q, and A S. If, now, the line S A be 

 produced to R, and A R be taken equal to A S, A R will 

 represent in magnitude and direction a force, R, equal to 

 S, which, if P and Q were removed, would counteract 

 the effect of the force represented by A S on A. A R, 

 therefore, represents, in magnitude and direction, a force 

 which would have the same effect on A, when it acted 

 upon it by itself, that the forces represented by A P and 

 A Q would have, if they both acted on A. 



The force R, represented by A R, is called the resultant 

 of the forces P and Q, represented by A P and A Q ; and 

 P and Q are called the components of R. 



Thus, for example, if three perfectly smooth pegs, B, C, 

 and D (Fig. 11), be inserted in a board standing in a ver- 

 tical position, and three strings be attached to a point, 



Fig. 11. 



or simply knotted together at 

 A, and weights of 21b., 31b., 

 and 4 Ib. fixed to their ex- 

 tremities, as in the figure, the 

 weights will balance each other; 

 and, neglecting the friction on 

 | the pegs, B, C, and D, and sup- 

 posing the strings perfectly 

 flexible, the whole will come to 

 rest in the position represented 

 by the diagram. The point or 

 knot A will be kept at rest by 

 the tensions of the weights 

 that is, by a force of 3 Ibs. 

 acting in the direction of A I!, 

 2 Ibs. in the direction of A D, 

 and 4 Ibs. in that of A C. Let 

 another peg, E, be inserted in 

 the board, somewhere in the 

 straight line C A produced, attach another string to A, 

 and pass it over A E, with a weight of 4 Ibs. at its other 



