﻿28 M. 0. E. Meyer on the Heating of a 



At every impulse upon the axis one part of the vis viva pre- 

 sent is lost ; for at each impulse the position of the axis of ro- 

 tation is changed ; hence, of the vis viva present, only that part 

 remains which corresponds to a rotation about the new axis ; all 

 the rest of the vis viva is lost, as far as rotation is concerned, and 

 is used in heating the disk. 



From this we can easily calculate the loss of vis viva and the 

 gain in heat occurring in each second. If we denote the angular 

 velocity of the disk by ty, the vis viva of the particles at the dis- 

 tance r from the axis is 



for the unit of mass. 



At this distance, however, there is an infinitely narrow zone of 

 the breadth dr, and the thickness of the disk S, which contains 

 the mass 



2irrBAdr t 



if A denotes the density of the disk of aluminium. This zone 

 has therefore the vis viva 



7rSA-\/r Wr ; 



and the entire disk the integral of this expression, 



where R denotes the radius of the disk, or 



iMR 2 f 



4 



by introducing the mass of the disk, 

 M = ttR 2 SA. 

 If, now, owing to one of the impulses in question, the axis of 

 the disk is deviated through the angle 'sfr, the residual vis viva 

 thereby becomes 



iMR 2 ^ 2 cos 2 (/>, 



and that which is lost for rotation and changed into heat is 

 iMR 2 ^ 2 sin 2 <£. 



This loss of vis viva and gain in heat occurs twice during each 



2 

 rotation — in the unit of time j~, if T denotes the time of one ro- 

 tation of the disk. The heat produced, therefore, in the unit of 

 time is equivalent to the vis viva, 



or, since 



, 2tt 



