276 



VII. DYNAMICS OF ROTATING BODIES. 



These are Euler's kinematical equations. They illustrate the statement 

 made in 76, about p, q, r as not being time derivatives, for it is 



easily seen that pdt, qdt, rdt do not 

 satisfy the conditions of being exact 

 differentials. 



The resultant of the weight of all 

 the parts of the body is Mg applied at 

 the center of mass. If this is at a 

 distance I from the fixed point the moment 

 of the applied force is Mglsmft about 

 the axis ON. 



L= Mglsin&coscp, 

 66) M = Mgl sin # sin cp, 



N= 0, 



so that Euler's dynamical equations are 

 A -~ = (B C) qr -f Mgl sin # cos <p, 



67) S -(C- 



Mgl sin & sinqp, 



Fig. 92. 



Multiplying respectively by p, q, r and 

 adding, we obtain 



68) Ap ^| 4- Bq ~ + Cr -^ = Mgl (p sin # cos gp ^ sin ^ sin 9) 



d# 



Integrating we get the equation of energy, 



69) Ap 2 + Bq* + Cr 2 = 2 (h - Mgl cos #). 



Since the moment of the applied forces has no vertical component, 

 the vertical component of the angular momentum is constant, or the 

 end of the vector H describes a plane curve in a horizontal plane. 

 Resolving H x = Ap,H y = I>q, H z = Cr on the vertical OZ', we obtain 



70) HJ = Ap sin 0* sin (p -f Bq sin # cos cp -f- Cr cos # == const. 



If the top is symmetrical about the 2T-axis, we have A = B. 

 Then the third equation 67) is 



C dr 

 G ~-> 



71) 



Cr = const. = H 2 . 



