194 V. OSCILLATIONS AND CYCLIC MOTIONS. 



and introducing these instead of the velocities 



210) ^ 



We have for the positional force 



This being negative denotes that a force P r toward the axis 

 must be impressed on the mass m in order to maintain the cyclic 

 state. This may be accomplished by means of a geometrical constraint, 

 or by means of a spring. The force or reaction P r which the 

 mass m exerts in the direction from the axis in virtue of the rotation 

 is the so-called centrifugal force. We see that if the motion is iso- 

 cyclic, the positional force increases with r, while if it is adiabatic, 

 as in the case worked out above, it decreases when r increases. The 

 verification of the theorems of 52 is obvious. The cyclic force 



vanishes when the rotation is uniform, and the radius constant. If, 

 the motion being isocyclic, that is, one of uniform angular velocity, 

 the body moves farther from the axis, P v> the cyclic force is positive, 

 that is, unless a positive force P 9 is applied, the angular velocity 

 will diminish. In moving out from r : to r 2 work will be done 

 against the positional force P r of amount 



r 2 r z 



212) - A = -Jprdr = my' *Jrdr = ^ (r 2 2 - r*) 9 



^ rj, 



while the energy increases by the same amount. 



Thus the first theorem of 53 is verified. If the motion is 

 adiabatic, 



o f 



If the body moves from the axis, cp' will accordingly decrease, 

 so that 



213) 

 The change in P r due to a displacement dr is, by 211), 



214) dP r = - m(y n dr + 2/V (V), 



of which the part containing dcp', 



215) 

 does the work 



