Equilibrium of a Particle

In the special case of two forces, we see that must coincide with 0, if the forces are to be in equilibrium. Clearly, this can only occur when the forces are equal and opposite, with the same line of action.

Triangle of Forces.-39.

If the particle is in equilibrium un der three forces, represented by OP2, the same argu ment shows that OP3 must be equal and opposite to OR In words, the three forces must be represented by the sides of a triangle taken in order.

Lamy's Theorem.-4o.

Fig. 13 illustrates this case. We have, as just stated, Transmissibility of Force.-41. Forces imposed upon the same particle are necessarily concurrent ; i.e., their lines of action must intersect at a common point. The same is not true of forces which act upon a body of finite size : to specify any such force completely, we must state not only its line of action, mag nitude and sense, but also its point of application. It will be realized that actual bodies dis tort when forces are applied, and that definite particles, in conse quence, alter their relative posi tions.

On the other hand, the distor tion is generally small, and for many purposes it may be neglected. Statics commonly treats of bodies as rigid, and it makes use of an assumption which can be regarded as intuitive, viz., that any point in a body, lying on the line of action of a force, may be regarded indifferently as the point of application. This is the principle of transmissibility of force; it enables us, in effect, to concentrate upon forces, without particular reference to the body upon which they act.

42. Thus, if three forces combine to maintain equilibrium in a body of any size, we may assert, quite generally, that they must be concurrent. This theorem will be seen to follow directly from the condition for two forces which was stated at the end of § 38: the third force must be equal and opposite to the resultant of the other two, and therefore it must act through their point of inter section. As fig. 14 indicates, the point at which three forces inter sect will not necessarily fall within the body upon which they act.

Parallel Forces.-43,

The principle of transmissibility of force can be employed to find the resultant of two parallel forces —a case which does not fall directly within the scope of the vector law. It is evident that the effect of a given system of forces will not be altered by the addition of two equal and opposite forces having the same line of action. If then P and Q (fig. 15) are the two parallel forces whose resultant is required, we may super pose two equal and opposite forces of magnitude S, acting, as finally, with forces P and Q acting through 0 in a direction parallel to the original lines of action of P and Q; and it follows that OC is the line of action of the required resultant.

Two cases demand examination, shown in diagrams (A) and (B), respectively, of fig. 15. In the first case, P and Q have the same sense, and C lies between A and B: in the second, P and Q are opposite in sense, and C lies outside AB, on the side of the greater force P. In either case we have The magnitude of the resultant force is evidently (P+Q) in the first case and (P—Q) in the second; its sense is in both cases that of i.e., the greater of the forces P and Q.

Thus one problem is solved, except in the special case in which P and Q are equal in magnitude and opposite in sense. In this case, according to (54), the point C is an infinite distance from A and B, and the magnitude of the resultant is zero : in effect, we cannot find a single force which will replace a pair of equal and opposite parallel forces whose lines of action are not coincident. A pair of forces of this nature is said to constitute a couple.

Moments.-44.

Equation (54) admits of interpretation in accordance with the concept, moment of a force, which was in troduced in § 30. If we draw a line EF through C, perpendicular to the lines of action of P and Q, we have at once, from the figure, and in this form it may be expressed in the statement, that the moments of P and Q about C, according to the definition of § 30, are equal and opposite; i.e., the resultant moment of P and Q about C is zero.