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
��Time in Seconds
�Position of Bob
�(start)
�10 left
�0.5
I
1-5
�
9-5 right
�2 (end of first period)
�9.0 left
�2-5
3
�8.6 right
�3-5
4 (end of second period) 6 (end of third period) 8 (end of fourth period)
�
8.1 left 7.3 left 6.6 left
��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 maximium 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
