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Now slide the ring r, Fig. 12, 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. 12, is a small post. This post is set up anywhere on a line drawn from the centre of the platform, 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 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.
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 below 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 1 is to 2, or, expressed in another way, 1: 2.
EXPERIMENT 14.—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 burn the thread, and the swinging bob will make this singular figure (Fig. 14).
EXPERIMENT 15.–From these directions you can go on and try all the simple ratios, such as 3 : 4, 4:5, 5 : 6, 6 : 7, :8, and 8 : 9. In each case raise the two figures to their squares, then multiply the smaller number by 1,000, and divide the product by the larger 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 = 16, and 9*1800 =562.5 millimetres.
The table (Fig. 15) gives, in the first and second columns, the rates of vibration, and in the third and fourth columns the corresponding lengths of the longer and shorter pendulums. Opposite these lengths are the figures which these double pendulums trace. In the sixth column are the names of the musical intervals (see page 49) formed by two notes, which are made by numbers of sonorous vibrations, bearing to each other the ratios given in the first and second columns.
FIXING THE CURVES ON GLASS.
EXPERIMENT 16.-These interesting figures, traced in sand by the double pendulum, may be fixed on glass in a permanent form; and, when framed, will make beautiful ornaments for the window or mantel, and will remind you that you are becoming an experimenter. Procure squares of clear glass about six inches on the sides, and buy at the painter's a small quantity of French varnish, or clear spirit-varnish. Hold one of these pieces of glass level in the left hand by one corner, and, with the right, pour
some of the varnish upon the glass. Let the varnish cover half the glass, and then gently tip the glass from side to side till the varnish runs into every corner ; then
and rest one corner in the mouth of the varnishbottle, and rock the glass slowly from side to side. This will give a fine, smooth coat of varnish to the glass, and we may put it away to dry. When the varnish is hard, lay the glass, varnished side up, on the stand, adjust the pendulum to make one of the figures, and then fasten it to the post. Burn the thread, and stop the motion of the bob as soon as the figure is finished.
Brush away any extra sand that may lie at the ends of the figure, and then take the glass carefully to a hot stove. Have some wooden blocks laid on the stove, and rest the glass on these. Presently the varnish will begin to melt, and then the glass may be lifted and carefully put away to cool, taking the utmost care not to disturb the sand. When the varnish is hard, the sand which has not stuck is removed by gently rapping the edge of the plate on the table. Then we shall have a permanent figure of the
To preserve it, lay small pieces of cardboard at each corner and narrow strips half-way along the edges, and then lay another piece of glass over these, and bind the two together with paper on the edges. The plate may now be placed on the lantern, and greatly magnified images of the curves may be obtained on the screen.
EXPERIMENTS IN WHICH WE COMBINE THE MOTIONS OF
TWO VIBRATING RODS.
We have just seen how the double pendulum combines into one movement the motions of two pendulums swinging at right angles to each other. Our experiments have