POTENTIAL ENERGY 167 



dW dW dW 



Since > , are the components of the force 



dx, 0^ dz, 



acting on particle (1), these equations indicate that particle (1) 

 is in equilibrium. Similarly, it follows that the other particles 

 are in equilibrium, giving the result. 



137. An important special case of these theorems arises when 

 the only forces which do any work in a displacement are the 

 weights of the bodies of which the system is composed. If M is 

 the mass of the whole system, and if h is the height of its center 

 of gravity above any standard horizontal plane, the potential 

 energy is, by 120, Mgh, and this is a maximum or a minimum 

 when h is a maximum or a minimum. Thus we have the theorem : 



In a system of bodies in which the only forces which perform 

 work in a, displacement are those of gravity, the configurations of 

 equilibrium are those in which the height of the center of gravity 

 is a maximum or a minimum. 



EXAMPLES 



1. Two uniform rods, each of length Z, are freely jointed at their extremities 

 and placed over a smooth cylinder of radius a of which the axis is horizontal. 

 Find the angle which the rods make with the horizontal when in equilibrium. 



2. An elliptic disk is weighted so that its center of gravity is halfway 

 between its center and one extremity of its major axis. Show that if its 



eccentricity is greater than there will be four positions of equilibrium in 



V2 

 which the disk stands vertical on a horizontal plane, but otherwise only two. 



3. A horizontal rod of weight W has its center pierced by a fixed vertical 

 screw on which it turns, one revolution raising or lowering it by \ inch. If 

 there is no friction, find the couple required to hold it at rest. 



4. A plug of weight W is made in the shape of a pyramid of square cross 

 section. It is placed with its axis vertical in a square hole of side c, the depth 

 of its vertex in this position being d below the plane of contact. Find the couple 

 required to hold it turned through an angle and still having its axis vertical. 



5. A smooth parabolic wire is placed with its axis vertical. Two beads are 

 strung on it, and are connected by a string which passes through a smooth ring 

 at the focus. Show that there are an infinite number of positions of equilibrium. 



6. A smooth bowl in the shape of an ellipsoid of semi-axes a, 6, c has one axis 

 vertical. Find the couple required to hold a rod of length I in a horizontal posi- 

 tion in the bowl, making an angle with a position of equilibrium. 



