String Galvanometer of Evnihoven, 213 



If we write a constant h s to abbreviate the expression 



^ = ™(l-cos^), (18) 



we have 



Ip 



and the equation (12) becomes 



'(ps + Jtfis + ntys^hsi (19) 



The Circuit Equation. 



When the circuit of the string is completed outside of the 

 galvanometer, the motion of the string of itself generates a 

 current which opposes the motion, giving rise to the well- 

 known electromagnetic damping. If R is the total resistance 

 and L the inductance of the circuit, and e' the back electro- 

 motive force generated by the movement of the string, we. 

 have 



e=Bi+Lj t +e'. 



Since the back electromotive force is proportional to the 

 velocity we may write 



e =h/(j) $} 



where hj is a constant quantity. The rate at which 

 mechanical energy is supplied to the moving string is equal 

 to the rate at which the electrical energy is converted into 

 beat. The former is the impressed force, hi times the 

 velocity <p s , and the latter is the back electromotive force 

 times the current. Hence 



h e i<p s = hjifaj 



and consequently the constant h s ' is the same constant h 8 be- 

 fore used in (18), and the prime may be suppressed. The 

 circuit differential equation becomes 



e=m+lf t +h4,=f(t) .... (20) 



The current, i, may now be eliminated from the pair of 

 simultaneous differential equations (19) and (20), giving the 

 following equation connecting the normal coordinate <\> 8 with 

 the time 



L ■■■ /R L,\ v /R 7 L , h\. R 2 , ,,,* m \ 



