COrPLES.] 



MECHANICAL PHILOSOPHY. STATICS. 



703 



from what it would when the force Q was positive, or 

 when both forces acted in the same direction. 



In the last case, that is, when P and Q act in oppo- 



I 



Fig. 44. 



N 



4 x 



P R\ 



site directions, if P and Q 

 are equal, we cannot repre- 

 sent their resultant by any 

 single force. 



For if we pursue the 

 mode of construction 

 adopted before, we shall 

 find the line BT (Fig. 44), \ 



in this case parallel to the 

 line A R ; and consequently 

 these lines, if produced, will never meet 



This admits of an easy proof, because B Q is parallel 

 and equal to A P, and K S and A S are equal to each 

 other, and are in the game line. 



Therefore the parallelogram APRS is similar and 

 equal to the parallelogram B Q T S, and the angle PAR 

 = the angle TBQ. 



Also because A P is parallel to B Q, the angle P A B = 

 angle QBA. 



And by addition, angle P A R + angle P A B = angle 

 TBQ + angle QBA. 



Or the angle R A B = the angle TEA. 



And, by Euc., B. I., P. XX VII., R A must be parallel 

 toTB. 



Our method in this case fails to discover a point D 

 anywhere in -the line AB produced, to which a single 

 ant can be applied. 



This is indicated by the formul which we have 

 already obtained of P Q for the magnitude of the 



O A Tl 

 resultant, and ,' ,=. for A D, the distance of its point 



of application from A, since when Q = P, they give us 

 P p or for the magnitude of the resultant, and 



P.AB P AB A , 



., ., or the algebraical sign of infinity for 



AD; a result which showi that our problem in this 

 case is impossible. 



t'ul'PLE. When two equal and opposite parallel 

 forces act at different points of a rigid body in the same 

 plane, their effect, as we have seen, cannot be counter- 

 acted by any single force applied Fig. 48. 



to the body, and their tendency r 

 will evidently be to twist the body 

 in the direction of the plane in 

 which they act. Thus P (Fig. 45) 

 acting on A in the direction A P, and P acting on B in 

 the direction BP will twist the body round in the 

 direction B A P or A B P. 



The term couple is applied to such a system of forces. 



ARM OF A COUPLE. The perpendicular distance 



Fig. 46. 



between the directions in which 



the forces producing a couple act, 



is called the arm of that couple. 



Thus if the line AB (Fig. 46; is 



perpendicular to AP and B P, 



which represent the directions in 



whicli the forces producing a couple act, A B is the arm 



of that couple. 



MOMENT OF A COUPLE. The product of the 

 arm of a couple and one of the forces producing it, is 

 called the moment of that couple. 



Thus if P be the force acting in the direction A P or 

 B P, then P X A B is the moment of the couple pro- 

 duced by the couple whose arm is A B. 



This moment is a measure of the tendency of the 

 couple to twist the body on which it acts, and it is 

 customary to indicate a couple by its moment. 



EQUILIBRIUM OF A COUPLE. Though no 

 single force can be found which can counteract the 

 effect of a couple on a rigid body, yet a couple may be 

 found to neutralise the influence of another. Thus 

 if a force Q (Fig. 47 1), equal and opposite to the force 

 1' represented by A Q be applied to A in the direction 

 AQ of the line PA produced, and a similar and equal 

 force at B in the line B P produced to O^ then the force 



Q at A being equal and opposite to that of P 

 but in the same straight line, will neutralise it. 



Fig. 47-(2). 



at A, 



Fig. 47-{l). 



1 



Fig. 48. 



t* 



Similarly, the forces P and Q acting at B will destroy 

 each other, and the body will be in a state of equili- 

 brium under the influence of two couples whose mo- 

 ments are P X A B and Q X A B, but which tend to 

 twist the body in opposite directions. (Fig. 47 2). 



A couple whose tendency is to twist the body in the 

 direction in which the hands of a watch move is called a 

 positive couple, such as Q x A B in the accompanying 

 diagram (Fig. 47 2), while the couple which would cause 

 the body to move in the opposite direction, such as 

 P X AB, is called a negative couple. (Fig. 48). 



It is also convenient to 

 designate a couple by its 



moments ; thus, when we .S c tuc P * % 



speak of the couple P X f 



AB, we mean the couple I * ! 



whose moment is P X A B, / 

 ^ B representing its arm, 1 pj, 

 lunlPoneof the equal forces 

 acting at its extremity. 



It is more convenient generally to represent the pro- 

 duct of P and A B, by the symbol P A B instead of 

 1'XAB. 



AXIS OF A COUPLE. The aj-is of a couple is a 

 straight line, which is supposed to be drawn perpendicular 

 to its plane, and proportional in length to its moment. 



Thus, if the arm of a couple be 4 inches in length, 

 and the forces acting at its extremities be both 5 pounds, 

 and the arm of another couple be inches, and the forces 

 at its extremities be both 8 pounds, the moments of 

 these couples will be represented by the numbers 20 and 

 48 ; and a line 20 inches in length, perpendicular to the 

 plane of the first, and another of 48 inches, perpendicular 

 to that of the second, will represent their respective axes. 



PROPOSITION XII. 



The arm of a couple maybe turned round any point in that 

 arm, in the plane of the couple, without altering the con- 

 ditions of equilibrium. 



Let A B represent the arm of the couple, P t and P 2 , 

 the forces acting at A and B. (Fig. 49). 



In A B take any point C, turn A B round C into the 

 new position A' C B'. 



Now at the points A' and B' we may apply equal and 

 opposite forces, P 3 , P 4 , 

 P 6 , and P 6 , perpendi- 

 cular to the line A'B', 

 and each equal to the 

 force P, or P 2 , with- 

 out disturbing the con- 

 ditions of equilibrium 

 of the body on which 

 the couple P'A B is 

 supposed to act. 



Produce the line 

 A P, to meet A'P 3 in 

 the point D, and P.,B 

 to meet B'P 5 in E. 

 Join C D and C E. 



Because in the triangles A' D C and ADC, the angles 

 at A and A' are right angles, the side D C common, and 

 A'C = AC. 



Therefore A' D= A D, and the angjes at D and C are 

 bisected by C D. 



Similarly, it may be shown that the angles at C and 

 E are bisected by C E. 



Fig. 49. 



