154 WOKK 



Thus the total work done against gravity is equal to the total 

 weight of the particles multiplied by the vertical height through 

 which the center of gravity of the particles has been raised. 



WORK PERFORMED AGAINST A COUPLE 



121. THEOREM. If a rigid body acted on by a system of forces 

 be given any small rotation through an angle e about any axis, the 

 work done is Ge, where G is the moment about this axis of the 

 forces opposing the motion. 



Let the axis of rotation be supposed to be a line perpendicular 

 to the plane of the paper, meeting it in the point L. Let a typical 



force be a force F acting on the particle 

 A of the body. 



As the result of the rotation, let A 

 move to a position A 1 , so that the angle 

 ALA 1 is equal to e, the angle through 

 which the body has been turned. 

 Then, during the rotation, the point 



FIG. 88 



of application of the force F moves 

 from A to A', and, therefore, the work done 



= F-AA f -cos<l>, 

 where < is the angle between F and AA r , 



= AA' x component of F along AA f 



= e X LA x component of F along AA 1 



= e x moment of F about the axis of rotation. 



If the rigid body is acted on by a number of forces applied to 

 its different particles, we find, on summation, that the total work 

 done 



= e x sum of the moments of all these forces 



about the axis of rotation 

 = Ge, where G is the moment about the axis of 

 rotation of all the forces. 



