148 



STATICS. 



or curve, which is of course equal and opposite to the reaction, 

 it can be found from a virtual displacement along the normal. 



241. If the equation (4) which expresses the principle of vir- 

 tual work be divided by the element of time &*, during which 

 the displacement 8s would take place, the factor s/$t=v repre- 

 sents a virtual velocity, and the equation becomes 



F 1 cos ! v -f F 2 cos 2 v -\ h F n cos a n v = o. 



On account of this form, the proposition is often called the 

 principle of virtual velocities. 



The product of a force into the virtual velocity of its point of 

 application in the direction of the force, Fcosct'V, is some- 

 times called the virtual moment of the force. 



242. The principle of virtual work can readily be extended to 

 the case of a rigid body acted upon by any number of forces. 



The forces acting on a rigid body can always be reduced to a 

 resultant R and a resulting couple H (Art. 200). This reduc- 

 tion is based on the supposition (Art. 84) that the point of 

 application of a force can be displaced arbitrarily along the line 



of the force. It can be shown that 

 such a displacement of the point of ap- 

 plication P of a force F (Fig. 75), from 

 P to Q along the line of the force, does 

 not affect the work done by the force 

 in any infinitesimal displacement of the 

 body. Let PF' = Ss be the displace- 

 ment of j?, QQ' = Ss' that of g; let p 

 and q be the projections of P and Q on 

 the line (the force F\ then, since the 

 body is rigid, P'Q' = PQ\ and conse- 

 quently Q^Kill ^differ from Pp only by 

 an infinitesimal of an order hi^HF than the order of the dis- 

 placement PP' = 8s. Hence, ^p 



F-Pp=F. Qq. 



Fig. 75. 



