250 . DYNAMICS OF ROTATING BODIES. 



CHAPTER VII. 



DYNAMICS OF ROTATING BODIES. 



8O. Dynamics of Body moving about a Fixed Axis. The 



simplest case of motion of a rigid body next to that of translation 

 is a movement of rotation with one degree of freedom, namely a 

 motion about a fixed axis. The centrifugal force exerted by the body 

 on the axis is Md& 2 where d is the distance from the axis of the 

 center of mass of the body, and since this is in the direction of d 7 

 which is continually changing, if a body is to run rapidly in bearings 

 the center of mass should be in the axis, otherwise the bearings are 

 subjected to periodically varying forces. At the same time, even if 

 this condition is fulfilled, there will be a centrifugal couple, also 

 tending to tear the body from its bearings, unless the axis is a 

 principal axis of inertia. It is worth noticing that the first condition 

 may be obtained in practice by statical means, by making the axis 

 horizontal, and attaching weights until the body is in equilibrium in 

 any position, but that the second condition is only obtained by 

 experiments on the body in motion. For this reason the former 

 condition is generally fulfilled in such pieces of machinery as the 

 armatures of dynamos, while the latter is not especially provided for. 

 Let us consider the motion of a heavy body about a horizontal 

 axis. The resultant of all the parallel forces acting on its various 

 particles is by 59 a equal to a single force equal to the weight of 

 the body Mg applied at the center of mass. The position of the 

 body is determined by a single coordinate which we will take as the 

 angle # made with the vertical by the perpendicular from the center 

 of mass on the axis. If the length of the perpendicular is h the 

 work done in turning the body from the position of equilibrium is 



The kinetic energy is 



2) T = 



a 6A 



The equation of energy accordingly is 



3) K\-r:\ + Mgh(l cos -fr) = const. 



