147-151] BODIES IN CONTACT. 133 



150. Forces which do no work. Constraints. Among forces which 

 do no work we have already had occasion to notice, 



(1) the tension of an inextensible string, 



(2) the internal forces between the parts of a rigid body. 



Both of these are of the nature of constraints, viz., they are forces required 

 to maintain geometrical conditions. 



Other examples of forces which do no work are 



(1) the pressure between smooth surfaces, 



(2) the reaction between rough surfaces which roll on each other. 

 These also are forces required to maintain geometrical conditions. 



In the case of smooth surfaces there is no relative displacement of the 

 points that come into contact in the direction of the pressure, and thus as 

 much work is done against the pressure on one body as is done by the pressure 

 on the other body at any point of contact. 



In the case of rough surfaces which roll on each other there is no relative 

 displacement of the points that come into contact in any direction, and thus as 

 much work is done against the pressure on one body as is done by the pressure 

 on the other body at any point of contact, and as much work is done against 

 the friction on one body as is done by the friction on the other body at any 

 point of contact. 



Care is required in the application of these results to the discussion of the 

 motion of one body in contact with another body. When both bodies are in 

 motion, relatively to the frame of reference, usually the pressure, or reaction, 

 on one body at a point of contact does work, and this work must not be 

 omitted from the work function (if there is one) of the forces acting on that 

 body. Thus it may well happen that there is no energy equation (Article 151) 

 for a single body, forming part of a system for which there is an energy 

 equation. 



When however a body slides in contact with a smooth, or rolls in contact 

 with a rough, surface which has a fixed position relative to the frame of 

 reference, then the pressure, or reaction, at a point of contact does no work. 



The notion of constraint, explained in Article 118, may be generalised. 

 We shall call forces which do no work constraints. 



151. Equation of Energy. As in Article 105 let n^ be the 



mass of any particle of a system, x lt y lt z^ its coordinates at time t, 

 X 1} T lt Z the resolved parts, parallel to the axes, of the resultant 

 of all the forces exerted upon it by particles outside the system, 

 Xi, YI, Z{ the resolved parts parallel to the axes of the resultant 

 of all the forces exerted upon it by the other particles of the 

 system. The equations of motion are such as 



X l + X i'. 



