578 Prof. A. L. Narayan on a Mechanical Illustration 



Therefore, if T represents the total kinetic energy 

 of the system, 



2 T = ,„ ! K,.(f) , + .»,K,.(f)%,,„K4.tt)- 



+ -{<&)' + »(%)' + '** $--+=*}- 



and Y the potential energy of the system (i. e, work done) 



= (m^JiY + m 2 a + m 3 b) g cos + m 2 gli 2 cos (f> + m^gli^ cos yfr 



+ K (a constant). 



.*. Equations of motion for small oscillations are : — 



(??i 1 K 1 2 + m 2 a 2 + m 3 b 2 ) -^ + m 2 ah 2 -p -f- m 3 bh~^ 



= (miAi 4- m 2 a + m 3 6) #0, . ( 1) 

 ^{m 2 K 2 2 )^ 2 + m 2 ah 2 C ^ 2 = m 2 gJi 2( p, ...... (2) 



m3Ks2 ~W + m * bJl3 ^2 = "hgh^- ...... (3) 



These equations show the analogy that exists between the 

 mechanical casein question and the case of coupled electrical 

 circuits, the equations of which we may write down as 

 follows : — 



^fr M *> M * ■•■•■» 

 ^t+^S^S 1 *- • • • w 



hs dt 2 + s 3 ~ M 'dt* +M - 1 ~W ' ' ' ' (6) 



where L's, </'s, and M's have their usual significance. 



* The equations of motion of the first arrangement, namely that 

 shown in photograph fig. I., will be of the form: 



a 2 Q-\-b 2 ^>-\-c 2 i\/ = d 2 <p, 



a^+b^ + c^ = d$. 



The essential difference between these equations and those of the 

 actual arrangement adopted finally being that the method of attachment 

 of B and C to A in the second case reduces the coefficients c 2 and bz to 

 zero. 



