CONDITIONS FOR EQUILIBRIUM 41 



If pq denote the angle between the lines of action of the forces 

 P and Q, we have pq = TT B, so that sin B = sin pq, and hence 



^- = ^-=^-. (ID 



sin qr sin rp sin pq 



The converse is true because, if the relation (11) is satisfied and 

 if the lines of action of the forces are in one plane, we can con- 

 struct a triangle of which the sides will represent the forces P, Q, R, 

 so that there is equilibrium. 



33. Analytical conditions for equilibrium. Expressed in an ana- 

 lytical form, the condition for equilibrium is that the resultant of all 

 the forces acting shall be zero. If the individual forces are known, 

 the resultant force can be obtained at once from the rules for the 

 composition of vectors, which have already been given in 14-16. 



If the forces all act in a plane, let their magnitudes be R v R v , 

 R n) and let their lines of action make angles e^ e 2 , , e n with the 

 axis of x. Then the resultant has components X, Y, where (cf. 14) 



X= RI cos e l + R 2 cos e 2 H ---- , 

 Y = R l sin j -j- R z sin e 2 + -. 



The magnitude of the resultant is Vj; 2 + F 2 , and this vanishes 

 only if X and Y vanish separately. Thus the condition for equi- 

 librium is that the components along the two axes shall vanish 

 separately, i.e. that the sum of the components of the separate 

 forces acting shall vanish when resolved along each axis. 



Similarly, if the forces do not all act in one plane, the condition 

 for equilibrium is that 'the sums of the components along three 

 axes in space shall vanish separately. 



EXAMPLES 



1. Forces of 12 and 8 pounds weight act in two directions which are at right 

 angles. Find the magnitude of their resultant. 



2. Three forces each equal to F act along three rectangular axes. Find their 

 resultant. 



3. The resultant of two forces PI and P 2 acting at right angles is R if P! 

 and P 2 be each increased by 3 pounds, R is increased by 4 pounds and is now 

 equal to the sum of the original values of PI and P 2 . Find P x and P 2 . 



