FOKCES IN COUPLES.] 



MECHANICAL PHILOSOPHY. STATICS. 



TOO 



Then without disturbing the equilibrium of the body: 

 we may introduce two equal and opposite forces P u re- 

 'uted in magnitude and direction by OP/ and 

 O PI", acting at O, and also transfer the point of appli- 

 cation of Pi from A! to a A . 



Hence, on the whole, the effect of the force P, 

 acting at A t , is equivalent to a force P, acting at O 

 in the direction O PI," parallel to AI P!, and two equal 

 and opposite forces represented in magnitude and 

 direction by OPi'". a, PI' acting at right angles to Oa, 

 and forming a couple Pj Oaj, or a couple whose mo- 

 ment is PI multiplied by the perpendicular from on 

 the line A t P t produced. 



In a similar manner the force P 2 (1) acting at A 2 may 

 be replaced by a force P 2 acting at O, parallel in direc- 

 tion to A, P a and a couple whose moment is P 2 multi- 

 plied by the perpendicular from O on A 2 P 2 produced. 



The force P 3 acting at A 3 (1), may be replaced by a 

 force P 3 acting at O parallel to A 3 P 3 , and a couple 

 whose moment is P 3 multiplied, by the perpendicular 

 from O on A 3 P 3 produced. 



And, lastly, P 4 acting at A 4 (1) may be replaced by a 

 force P 4 acting at O parallel to A 4 P 4 , and a couple 

 whose moment is P., multiplied by the perpendicular 

 from O on A 4 P 4 produced. 



On the whole, therefore, the forces P., P 2 , P s , and 

 P, acting at the points A,, A 2 , A 3 , and A,, may be 

 replaced by forces P,, P 2 , P 3 , and P 4 , all acting at O 

 in directions parallel to A 1 P,, A 2 P 2 , A 3 P 3 , and A, P,, 

 whose resultant will be a single force whose magnitude 

 and direction may be found by Prop. II. ; and four 

 couples whose arms will have a common extremity O, 

 and whose moments will be equal to P 1 , P 2 , P s , and 

 P,, each icspectively multiplied by the perpendicular 

 from O on the original direction of the force ; or on A, 

 P,, A 2 P 2 , A 3 P 4 , and A, P, produced. 



l!y Prop. XVI., these four couples will be equivalent 

 to a single couple, whose moment is equal to the alge- 

 braical sum of their moments. 



The single resultant force acting at O shows the ten- 

 dency of the four forces P Jt P : , P 3 , and P,, acting at 

 O to move O in some rectilinear direction, while the re- 

 inltant couple gives their, tendency to twist the body in 

 tome direction round the point O. 



Jn order, therefore, that the rigid body should be in 

 equilibrium, when acted on by these forces, these re- 

 niltants must each be equal to nothing, since if the re- 

 sultant force alone were equal to nothing, the resultant 

 couple would twist the body round O, as a fixed point ; 

 or if the moment of the "resultant couple alone were 

 equal to nothing, the body would move so as to keep O 

 in a straight line. 



Hence the condit'uma of equilibrium of four forces P,, 

 P z , P s , and P 4 , acting on a rigid body in the same 

 plane at the points A lt A 2 , A 3 , and A 4 in the directions 

 P, A,, P 2 A 2 , P 3 A 3 , and P 4 A 4 are two. 



Firit.fhe resultant of four forces, respectively equal 

 to P t , P 4 , P 3 , and P 4 , acting on any point O of the 

 body, in "directions respectively parallel to A, Pj, A 2 

 P 2 , A 3 P 3 , and A 4 P 4 , must be equal to zero. 



Second. The algebraical sum of the moments of the 

 four couples, whose arms are the perpendiculars drawn 

 from O on A, P,, A 2 P 2 , A, P 3 , and A 4 P 4 produced, 

 and whose forces are respectively equal to P,, P 2 , P 3 , 

 and P 4 , must likewise be equal to zero. 



This latter condition is technically called taking the 

 moments about the point O. 



The reasoning above used for four forces may readily 

 be extended to any number ; it must also be observed 

 that the position of the point O is perfectly arbitrary in 

 the solution of problems. It is generally so chosen as 

 to facilitate the solution, and to avoid unnecessary labour. 



Instead of pursuing the preceding method, it is fre- 

 quently advisable to resolve each of the forces P,, !'. 

 P 3 , and P, into two, acting in directions parallel to the 

 arbitrary axes O X and O Y (1), as in Props. VIII. and 

 XI. 



Confining, as before, onr attention first to the force 

 P u acting at A, in the direction A! PI. 



Through Aj (3> draw A, X t and A x YI parallel to 



Fig. 53 -(3). 



O X and O Y respectively ; and through P 1} P t Xj , 

 and Pj YI perpendicular to A! X,, and A t Y t . Again 

 through A, draw A, N\ perpendicular to O Y, and 

 AI M t perpendicular to O X. 



The force HI acting at A! may be replaced by two 

 forces Xi and Y\ at right angles to each other, repre- 

 sented in magnitude and direction by A! X t and Ai Y t 

 (Prop. III.) 



Without disturbing the conditions of equilibrium, two 

 equal and opposite forces represented in magnitude and 

 direction by O X/ and O X,", each equal to AI X!, 

 may be applied to O in the direction O X. 



And two equal and opposite forces O Y,', O Yj", each 

 equal to A! YI may be applied to O in the direction 

 O Y. 



Fi *- M Also- (4) the 



force X, may be 

 transferred from 

 Ai to N,, and 

 the force Y, from 

 A, toM,. 



The forces now 

 acting on the 

 body as in (4), 

 may be divided 

 into two groups, 

 one as ill (a), 

 consisting of the 

 forces Xj and Y, 

 acting at O, and 



magnitude and direction by OX/ and 

 (6), of the couples whose 



Fig. 58-(5.) 



represented in o 



O Y/ ; and the other as in 



moments are X, 



multiplied by ON, , 



and Y, multiplied 



byOM,. 



The tendency of 

 the couple X t "ONj 

 is evidently to twist 

 the body in the 

 opposite direction 

 to that in which the 

 couple Y l < ON, has 

 a tendency to twist 

 it. Hence, if one 

 be considered positive, the other must be negative. By 

 construction A, M, = y,, and A, N, = M, O = ,. 



Hence the resultant moment of these two couples, 

 Prop. XVI., will be 



On the whole, therefore, we have replaced the force 

 P, acting at A, by the forces X, and Y. acting on O in 

 the direction of the axes O X and O Y, and a couple 

 whose moment is equal to Y, ' x t X, i/, . 



Referring to (3), we see that AX Xi = AI PI cos. 

 1( and Ai.Y x = A x P^ sin. ,. 



