Popular Science Monthly 



111 



by fastening to it, near the bottom, a 

 fairly large piece of cardboard, in such 

 a way that it will act as a brake. The 

 swings of the pendulum will die away 

 much faster than before; that is, the 

 damping will be increased on account of 

 the increased frictional resistance of the 

 fan. On such a more strongly damped 

 pendulum (assuming that the oscillation 

 is started by letting the pendulum 

 begin from the point lo), the successive 

 maximum swings to the left (at the ends 

 of the first, second, third and later 

 periods), may be as follows: 8.0 (end 

 of 1st); 6.4 (end of 2nd); 5.1 (end of 

 3rd); 4.1; 3.3; 2.6; 2.1, etc. It is 

 seen at once that now the swings decrease 

 much more rapidly. This is even more 

 vivid when Fig. 3, which shows the 

 motions Oi the second pendulum, is 

 inspected; the rapid fall of the broken 

 line along the top, which indicates the 

 damping, should be noted especially. 

 The constant ratio or damping factor, 

 whose value is an indication of the damp- 

 ing, may be found as before by dividing 

 the first maximum amplitude by the 

 second, the second by the third, etc. 

 This gives us: 10/8 = 8/6.4 = 6.4 = 5.1 

 = etc. = 1.25. Since this ratio is larger 

 than before the brake was added to the 

 pendulum, we have an arithmetical 

 proof that the damping is larger. 



So far we have considered only the 

 "damping" of the oscillation system; 

 what is the "logarithmic decrement?" 

 Nothing more nor less than the natural 

 logarithm of the constant ratio which 

 has just been figured out. These 

 logarithms, or special numbers, for 

 several different ratios, are given in the 

 following table: 



Ratio 

 I 



1.05 

 I. II 

 1. 16 

 1.22 

 1-25 

 1.28 



1-35 



Logarithm 

 0.00 

 0.05 

 O.IO 



0.15 



0.20 

 0.22 

 0.25 

 0.30 



By looking up the ratio i.i, which was 

 that of the first pendulum, in the table 

 it is seen that the logarithmic decrement 

 of that arrangement was a trifle under 

 0.1 per period; similarly, for the second 

 pendulum (which had a damping factor 



of 1.25), the decrement is found to be 

 0.22 per complete period. 



Although the examples just given are 

 purely mechanical, damping in electric 

 circuits is of the same character. Let 

 us consider the circuit of Fig. 4, which 

 has connected in series a condenser C, 



/fid/if\ ^^S- 3. Swings of second pendulum 



an inductance L, a resistance R and a 

 special current indicator /. This indica- 

 tor is of the sort which will show the 

 amount and direction of the current 

 flowing through the circuit at any 

 instant, as would a Braun-tube oscillo- 

 graph. If C is charged to a certain 

 potential and then is allowed to dis- 

 charge through the oscillation circuit 

 by the sudden closing of switch S, the 

 result will be a free oscillating current 

 through L, I and R. As was shown in 

 the March article of this series, the 

 frequency and time period of this free 

 oscillation can be figured out from a 

 simple rule, if one knows the inductance 

 and capacity of the circuit. The thing 

 important to this discussion is not the 

 period of frequency, however, but the 

 rate at which the free oscillation dies 

 away. If the oscillograph / is arranged 

 to make an actual photograph of the 

 oscillation current-effects (which is en- 

 tirely feasible, even on very high 

 frequencies), the result will be a curve 

 of the sort shown in Figs. 2 and 3; if 

 the capacity and inductance, or either 

 of them, are in- 

 creased, the time 

 period will be 

 lengthened and the 

 curves will spread 

 out more along the 

 horizontal line. If 

 the voltage applied 

 to the condenser 

 before the switch S is closed is made 

 larger, the current flowing will be in- 

 creased and the highest and lowest 



O 



J 



Fig. 4. Oscillograph 

 Circuit 



