32 Induction of Electric Currents. 



The equations which express the mechanical properties of 

 the system are readily found, and are precisely the same as 

 those applicable in the electrical problem. Since the potential 

 energy vanishes, everything turns upon the expression for the 

 kinetic energy. If x and y denote the circumferential velo- 

 cities in the same direction of the pulleys A, B where the 

 cord is in contact with them, \{x-\-y) is the vertical velocity 

 of the pendent pulleys. Also \{x—y) is the circumferential 

 linear velocity of C, D, due to rotation, at the place where the 

 cord engages. If the diameter be here 2a, the angular velocity 

 is(x—y)/2a. Thus, if M be the total mass of each pendent 

 pulley and attachment, Mk 2 the moment of inertia of the revol- 

 ving parts, the whole kinetic energy corresponding to each is 



For the energy of the whole system we should have the 

 double of this, and, if it were necessary to include them, terms 

 proportional to x 2 and y 2 to represent the energy of the fixed 

 pulleys. The reaction between the pulleys A, B depends upon 

 the presence of a term xy in the expression of the energy. 

 We see that this would disappear if ¥ = a 2 ; as would happen 

 if the whole mass of the pendent pulleys and attachments 

 were concentrated in the circles where the cord runs. The 

 case discussed above, as analogous to electric currents, occurs 

 when k 2 < a 2 , a condition that will be satisfied, even without 

 non-rotating attachments, if the cord run near the circum- 

 ference of the rotating pulleys. The opposite state of things, 

 in which F>a 2 , would be realized by carrying out masses 

 beyond the groove, and thus increasing the rotatory in com- 

 parison with the translatory inertia. In this case the mutual 

 action between A and B is reversed. If when all is at rest 

 A be suddenly started, B moves forward in the same direction. 

 Otherwise C and D would have to rotate, and this in their 

 character of fly-wheels they oppose. 



Generally, if L, N be the coefficients of self-induction, and 

 M mutual induction, we have (constant factors being omitted) 



M = a 2 -P. 

 In order to imitate the case of two circuits coiled together in 

 close proximity throughout, we must have in the mechanical 

 model k 2 = ; that is, the rotatory inertia of the pendent pulleys 

 must be negligible in comparison with the translatory inertia. 

 Also the energy of the fixed pulleys, not included in the above 

 expressions, must be negligible. If these conditions be 

 satisfied, a sudden rotation imposed upon A generates an 

 equal and opposite motion in B. 



