178 



WORK 



to one in which it increases, we pass through a minimum. Thus 

 maximum and minimum values of W must occur alternately, or, 

 what is the same thing, configurations of stable and unstable 

 equilibrium must occur alternately. 



150. Examples of these two kinds of equilibrium can be found 

 in the illustrations already employed. 



1. Locomotive moving on a pair of rails. Let h be the height of the center 

 of gravity in any position, let x denote distances measured horizontally 

 along the track, and let M be the mass of the locomotive. The potential 



energy is then Mgh. The con- 

 dition for equilibrium in the 

 configuration x is 



h maximum 

 EquiLunstable 



h minimum 

 Equil. stable 



FIG. 97 



or =0, expressing that the 

 dx 



value of h must be either a maxi- 

 mum or a minimum. The table 

 on page 177 shows that if h is a 

 minimum, i.e. if the center of 

 gravity is at its lowest point, 

 the equilibrium will be stable. 

 Thus, if the locomotive is moved slightly from this position, it will roll 

 back to it again. If h is a maximum, i.e. if the center of gravity is at 

 its highest point, the equilibrium will be unstable. The locomotive is 

 now at the summit of a hill, and if displaced to either side of the summit, 

 will continue rolling down the hill. 



NOTE. If the moving parts of the engine are not " balanced " properly, the center 

 of gravity may not always be at the same height above the rails, so that the maxima 

 and minima of h do not necessarily occur at points where the height of the track is a 

 maximum or a minimum. For instance, a position of equilibrium might occur where 

 the track was not level, or again a position of stable equilibrium might occur at a 

 point at which the track was at its highest point, the height of the center of gravity 

 above the rails being of course a minimum at this point. Thus if the engine were dis- 

 placed to a point slightly lower on the track, and set free, it would return of itself to 

 the highest point. The principle here is the same as that of mechanical toys which, 

 on being placed at rest at the foot of an inclined plane, start to roll up the plane as 

 soon as set free. 



We notice that positions of stable and unstable equilibrium must occur 

 alternately, as already proved in 149. 



2. Door turning on hinges. Here again the potential energy is Mgh, where 

 h is the height of the center of gravity of the door above any standard 



