776 



Popular Science Monthly 



be lengthened or shortened, as required, 

 until, by a number of successive measure- 

 ments, it is shown that the time of 

 making one complete swing is two 

 seconds. To measure the damping it 

 is not necessary to have the period any 

 specific length of time, but the plotting 

 of oscillation curves of the pendulum is 

 made easier if some simple number is 

 chosen. 



This plotting of the oscillation is an 

 * interesting and useful preliminary to 

 the determination of the damping of 

 the pendulum. Suppose that the period 

 has been adjusted to 2 seconds, and 

 that the scale along which the pendulum- 

 bob swings has been marked off into 

 10 equal parts on each side of the middle 

 or zero position. Let the bob be drawn 

 to the left and held at the tenth division ; 



points. For a certain pendulum these 

 may be as follows : 



Fig. 2. Swings of first pendulum 



if it is released it will reach the lowest 

 point (zero) in exactly ^4 second and 

 will swing out to the right side. At the 

 end of I second it will reach the end of 

 the swing to the right and will be on the 

 point of returning. At the end of 

 i}4 seconds the bob will be again 

 opposite the zero point, and at the end 

 of 2 seconds it will be at the end of its 

 first complete period and about to 

 swing to the right in beginning the 

 second period. The important thing to 

 note is that although the bob started at 

 10 on its scale, it did not swing so 

 far to the right but instead commenced 

 to return at the point indicated by 

 about 9.5 on the scale. At the end of the 

 first complete period it swung out only 

 about as far as 9 on the left; at the 

 next complete period it swung only a 

 little beyond 8. If one watches the 

 extreme reach at the end of each swing 

 very carefully, it becomes possible to 

 make a table of the successive turning 



and so on. At the end of each half 

 second the bob would be at zero, and at 

 the ends of the fifth, sixth and later 

 periods, at the following values of the 

 scale to the left: 5.9; 5.3; 4.8; 4.3; 

 3-9; 3-5; 3-1 ; 2.8; 2.5; 2.3; 2.0; etc. 

 By drawing a horizontal line to represent 

 time in seconds and by dividing the 

 space above and below it into ten equal 

 zones, above for swings to the left and 

 below for swings to the right, the 

 diagram of Fig. 2 may be drawn by 

 measuring off the points given in the 

 table (or those measured from your 

 own pendulum). This diagram repre- 

 sents the actual movements of the 

 suspended weight, and by drawing a 

 broken line through the highest points 

 one can get a good idea of how fast the 

 swings die away, or, in other words, of 

 how great the damping is. 



The most interesting thing about the 

 figures determined by the above experi- 

 ment is that the ratio of the successive 

 measurements or amplitudes of swing 

 remains a constant quantity. This may 

 be proved by taking the ratios of the 

 swings at the ends of each period; the 

 first ratio is 10/9=1.1. The second is 

 9/8.1 = 1.1. The third, 8.1/7.3=1.1. 

 Likewise, all the others may be found to 

 be equal to i.i, since it is a law of nature 

 that all simple free oscillations in any 

 vibrating system (whether mechanical 

 or electrical) will die away or be damped 

 out at such a rate that the ratio of their 

 successive maxirrium amplitudes remains 

 constant. This ratio of amplitudes is 

 a measure of the damping, and is called 

 the damping factor. The larger the 

 ratio the higher the damping. 



Suppose that the wind friction of the 

 pendulum shown in Fig. i is increased 



