STABILITY AND INSTABILITY 



179 



level. As the door turns on its hinges, its center of gravity describes a 



circle about the line of hinges. If this line is perfectly vertical, the circle 



described by the center of gravity 



lies entirely in a horizontal plane, so 



that every position is one of equilib- 



rium, and the question of stability or 



instability does not arise. If, how- 



ever, the line of hinges is not per- 



fectly vertical, the circle will lie in 



an inclined plane. The points at 



which the height above the standard 



horizontal plane is a maximum or 



minimum are two in number : 



P, the highest point of the circle, FIG. 98 



at which equilibrium is unstable ; 



Q, the lowest point of the circle, at which equilibrium is stable. 



3. Bead sliding on wire. To obtain a definite problem, let us suppose 

 that the bead P slides on an elliptic wire placed so that its major axis A A' 

 is vertical, and let it be acted on by its weight, and 

 also by the tension of a stretched elastic string of 

 which the other end is tied to the center of the 

 ellipse. Let a, b be the semi-axes of the ellipse, and 

 let I, X be the natural length and modulus of the 

 string, I being greater than a, so that the string is 

 always stretched. Let w be the weight of the bead. 

 The first step is to calculate the potential energy 

 in any configuration. Let the configuration be 

 specified by the eccentric angle <f> of the point on 

 the ellipse occupied by the bead. The height of 

 the bead above the center of the ellipse is then 

 acos^, so that that part of the potential energy 

 which arises from gravitational forces is wa cos <f>. 

 The length of the string r is given by 



r 2 = a 2 cos 2 -f & 2 sin 2 0, (a) 



and the work done in stretching the string from length I to length r is ( 113) 



This may be taken to be the part of the potential energy which arises 

 from the stretching of the string. Thus the total potential energy will be 



W = wa cos 



+ ^(r-0 2 - 



