82 Mr Gallop, On the rise of a Spinning Top. 



On the rise of a Spinning Top. By E. G. Gallop, M.A., 

 Gonville and Caius College. 



[Received 29 January 1903.] 



(Abstract.) 



In order to explain the way in which the axis of a top in 

 rapid rotation rises until it becomes sensibly vertical, it is 

 necessary to take account of the fact that the lower end of the 

 top is a surface of small extent and not a mere point. On the 

 assumption that the lower end is rounded off into a portion of 

 a sphere, and that the effect of friction between the top and 

 ground can be represented by a single force through the point 

 of contact, it is proved that when the initial spin about the axis 

 of figure exceeds a certain limit, the inclination of the axis to the 

 vertical can never exceed a certain limiting value. This limit de- 

 pends on the amount of energy possessed by the top, and as energy 

 is dissipated by sliding friction this limiting value diminishes till 

 when the energy reaches a certain value the limiting inclination 

 is reduced to zero and the top is left spinning with its axis 

 vertical. Whether or not the amount of energy dissipated is 

 sufficient to reduce the limiting inclination to zero, will depend 

 on the coefficient of friction and other circumstances, but in any 

 case it is proved that under the conditions assumed the top will 

 never fall to the ground if the initial spin exceeds a certain value. 

 Thus the ultimate fall of the top, which takes place under actual 

 conditions, must be attributed to the resistance of the air and 

 the friction couple which, though probably not producing much 

 serious effect on the motion at first, eventually diminish the spin 

 till it is unable to counterbalance the ordinary effects of gravity. 



The method is also applied to explain the way in which a 

 heterogeneous sphere spinning on a horizontal plane tends to 

 raise its centre of mass until it is vertically above the centre 

 of figure. 



Some numerical results are added for the case of a top 

 resembling a hollow sphere of diameter 4 cm. mounted on an axis 

 with its centre 4 cm. above the lower end, which is rounded off 

 into a sphere of radius 0*1 cm. It is proved that if this top is 

 started with its axis inclined at 30° to the vertical with a spin 

 of 32 revolutions per second, it would attain the state of steady 

 motion with axis vertical as soon as the amount of energy 

 dissipated by sliding friction amounted to 25^* n °f the original 

 kinetic energy; and that the inclination of the axis could never 

 reach 32° 23'. A smaller angular velocity would, of course, suffice 

 to prevent the axis of a larger top of similar construction from 

 falling to this inclination. 



