the Problem of Uniform Rotation. 323 



directly. Thus, when co becomes co 4- dco, the increase in 

 kinetic energy differs from what it would be if I were 

 constant by ^co 2 dI, and hence^ if we consider that the change 

 in internal energy of the disk is ^co 2 dl, we conserve our total 

 energy. _ ; 



Hence if we write P as the internal energy of the Tin- 

 contracted disk moving with angular velocity co, and P as 

 the internal energy of the contracted disk having the same 

 angular velocity, we have 



p-p (J =r ico^dco 



p lft)2 <n 



Jo 2 dco 



Now for the value of I for our rotating contracted disk, 

 taking into account the fact that the mass of each element 

 is altered in the ratio 



l+-r:l, 



we have 



f 



ds 



r 



To shorten our calculations, we will obtain P — P , not 

 from the integral expression for it given above, but from 

 the fact that 



P-P =E -E, 



where E and E are kinetic energies corresponding to P 

 and P, since 



e?E=E and I Icodco = E . 



