MOTION OF A TOP 311 



The kinetic energy of the top is, by 248, 



while the potential energy is Mgh cos 6, where h is the distance 

 of the center of gravity of the top from 0. Thus the equation of 

 energy is 



+ B (o>; + a>l) + 2 Mgh cos 6 = E, (139) 



where E is a constant. This may be put into a different form. 

 For o>2 + >l i g th e square of the angular velocity of the axis of 

 the top : it is therefore the square of the actual velocity of the 

 point 1, 6, <f> on the unit sphere, and hence we have 



The equation of energy now assumes the form 



AW + B % sin 2 /Yl + 2 Mgh cos 6=E. (140) 



We can obtain a third equation from the fact that the angular 

 momentum about Oz, the vertical, is constant. The angular momen- 

 tum may be regarded as compounded of 



(a) the momentum due to the rotation fl about axis 3 ; 



(&) the momentum due to the motion of the axis of the top. 



The rotation fl about axis 3 may be further decomposed into 

 rotations H cos 6, fl sin 6 about the horizontal and vertical, giving 

 moments of momenta Al cos 9, Al sin 6 about the horizontal and 

 vertical. Thus the moment of momentum contributed by part (a) 

 is Al cos 6. 



The motion of the axis of the top may be resolved into a rota- 



tion of angular velocity sin 6 about an axis making an angle 



d() 



6 with the vertical, and one of angular velocity about a 



dt 



