The resultant of two parallel forces P and Q is R. If the force P shifts by a distance x, parallel to itself, prove that R will shift by (Px)/(P+Q).

The resultant of two parallel forces P and Q is R. If the force P shif

1.	The resultant of two parallel forces P and Q is R. If the force P shifts by a distance x, parallel to itself, prove that R will shift by (Px)/(P+Q).
Answer» Solution :Suppose the parallel forces P and Q act at the points A and B the resultant R acts at C Fig. `:. PxxAC =QxxBC` or, `" " (P)/(Q)=(BC)/(AC) or, (P+Q)/(Q) =(BC+AC)/(AC) =(AB)/(AC)` `:. AC = AB xx(Q)/(P+Q)` Now if the force P acts at point D such that AD =x, then the resultant also SHIFTS and acts at E instead of C . Hence, `PxxDE =QxxBE` or, `(P)/(Q)=(BE)/(DE) or, (P+Q)/(Q)=(BE+DE)/(DE) =(BD)/(DE)=(AB-x)/(AE-x)` `:. (AE-x) (P+Q) = (AB-x) Q` `:. AE = (Q.AB+P.x)/(P+Q)` `:.` Displacement of the line of action of the resultant, `CE = AE -AC= (Q.AB+P.x)/(P+Q)-(Q.AB)/(P+Q) =(Px)/(P+Q)`