161. Rotors in a plane. Given a number of rotors in a plane, it is required to find their resultant.

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Let a, b, c, be the axes of the rotors, a, B, Y, the vectors. If there be two rotors which meet as a and B at 0, we can find their resultant by finding their resultant

vector a+B and drawing through O, a line parallel to a B. Then the resultant rotor acts along this line and its vector is a + B. If this rotor a+ B meets a third Υ at O2 we can combine these again by finding the resultant vectora+B+y and drawing through O, a line parallel to it. Then the resultant of the rotors a, B, and y lies in this line and its vector is a +B+y.

If this process can be continued until we have combined all the rotors, then we shall have found the resultant vector and the axis of the resultant rotor.

In the figure, σ is the resultant vector and d is its line of position.

The polygon which gives the resultant vector is called the vector polygon.

The rotor found by this construction must be the resultant of the given rotors, because

(1) The theorem of § 159 holds at each step,

(2) The momental areas being vectors their order of addition is immaterial.

This process cannot, however, always be employed, for it may happen that after combining a number of rotors their resultant does not meet any of the remaining rotors, or it may be that all the given rotors are parallel.

EXERCISES IX.

(1) A system of rotors should be drawn at random and their magnitudes specified, they should be combined as above step by step until either the resultant is found or the construction fails. In the latter case a different order of combination should be tried to see if the difficulty can be overcome.

(2) Find the resultant of the rotors given in figure, the

magnitudes and directions being as indicated. The points A, B, C, D forming a square of side 2".

162. Theorem. Two parallel rotors equal in magnitude and opposite in sense have no resultant.

Let a and a be the vectors of the rotors, P1, P2 position vectors giving the axes.

the

But [oa] gives the area of the parallelogram of which a and -a are opposite sides and is therefore independent of the position of the pole.

No single rotor can be found whose momental area is the same for all poles (see § 155).

163. Couples.

Definition 4. Two rotors whose axes are parallel and whose vectors are equal and opposite are called a Couple of Rotors or simply a Couple.

Definition 5. The momental area of a couple is the sum of the momental areas of its two rotors.

By § 162 it follows that the momental area of a couple is equal to the area of the parallelogram formed by the two rotors.

Corollary 1. All couples which have the same momental area are equivalent (see Definition 1).

A couple remains the same as long as its magnitude and aspect remain unaltered.

A couple therefore is a vector-quantity represented by a vector, drawn in the proper sense (see $$ 22, 23, and 112) perpendicular to the plane of the rotors, whose length represents to scale the magnitude of the couple.

The two rotors forming the couple may therefore be replaced by any other two in the same or any parallel plane, so long as they form a couple of the same momental

area.

In the figure all the couples are equivalent if

[σa]=[eß]=[ny].

Moreover the momental area will remain unaltered if any one of these be turned in the plane of the paper, or moved to another part altogether or taken to any plane parallel to the plane of the paper.

Corollary 2. Any number of couples are equivalent to a single couple (called the Resultant Couple).

For the momental area of a couple is represented by a vector perpendicular to the plane of the couple, whose length represents to scale the area, and whose sense is given by our old convention of § 24.

The sum of a number of such vectors is formed in the ordinary way (see § 126 a and b), and represents a couple in a plane perpendicular to the resultant vector, the momental area being proportional to the length of this vector and the sense given as before.

If the couples are coplanar, the vectors representing them are all parallel, i.e. are like vectors, and are added together by adding their magnitudes.

In a plane therefore couples are added as scalars.

164. We now give a general method for the combination of coplanar rotors by means of the link and vector polygons. For the sake of simplicity, only four rotors are taken, but the reasoning it will be seen holds for all cases. The construction will first be given and afterwards the proof of its validity and generality.

Let a1, a, a, a, be the lines of position of the rotors, А ̧ ÂÂ1⁄2  ̧ à ̧А the vectors of the rotors. The

vectors are placed so as to form the vector-polygon, and AA, is their resultant.

Choose a point P in the vector-polygon (called the pole of the vector-polygon) and join it to all the vertices as in figure. Next connect the lines of position of rotors by a polygon, whose sides are parallel to PA, PA1... respectively, as follows. Through some convenient point B1 in a draw BB parallel to PA, and through the same point B draw BB, parallel to PA, cutting a, in B2. Then through B2 draw BB, parallel to PA, and cutting a, in B, and so on, until the last line BB is drawn parallel to PA. The point B where the last line cuts the first is a point in the resultant rotor, AA, is the vector of this rotor, the resultant lies therefore in a line through B parallel to AA,.

The polygon just drawn is called the Link-Polygon corresponding to the already constructed vector-polygon. Proof that this construction does give the resultant (if there is one).

Notice first that the resultant is unaltered if we add two rotors which are equal and opposite, because the sum of their momental area vanishes. Notice also that we may add the momental areas in any order we please.

Add then a rotor in the line b with vector ß= PA. The resultant of this rotor with that in a, has the