316 MOTION OF EIGID BODIES 



the top remains strictly vertical, in common language, the top 

 is " asleep." 



If the conditions were the ideal conditions supposed, this motion 

 would continue forever, but in nature such ideal conditions can- 

 not exist. The region of contact between the peg and the surface 

 on which it spins is not strictly a point, but a small circle or 

 ellipse, on account of the small compression which takes place at 

 the point of contact. By making the peg of hard steel and spin- 

 ning on a hard surface, this region is very small, but is still of 

 finite dimensions. The consequence is that the reactions on the 

 peg do not all meet the axis. There is a small frictional couple 

 resisting the rotation of the top, and O gradually decreases. 



When H has so far decreased as to be less than H , the ranges 

 of oscillation are 6 and = . The top is no longer asleep, 

 but is now wobbling through an angle . As fl continues to 

 decrease, continually increases, as is clear from equation (147), 

 and finally reaches so large a value that the top rolls against 

 the ground and so falls over. 



257. The interest of these results will perhaps be enhanced by exam- 

 ining the form they assume when the top is of a very simple kind. 

 Let us suppose that a top is formed by running 

 a pin through the center of a uniform disk of 

 mass M, radius a. Let the length of the pin 

 which protrudes through the disk on its lower 

 side be h, and let the mass of t*he pin be 

 neglected in comparison with that of the disk. 



The h is the same as the h of our previous analysis. The values of 

 A and B are 



When spinning at the critical velocity O at which wobbling sets in, the 

 velocity of a point on the rim is O a or 2 V^A. Thus wobbling begins when 

 the velocity of a point on the rim is reduced to 2 V^A, a velocity which 

 depends only on the height of the disk and not on its radius. We see that 

 the lower the disk is, the slower it can spin without wobbling. If we take 

 h = 2 inches, we find that wobbling begins when the rim velocity is about 

 4.7 feet a second. 



I 



