314 MOTION OF E1GID BODIES 



255. Let us find what is the least angular momentum which the 

 top must have so as to spin without falling over. To do this, we 

 may assume that falling will occur if ever 6 exceeds a certain limit 

 3 , either through the peg slipping or through its side touching 

 the ground. The condition that the top shall not fall over is that 

 # 2 must be less than 3 , and hence that /(cos 3 ) must be positive. 

 Thus the values of E, G, and fl must be such that 



B sin 2 8 (AW + 2 Mgh cos 3 - E) + ( G -An cos 8 ) 2 



is positive. 



Suppose that the top is started at an inclination to the ver- 

 tical, having no motion except one of rotation H about its axis. 

 We then have, from equations (140) and (141), 



E = AW + 2 Mgh cos , 



G=Al cos . 

 Thus 



/(cos 8 ) = B sin 2 3 (AW + 2 Mgh cos 8 - E) + ( G - Aft cos 3 ) 2 

 = B sin 2 3 2 Mgh (cos 3 - cos ) + ^ 2 ft 2 (cos 3 - cos ) 2 

 = (cos 3 - cos ) [2 MghB sin 2 3 + ^ 2 H 2 (cos 3 - cos )]. 



(144) 



Since the top is necessarily started in a position in which it can 

 spin, the value of cos 3 cos is necessarily negative. Thus in 

 order that /(cos 3 ) may be positive, we must have 



A*W (cos - cos 3 ) - 2 MghB sin 2 3 (145) 



2 MghB sin 2 3 



positive, or W > y -^-. (146) 



A 2 (cos cos 3 ) 



We notice that if A is very small, the value of fl required to 

 keep the top from falling is very large. It is therefore very hard 

 to spin a top of small cross section, such as a lead pencil or a 

 pointed wire*. 



If we can choose the angle at which we start the top, cos is 

 at our disposal. We see that the necessary value for H is least 



