70 Mr Bennett, The Rotation of the Non-Spinning Gyrostat 



The Rotation of the Non-Sf inning Gyrostat. By G. T. Bennett, 

 M.A., F.K.S., Emmanuel College, Cambridge. 



[Read 8 March 1920.] 



§ 1. The following extract is taken from an old examination 

 paper* : 



"A symmetrical wheel free to rotate about its axle is 

 moved from rest in any position by means of the axle and is 

 finally restored to a position in which the axle again points 

 in the same direction as formerly. Shew that the wheel, again 

 at rest, will have rotated through a plane angle equal to the 

 solid angle of the cone described by the varying directions of 

 the axle." 



The proof of this result may be put briefly in a geometrical form. 

 Translational and rotational movements being independent, the 

 centroid of the wheel may be treated as stationary. As the gyrostat 

 has no component rotation about its axis, the axis of rotation is at 

 any moment some diameter of the wheel. This line has the central 

 plane of the wheel as locus for the body-axode, and has a closed 

 cone of arbitrary form as locus for the space-axode. The angular 

 movement is therefore representable by the rolling of the plane 

 on the cone. The angle of ultimate rotation of the wheel is thus 

 (for cones of ordinary type) the excess of the four right angles of 

 the plane surface above the total surface-angle of the cone. This 

 difference is equal to the solid angle of the reciprocal cone described 

 by the axis of the wheel. And hence follows the result quoted; 

 namely, that the solid angle described by the axis of the wheel is 

 equal to the circular measure of the plane angle of the resultant 

 displacement of the wheel about its axis. Further, the sense of the 

 displacement accords with the sense of circulation associated with 

 the solid angle. 



§ 2. The result may be extended to the case in which the initial 

 and final directions of the axis are different, say a and 6. For the 

 axis may be restored to its original direction a by a subsequent 

 movement in the plane ha; and this latter movement, which is a 

 rotation about the normal to a and 6, leaves unaltered the angle 

 that any diameter of the wheel makes with the plane ah. Hence 

 the original movement, shifting the axis of the wheel from a to 6 



* Emmanuel and other Colleges, Second Year Problems, Wed. Jmie 8, 1898 

 Question 11. 



