150 STATICS. [244. 



the sum of the works of R and H must vanish by hypothesis, 

 it follows that H=o. 



The two conditions R = o, H=o, are, therefore, both fulfilled. 



244. The following examples may serve to illustrate the appli- 

 cation of the principle of virtual work. 



To find the force just necessary to move a cylinder of radius r and 

 weight W up a plane inclined at an angle a to the horizon by means of a 

 crow-bar of length \ set at an angle (3 to the horizon (Fig. 76). 



Let s be the distance from the fulcrum A of the crow-bar to the 



point of contact B of the cylinder 

 with the plane. 



Turning the crow-bar about A by 

 an angle 8ft, the work of the force F 

 acting at the end of the bar is F- 1 8ft. 

 The corresponding displacement of 

 the centre C of the circle, which is 

 the point of application of the force 

 W, is parallel to the inclined plane, 

 and may be regarded as the differen- 

 tial 8s of the distance AB = s. The 

 work of W is, therefore, W8s sin a. This gives the equation of work 



hence 



"i i- 



The relation between s and ft can be found by projecting ABCDA 

 on the vertical line ; this gives 



r cos a -f- s sin a = r cos ft + J sin/3, 



whence , = r co0-cos. 



sin a sin ft 



Differentiating the former equation, we find 



sin a8s = - r sin (38ft + s cos ft 8ft + sin ft 8s, 



. 8s __ scosft r sin ft _ ^cos 2 /? cos cos ft sin a sin/? + sin 2 /? 

 8ft since sin/3 (since sin ft)' 2 



