43. PROP. The effect of a couple is not altered if we transfer the couple to any plane parallel to its own, the arm remaining parallel to itself. Let AB be the arm, A'B' its new position parallel to AB. P 1 or P, and parallel PÅ PÅ P ̧; this will Join AB', A'B bisecting each other in G. 1 49 But P, and P are equivalent to 2P, acting at G in direction Ga, and P, and P, are equivalent to 2P, acting at G in direction Gb. 2 3 Hence P1, P, P, P are in equilibrium with each other; therefore the remaining forces P and P, acting at A' and B', produce the same effect as P, and P, acting at A and B. Hence the proposition is true. 1 2 44. PROP. The effect of a couple will not be altered if we replace it by another of which the moment is the same; the plane remaining the same, and the arms being in the same line, and having a common extremity. P=Q+R and parallel to P; this will not alter the effect of the couple. Now Rat A and Q at C will balance Q+ R at B, if AB: BC :: Q: R, Art. (36), or if AB: AC :: Q: Q + R, that is, if Q.b = P.a; we have then remaining the couple Q, Q acting on the arm AC. Hence the couple P, P acting on AB, may be replaced by the couple Q, Q acting on AC, if Q.b = P.a, that is, if their moments are the same. 45. From the last three articles it appears that, without altering the effect of a couple, we may change it into another of equal moment, and transfer it to any position, either in its own plane or in a plane parallel to its own. The couple must remain unchanged so far as concerns the direction of the rotation which its forces would tend to give the arm, supposing its middle point fixed as in Art. (41). In other words, the line which we have called the axis, measured as indicated in that article, must always remain on the same side of the plane of the couple. 46. We may infer from Art. (44) that the effects of couples are proportional to their moments. Let there be two couples, one in which each force = P, and one in which each force Q, the arms of the couples being equal; these couples will be in the ratio of P to Q. For suppose, for example, that P is to Qas 3 to 5; then each of the forces P may be divided into 3 equal forces and Q into 5 such equal forces. Then the couple of which each force is P may be considered as the sum of 3 equal couples of the same kind, and the couple of which each force is Q as the sum of 5 such equal couples. The intensities of the couples will therefore be as 3 to 5. Next, suppose the arms of the couples unequal, and denote them by p and q respectively. The couple which has each of its forces = Q and its arm = q is equivalent to a couple having each of its forces and its arm = = P, by Art. (44). The intensities of the couples are therefore by the first case in the ratio of P to Qq that is of Pp to Qq. Since two couples have the same ratio as their moments, it follows that the moment of a couple is the measure of its intensity. For if we take as our unit couple that which has each of its forces equal to the unit of force and its arm equal to the unit of length, the couple which has P for each of its forces and p for its arm will contain the unit of couple as many times as P.p contains 1.1. 47. With respect to the effect of a couple, we may observe that it is shewn in works on rigid dynamics that if a couple act on a free rigid body it will set the body in rotation about an axis, passing through a certain point in the body called its centre of gravity, but not necessarily perpendicular to the plane of the couple. 48. PROP. To find the resultant of any number of couples acting upon a body, the planes of the couples being parallel to each other. First, suppose all the couples transferred to the same plane Art. (43); next, let them be all transferred so as to have their arms in the same straight line, and one extremity common, Art. (42); and lastly let them be replaced by others having the same arm, Art. (44). we shall have them replaced by the following forces (supposing a the common arm), Hence their resultant will be a couple of which the force equals Hence, the moment of the resultant couple is equal to the sum of the moments of the original couples. If one of the couples, as Q, Q, act in a direction opposite to the couple P, P, then the force at each extremity of the arm of the resultant couple will be or the algebraical sum of the moments of the original couples; the moments of those couples which tend in the direction opposite to the couple P, P being reckoned negative. 49. PROP. To find the resultant of two couples not acting in the same plane. Let the planes of the couples intersect in the line AB, R R B which is perpendicular to the plane of the paper, and let the couples be referred to the common arm AB, and let their forces thus altered be P and Q. In the plane of the paper draw Aa, Ab perpendicular to the planes of the couples P, P and Q, Q; and equal in length to their axes. Let R be the resultant of the forces P and Q at A, acting in the direction AR; and of P and Q at B, in the direction BR. Since AP, AQ are parallel to BP, BQ respectively, therefore AR is parallel to BR. Hence the two couples are equivalent to the single couple R, R acting on the arm AB. Draw Ac perpendicular to the plane of R, R, and in the same proportion to Aa, Ab that the moment of the couple R, R is to those of P, P and Q, Q respectively. Then Ac is the axis of R, R. Now the three lines Aa, Ac, Ab make the same angles with each other that AP, AR, AQ make with each other; also they are in the same proportion in which or in which AB.P, AB.R, AB.Q are, |