302 VH. DYNAMICS OF ROTATING BODIES. 



By means of observations of the value of the precession, we 



tQ ^\ 



may thus obtain the ratio of - ~n~^' We see that ^ ne forces causing 



precession are proportional to s - On account of the nearness of the 



moon, therefore, and in spite of its small mass, the precession 

 produced by the moon is somewhat greater than that due to the 

 sun. Since the moon's orbit departs but little from the plane of the 

 ecliptic the precession due to the moon may be calculated approxi- 

 mately by the above formulae, and compounded with that due to 

 the sun. 



97. Top on smooth Table. Having treated in detail the 

 motion of a body with one point fixed, and three degrees of freedom, 

 it remains to consider the motion of bodies which, like the ordinary 

 top, spin upon a table or other surface. We must now consider 

 the reaction between the body and the surface, and we have to 

 distinguish between the ideal case of perfectly smooth, or frictionless 

 bodies, where the reaction is normal, and bodies between which there 

 is friction, so that the reaction is not normal. We will consider the 

 first case. Let us examine the motion of a symmetrical top, spinning 

 on a sharp point resting on a smooth horizontal plane. The top has 

 five degrees of freedom, its position being defined as before by the 

 three angles -O 1 , ^, (p, and in addition, by the coordinates x, y, of the 

 center of mass, the #- coordinate being given by 



= I COS ft. 



Since the only force which we have not already considered is 

 the reaction, which has no horizontal component, the horizontal 

 component of the acceleration of the center of mass vanishes, so that 

 its motion is in a straight line with constant velocity. It therefore 

 remains only to determine the motion of rotation. This being in- 

 dependent of the horizontal motion just found, we may consider the 

 latter to vanish, so that the center of mass will be supposed to move 

 in a vertical line. The motion thus becomes one of three freedoms, 

 and we shall treat it by Lagrange's method as before. By the 

 principle of 32, 50), the kinetic energy is equal to that which the 

 body would have if concentrated at its center of mass, 



plus that which it would have if it performed its motion of rotation 

 about the center of mass supposed at rest. If then A and C denote 

 the moments of inertia about the center of mass (in 90 they were 

 the moments about the fixed point), we have 



175) T=~ [M Z 



