VIRTUAL WORK 157 



125. Again, the work performed against the pair of forces which 

 constitute action and reaction between two smooth surfaces can 

 be seen to vanish. 



First consider the case in which one body is held at rest while 

 the second is made to slide over its surface. In such a displace- 

 ment the work performed, if any, is performed against the reaction 

 which acts on the moving body. Since the 

 force acts along the normal, while its point of 

 application necessarily moves in the tangent 

 plane, i.e. at right angles to the normal, 

 we see that the work done is nil. 



The most general motion possible for the two 

 surfaces is compounded of a motion of the kind 

 just described and a motion in which the two 

 surfaces move as a rigid body. The work done 

 in the first part of the displacement has just been seen to be zero, 

 the work done in the second part of the displacement vanishes by 

 124; hence the total work vanishes, proving the result required. 



126. The results just proved are not true if the contact between the 

 surfaces is rough. The work done in such a case depends on the magni- 

 tude of the frictional forces, and as it is generally as difficult to determine 

 the amount of these forces as to solve the whole problem, the method of 

 virtual work is not of any value in such cases. 



127. We have now seen that a large number of forces may be 

 left out of account altogether in calculating the work done in a 

 small displacement, and the principle of virtual work, which states 

 that when a system is in equilibrium the work done in any small 

 displacement is zero, requires us only to calculate the work per- 

 formed against external forces, and not that performed against the 

 internal actions and reactions of rigid bodies. 



128. Systems of pulleys. An important application of the prin- 

 ciple of virtual work is the following : Let us suppose that we have 

 any arrangement of pulleys and inextensible ropes, the ropes hav- 

 ing two free ends, to one of which the weight to be raised is 

 attached, and to the other of which the power is applied. Let these 



