CHAPTER III.

ON FORCES WHICH ACT IN ONE PLANE BUT NOT UPON THE SAME POINT OF A RIGID BODY.

39.

THE THEORY OF COUPLES.

REMARK. It has been stated in Art. 21, that the effect of a force is not altered by supposing it to be transferred from one point of the body in the line of the direction of its action to another: from this it follows that if the directions of the forces which act at different points of a rigid body, all pass through a point, we may fictitiously transfer them to that point, and then by the preceding Chapters find their resultant, which in its turn we may transfer to any convenient point of the rigid body which happens to lie in the line of its direction. It is obvious, that when any two forces in the same plane act upon a rigid body at different points, their directions unless parallel being produced will meet, and therefore after the statement just made it will not be necessary to include the consideration of two non-parallel forces in the present Chapter, we shall therefore begin with the following.

40. Two forces act in parallel directions upon different points of a rigid body, to find their resultant.

CASE 1. Let F, F be the two forces, and let us, first, suppose them to act in the same direction.

Let A, B (fig. 8) be any two points of the rigid body in the lines of direction of the respective forces: join A, B ; at these points in opposite directions along the line AB apply any two equal forces f, f. These being in equilibrium produce no effect.

Now F and f (by Art. 26) and F" and f' will have resultants (m, n suppose) acting in certain directions Am, Bn within the angles FAƒ, and F'Bƒ': these lines being produced will meet in some point P to which let m, n be transferred: and let them there be resolved into their original components; viz., m into f and F, acting at P in the directions Pf and PR; (PR being drawn parallel to AF); and n into ƒ and Facting at P in the directions Pƒ' and PR, which is also parallel to BF. The forces f and f' at P being in equilibrium may be removed, and there remain the original forces F, F' both acting at P along the line PR parallel to their direction at A and B. Hence the resultant of F and F is a force, equal to their sum F + F', acting at any point in the line PR; the position of which we find as follows.

Let PR cut AB in Q. Then because m is the resultant of F and ƒ, a force equal to m applied at A in the direction AP would keep the two forces F, f in equilibrium; and the three being parallel to the sides of the triangle APQ taken in order, are proportional to those sides (Art. 27), .. Ff PQ AQ

Similarly

f' F':: BQ: PQ

:

.. F : F':: BQ : _AQ; •:•ƒ= ƒ'.

Consequently Q divides AB into two parts which are inversely proportional to the forces adjacent to which they lie.

41. CASE 2. Let us now suppose the two forces F, F to act in contrary directions, as in fig. 9, and that they are unequal, F being the greater.

The forces f and f' being introduced as before, we remark that since F is greater than F" the directions of the forces m and f will make a greater angle with each other, than the angle which is contained between the directions of n and f; that is, the fAm is greater than f'Bn, or ABP. Consequently the lines nB, Am being produced will meet on the side towards the greater force F, as is represented in the figure. From this point proceeding as in the former case we find that the forces F, F", preserving their

Hence

proper directions, may be removed to the point P. their resultant R is equal to F-F', the algebraic sum of the forces, and acts in the direction of the greater force. The word sum is used in the statement of this result, because F being assumed positive, F' acting in the contrary direction must be accounted a negative force. (See Art. 23).

The position of the point Q is found as before from the proportion

FF BQ AQ,

and it is to be noticed particularly that Q lies in BA produced; and is situated nearer to A, (the point of application of the greater force), than to B.

42.

CASE 3. Let us lastly suppose the two forces F, F acting in contrary directions, to be equal.

In this case the angles fAm, ABP are equal; and consequently the lines Am, nB are parallel, and there is no point of concourse. It would appear then, that the former mode of finding the resultant of F and F fails entirely in this case. The present case may, however, be considered the ultimate state of Case 2, at which we arrive by supposing the magnitude of F to approach continually nearer to that of F, until at length their difference becomes less than any assignable quantity. Let us then reconsider Case 2. We have found

R=FF',

Hence, we see that as F" increases, the point Q moves continually farther from B, and BQ becomes infinite in the ultimate state; and at the same time the resultant R diminishes in such a manner that the product R.BQ never changes, and becomes zero in the limit. Hence, in the ultimate state, that is, when F" differs from F by less than

any assignable quantity, we have a resultant zero acting at a point infinitely distant from A or B; yet even then the product R. BQ remains finite, which apart from any other consideration would induce us to conjecture, that some finite effect is due to the action of F and F in this case, although not such an one as can be represented by a single force.

Upon these grounds we conclude, that a system of two equal forces acting in contrary directions on different points of a rigid body is not reducible to a single resultant.

43. DEFS. Such a system of two equal forces acting in opposite directions but not in the same straight line, is denominated A COUPLE. A plane which passes through the two lines in which the forces of a couple act, is called the plane of the couple.

When the line AB (fig. 9) is drawn at right angles to the directions of the forces of the couple, it is called the arm of the couple; and the product F. AB is then called the moment of the couple.

44. REMARK. It is obvious from an examination of fig. 9, that one effect of a pair of forces, acting in contrary directions at different points of a rigid body, whether they be equal or unequal, is the communication of a rotatory motion (see Art. 6) to the body on which they act; what other effect they would produce is not so obvious, nor indeed does it belong to us, in treating of the present subject, to consider what is the effect of unbalanced forces in any case. For the satisfaction of the reader, however, and for convenience in what follows, it may be stated, that it is proved in Dynamics, that the sole effect of a couple is to communicate an angular motion about an axis perpendicular to the plane of the couple, the axis passing through a certain point in the body, called the centre of gravity.

45. DEF. If the forces of the couple act so as to tend to turn the body round in the direction of the motion of the hands of a watch, it is called a right-handed couple, and more frequently a positive couple; but if, as in fig. 9, the forces act so as to turn the body in the contrary direction, the couple is styled left-handed, or negative.

These terms, to prevent confusion, will be used in this book as here defined; but the reader will observe that, in Statics as in Algebra, the terms positive and negative are only relative, and may be applied, at discretion, to any two forces acting in contrary directions, or to any two couples which tend to communicate opposite angular motions to the body on which they act.

46. The reflecting reader will have remarked that a couple, though positive when viewed by a spectator looking at it from one position, appears negative to a spectator looking at it from a position on the other side of its plane. A couple is therefore positive or negative, according to the situation of the spectator, with respect to its plane. It will prevent confusion, if we call that face of the couple's plane the positive face, upon which the spectator looks when the couple appears to him to be positive: the other face of the plane must then be considered negative.

47. DEFS. A straight line, in length proportional to the moment of a couple, being drawn perpendicular to the plane of the couple, is called "the axis of the couple."

And it is said to be the positive, or the negative axis, according as the perpendicular stands on the positive, or on the negative face of the couple's plane.

If the axis of a couple is mentioned without it being stated whether it is positive or negative, we are to understand that the positive axis is alluded to.

The angle between the planes of two couples is measured by the angle between their positive axes.

48. The effect of a couple is not altered by turning its arm through any angle in the plane of the couple.

Let F and F be the equal forces of a couple acting at two points in the lines FA, FB (fig. 10), and having the arm AB. From A, any point in the line in which F acts, draw AB' = AB; and at A, B' in the plane of the couple F, F, and in directions at right angles to AB′ apply two pairs of opposite forces f, g; f', g', each force being equal to F. These being in equilibrium, produce no effect.