I 4 6 STATICS. [237. 



(5) A mass of 12 Ibs. slides down a smooth plane inclined at an angle 

 of 30 to the horizon, through a distance of 25 ft. ; what is the work 

 done by gravity ? 



237. It follows from the definition of work that, if any num- 

 ber of forces F v F 2 , . . ., F n act on a particle P, the sum of their 

 works for any displacement PP' = ds is equal to the work of their 

 resultant R for the same displacement. For, the resultant R 

 being the closing line of the polygon constructed by adding the 

 forces F v F 2 , ..., F n geometrically, the projection of R on any 

 direction, such as PP', is equal to the sum of the projections of 

 the forces F on the same line (Art. 89) ; that is, if <* 15 2 , . . ., 

 n be the angles made by F v F 2 , . . ., F n with PP', and a the 

 angle between R and PP', we have 



F 1 cos ! + F 2 cos 2 H ----- \-F n cos n = R cos a ; 



multiplying this equation by ds, we obtain the above proposition 



F 1 cos oL^ds + F 2 cos v^ds -\ ----- \- F n cos a n ds = R cos ads, 



which expresses the so-called principle of work for a single 

 particle. 



238. When the particle is in equilibrium, so that the forces 

 do not actually change the motion, we may derive from this 

 proposition a convenient expression for the conditions of equi- 

 librium by considering displacements that might be given to 

 the particle. Such displacements are called virtual, and the 

 corresponding work of any of the forces is called virtual work. 



It is customary to denote a virtual displacement by Ss, the 

 letter B being used to distinguish from an actual displacement 

 ds ; this distinction becomes of importance in kinetics. 



239. The resultant being zero in the case of equilibrium, the 

 sum of the virtual works of all forces acting on the particle must 

 be zero for any virtual displacement, i.e. 



^i cos !&$ -f F z cos aJBs -\ ----- h F H cos a n $s = o. (4) , 



As the resultant must vanish if its three projections vanish 

 for any three axes not lying in the same plane, the necessary 



