594 



NATURE 



{Oct 3, 1878 



ON THE NATURE OF VIBRATORY MOTIONS^ 



II. 



Blackburn's Double Penduhan. 



fpXPERIMENT 7. — Let us return to our sand-pendu- 

 -" lum. We have examined the vibrations of a single 

 pendulum, let us now examine the vibrations of a double 

 pendulum, giving two vibrations at once. The little copper 

 ringr, in Fig. 7, on the cord of our pendulum, will slip up 

 and down, and by moving it in either direction we can com- 

 bine two pendulums in one. Slide it one quarter way up 

 the cord, and the double cord will be drawn together below 

 the ring. Now, if we pull the bob to the right or left, we 

 can make it swing from the copper ring just as if this 

 point were a new place of support for a new pendulum. 

 As it swings, you observe that the two cords above the 

 ring are at rest. But the upper pendulum can also be 

 made to swing forward and backward, and then we shall 

 have two pendulums combined. Let us try this and see 

 what will be the result. 



Just here we shall find it more convenient to use the 

 metric measure, as it is much more simple and easy to 



remember than the common measure of feet and inche?. 

 If you have no metric measure you had best buy one, or 

 make one. Get a wooden rod just 39i%^u inches long, and 

 divide this length into 100 parts. To assist you in this 

 you may remember that i inch is equal to 25 j^ milli- 

 metres. Ten millimetres make a centimetre, and 100 

 centimetres make a metre. 



Now slide the ring r, Fig, 7, up the cords till it is 25 

 centimetres from the middle of the thickness of the bob. 

 Then make it exactly 100 centimetres from the under side 

 of the cross-bar to the middle of the thickness of the bob, 

 by turning the violin-key on the top of the apparatus. 



At D, Fig. 7, is a small post. This post is set up 

 anywhere on a line drawn from the centre of the plat- 

 form, and making an angle of 45° with a line drawn from 

 one upright to the other. Fasten a bit of thread to 

 the string on the bob that is nearest to the post, and draw 

 the bob toward the post and fasten it there. "When the 



' From a forthcoming work on " Sound : a Series of Simple, Entertaining, 

 and Inexpensive Experiments in the Phenomena of Sound, for the Use of 

 Students of every Age." By Alfred Marshall Mayer, Professor of Physics 

 in the Stevens Institute of Technology. Communicated by the Author. 

 (Continued from p. 574 ) 



bob is perfectly still fill the funnel with sand, and then 

 hold a lighted match under the thread. The thread will 

 burn, and the bob will start off on its journey. Now, in 

 place of swinging in a straight line, it follows a curve, and 

 the sand traces this figure over and over. 



Fig. 8. " "• 



Here we have a most , singular result, and we may 

 well pause and study it out. You can readily see that we 

 have here two pendulums. One-quarter of the pendulum 

 swings from the copper ring, and, at the same time, the 

 whole pendulum swings from the cross-bar. The bob 

 cannot move in two directions at the same time, so it 

 makes a compromise and follows a new path that is made 

 up of the two directions. 



The most important fact that has been discovered in 

 relation to the movements of vibrating pendulums is that 

 the times of their vibrations vary as the square roots of 

 their lengths. The short pendulum above the ring is 25 

 centimetres long, or one-quarter of the length of the longer 

 pendulum, and, according to this rule, it moves twice as 

 fast. The two pendulums swing, one 25 centimetres and 

 the other 100 centimetres long, yet one really moves twice 

 as fast as the other. While the long pendulum is making 

 one vibration the short one makes two. The times of 

 their vibrations, therefore, stand as i is to 2, or, expressed 

 in another way, 1:2.: 



Experiment 8.— -Let us try other proportions and see 

 what the double pendulum will trace. Suppose we wish 

 one pendulum to make 2 vibrations while the other makes 

 3. Still keeping the middle of the bob at 100 centimetres 

 from the cross-bar, let us see where the ring must be 

 placed. The square of 2 is 4, and the square of 3 is 9. 

 Hence the two pendulums of the double pendulum must 

 have lengths as 4 is to 9. But the longer pendulum is 

 always 1,000 millimetres. Hence the shorter pendulum 

 will be found by the proportion 9:4:: 1,000 : 444 '4 

 millimetres. Therefore we must slide the ring up the 

 cord till it is 444" 4 millimetres above the middle of the 

 thickness of the bob. 



Fasten the bob to the post as before, fill it with sand> 

 and bum the thread, and the swinging bob will make this. 

 singular figure (Fig. 9). 



Fig. 9. 



Experhnent 9.— From these directions you can go on 

 and try all the simple ratios, such as 3:4, 4 : 5> 5:6,. 

 6 : 7, 7 : 8, and 8:9. In each case raise the two figures 

 to their squares, then multiply the larger number by 

 1,000, and divide the product by the smaller number ; the 

 quotient will give you the length of the smaller pendulum 

 in millimetres. Thus the length for rates of vibration, as 

 3 is to 4, is found as follows : 3 X 3 = 9^ 4 X 4 = ^6, and 

 9j<_i,ooo _ ^g^.^ millimetres. 



The table (Fig. 10) gives, in the first and second 

 columns, the rates of vibration, and in the thud and 



