CHAPTER V

COUPLES

30. Couples Defined. In Art. 20, Case (c), it was shown that the resultant of two parallel forces in a plane was equal to the algebraic sum of the two forces. The consideration of the case when the forces are equal and opposite in direction, that is, where the resultant is zero, will

now be considered. It is easy to see that since the resultant is zero, the two forces tend to produce only a rotation of the rigid body about a gravity axis perpendicular to the plane of the forces. Such a pair of equal and opposite parallel forces is called a couple. Let it be represented as in Fig. 47 (a), the two forces being P, and d the distance between the lines of action of the forces. This distance d is called the arm of the couple; one of the forces times

the arm is called the moment of the couple. It was found in Art. 20 that the algebraic sum of the moments of the resultant and the system of parallel forces with respect to any point in their plane, is zero. In this case, since the resultant is zero, the moment of the forces of the couple with respect to any point in the plane is equal to the sum of the moments of the two forces with respect to that point. Let the point be C, Fig. 47 (a), distant x from the force P; then - Px-P(-d-x) represents the sum of the moments of the two forces with respect to the point C (calling distance below negative). This sum is equal to Pd, the moment of the couple. The student should take C in different positions and show that the moment of the two forces with respect to any point in the plane is always Pd. Since the moment consists of force times distance, it is measured in terms of the units of force and distance; that is, foot-pounds or inch-pounds, usually. If . the couple tends to produce rotation in the clockwise direction, the moment is said to be negative; and if counterclockwise, positive.

31. Representation of Couples. - -The couple involves magnitude (moment) and direction (rotation), and may, therefore, be represented by an arrow, the length of the line being proportional to the moment of the couple, and the arrow indicating the direction of rotation. In order to make the matter of direction of rotation clear, the agreement is made that the arrow be drawn perpendicular to the plane of the couple on that side from which the rotation appears counter-clockwise. This means that if we look along the arrow pointing toward us, the rotation

appears counter-clockwise. Thus, the couple of Fig. 47 (a), whose moment is Pd, may be represented by the arrow in Fig. 48 (a), where the length of line AB is proportional to Pd and the couple is in a plane through B and perpendicular to AB. The line AB is sometimes called the axis of the couple; it may be drawn perpendicular

in Fig. 47 whose moment is P1d, is represented completely by the arrow (b), Fig. 48, the length CD being proportional to Pidi.

NOTE. The arrows are placed slightly away from the ends, so that the moment arrows may not be confused with force arrows. These arrows, like force arrows, may be added algebraically when parallel, resolved into components and compounded into resultants; the principle of transmissibility holds and also the triangle and polygon laws as seen for force arrows. Several important conclusions follow easily as a result of this arrow representation.

Since a moment arrow represents both force and distance and direction of rotation, it is evident that it cannot be balanced by a single force arrow even though they have the same line of action and are opposite in direction. Hence, we conclude that a single force cannot balance a couple.

32. Couples in One Plane. If the couples are all in the same plane, their moment arrows are all parallel, and may be added algebraically, so that the resultant couple lies in the same plane and its moment is the algebraic sum of the moments of the individual couples.

a

For example, in Fig. 49, the couples Pidi, Pad2, Pgdg, P4d4, Pd, Podo, are all in the plane (ab); their resultant couple must also be in this plane, and its moment must be equal to the algebraic sum of the moments of these couples.

It is evident since the above is true that a couple may be transferred to any part of its plane without changing its effect upon the rigid body upon which it acts. This means, when applied to some particular rigid body, as a closed book, that the effect of

d3

di

P

P

d4

de

FIG. 49

a couple acting in the plane of one of the covers of the book (book remains closed) tends to produce rotation about an axis, perpendicular to the cover through the center of gravity of the book; and that this rotation is the same no matter where the couple acts, provided it remains always in the same plane. The moment arrow of the resultant couple will be perpendicular to the cover of the book and on the side from which the rotation appears counter-clockwise. The student should endeavor to see the application of the above theorem and to see that it agrees with his observations.

33. Couples in Parallel Planes. The moment arrow represents a couple in magnitude and direction of rotation and shows that the plane of the couple is perpendicular to its line. This moment arrow represents any couple of given moment and direction in any plane perpendicular to its line. It is evident, then, that a couple may be transferred to any parallel plane without changing its effect upon the rigid body upon which it acts. Applied to the case of the book in the preceding article, it may be said that the effect of the couple would be unchanged if it acted in the plane of the other cover or in any of the leaves.

34. Couples in Intersecting Planes. - Suppose all the couples in a plane (1, 2) be added and let AB (a) (Fig. 48) represent the moment arrow of the resultant couple, and let the sum of all the couples in the plane (2, 3) intersecting (1, 2) be represented by CD (b) (Fig. 50).

Move A and C to some point on the line of intersection of the two planes. The resultant moment arrow is now found by the parallelogram law. The resultant couple has a moment represented by AE and acts in a plane perpendicular to AE and making an angle a with the plane (2, 3).