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of sound are produced. Therefore, the knowledge which we now desire to give the reader lies at the

very

foundation of a correct understanding of the subject of this book.

An experiment is the key to this knowledge. It is the experiment with

THE CONICAL PENDULUM.

An ordinary pendulum changes its speed during its swings right and left exactly as a ball appears to change its speed when this ball revolves with a uniform speed in a circle, and we look at it along a line of sight which is in the plane of the circle.

EXPERIMENT 6.-Let one take the ball and wire to the farther end of the room, and by a slight circular motion of the end of the wire cause the ball to revolve in a circle. Soon the ball acquires a uniform speed around the circle, and then it forms what is called a conical pendulum; a kind of pendulum sometimes used in clocks. Now stoop down till your eye is on a level with the ball. This you will know by the ball appearing to move from side to side in a straight line. Study this motion carefully. It reproduces exactly the motion of an ordinary pendulum of the same length as that of the conical pendulum. From this it follows that the greatest speed reached during the swing of an ordinary pendulum just equals the uniform speed of the conical pendulum. That the apparent motion you are observing is really that of an ordinary · pendulum, you will soon prove for yourself to your entire satisfaction; and here let me say that one principle or fundamental fact seen in an experiment and patiently reflected on is worth a chapter of verbal descriptions of the same experiment.

Suppose that the ball goes round the circle of Fig. 5 in two seconds; then, as the circumference is divided into 16 equal parts, the ball moves from 1 to 2, or from 2 to 3, or from 3 to 4, and so on, in one-eighth of a second. But to the observer, who looks at this motion in the direction of the plane of the paper, the ball appears to go from 1 to 2, from

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2 to 3, from 3 to 4, etc., on a line A B, while it really goes from 1 to 2, from 2 to 3, from 3 to 4, etc., in the circle. The ball when at 1 is passing directly across the line of sight, and, therefore, appears with its greatest velocity; but when it is in the circle at 5 it is going away from the observer, and when at 13 it is coming toward him, and, therefore, although the ball is really moving with its regular speed when at 5 and 13, yet it appears when at these points momentarily at rest. From a comparison of the similarly numbered positions of the ball in the circle and on the line A B, it is evident that the ball appears to go from A to B and from B back to A in the time it takes to go from 13, round the whole circle, to 13 again. That is, the ball appears to vibrate from A to B in the time of one second, in which time it really has gone just half round

the circle. A comparison of the unequal lengths, 13 to 12, 12 to 11, 11 to 10, etc., on the line A B, over which the ball goes in equal times, gives the student a clear idea of the varying velocity of a swinging pendulum.

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Fig. 6 represents an upright frame of wood standing on a platform, and supporting a weight that hangs by a cord. A A is a flat board about 2 feet (61 centimetres) long and 14 inches (35,5 centimetres) wide.

B B are two uprights so high that the distance from the under

side of the cross-beam C to the platform A A is exactly 411 inches (1 metre and 45 millimetres). The cross-beam C is 18 inches (45.7 centimetres) long. At D is a wooden post standing upright on the platform. Get a lead disk, or bob, 34 inches (8 centimetres) in diameter and inch (16 millimetres) thick. In the centre of this is a hole 1 inch (25 millimetres) in diameter. This disk may easily be cast in sand from a wooden pattern. At the tinner's we may have made a little tin cone 1}ç inch (30 millimetres) wide at top and 24 inches (57 millimetres) deep, and drawn to a fine point. Carefully file off the point till a hole is made in the tip of the cone of about 15 inch in diameter. Place the tin cone in the hole in the lead disk, and keep it in place by stuffing wax around it. A glass funnel, as shown in the figure, may be used instead of the tin cone. With an awl drill three small holes through the upper edge of the bob at equal distances from each other. To mount the pendulum, we need about 9 feet (271.5 centimetres) of fine strong cord, like trout-line. Take three more pieces of this cord, each 10 inches (25.4 centimetres) long, and draw one through each of the holes in the lead bob and knot it there, and then draw them together and knot them evenly together above the bob, as shown in the figure. On the cross-bar, at the top of the frame, is a wooden peg shaped like the keys used in a violin. This is inserted in a hole in the bar—at Fin the figure. Having done this, fasten one end of the piece of trout-line to the three cords of the bob, and pass the other end upward through the hole marked E; then pass it through the hole in the key F; turn the key round several times; then pass the cord through the hole at G, to the bob, and fasten it there to the cords. Then get a small bit of copper wire and bend it once round the two cords just

above the knot, as at r in the figure. This wire ring, and the upright post at the side of the platform, we do not need at present, but they will be used in future experiments with this pendulum.

Tack on the platform A A a strip of wood I. This serves as a guide, along which we can slide the small board m, on which is tacked a piece of paper.

EXPERIMENT 7.-Fill the funnel with sand, and, while the pendulum is stationary, steadily slide the board under it. The running sand will be laid along L M, Fig. 7, in a straight line. If the board was slid under the sand during exactly two seconds of time, then the length of this line may stand for two seconds, and one-half of it may stand for one second, and so on. Thus, we see how time may be recorded in the length of a line.

Brush off the heaps of sand at the ends of the line, and bring the left-hand end of the sand-line directly under the point of the funnel, when the latter is at rest. Draw the lead bob to one side, to a point which is at right angles

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to the length of the line, and let it go. It swings to and fro, and leaves a track of sand, a b, which is at right angles to the line L M, Fig. 7.

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