THE COMPOUND PENDULUM AND THE COMPOSITION OF VIBRATIONS

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LECTURE XIX. THE COMPOUND PENDULUM AND THE COMPOSITION OF VIBRATIONS.

The Compound Pendulum.—The Centre of Oscillation.—The Centre of Percussion.—The Conical Pendulum.—The Composition of Vibrations.

THE COMPOUND PENDULUM.

Fig. 87.

623. Pendulous motion must now be studied in other forms besides that of the simple pendulum, which consists of a weight and a cord. Any body rotating about an axis may be made to oscillate by gravity. A body thus vibrating is called a compound pendulum. The ideal form, which consists of an indefinitely small weight attached to a perfectly flexible and imponderable string, is an abstraction which can only be approximately imitated in nature. It follows that every pendulum used in our experiments is strictly speaking compound.

624. The first pendulum of this class which we shall notice is that used in the common clock (Fig. 87). This consists of a wooden or steel rod a e, to which a brass or leaden bob b is attached. This pendulum is suspended by means of a steel spring c a, [Pg 300]which being very flexible, allows the vibration to be performed with considerable freedom. The use of the screw at the end will be explained in Art. 664. A pendulum like this vibrates isochronously, when the amplitude is small, but it is not easy to see precisely what is the length of the simple pendulum which would oscillate in the same time. In the first place, we are uncertain as to the virtual position of the point of suspension, for the spring, though flexible, will not yield at the point c to the same extent as a string; thus the effective point of suspension must be somewhat lower than c. The other extremity is still more uncertain, for the weight, so far from being a single point, is not exclusively in the neighbourhood of the bob, inasmuch as the rod of the pendulum has a mass that is appreciable. This form of pendulum cannot therefore be used where it is necessary to determine the length with accuracy.

  625. When the length of a pendulum is to be measured, we must adopt other means of supporting it than that of suspension by a spring, as otherwise we cannot have a definite point from which to measure. To illustrate the mode that is to be adopted, I take here an iron bar 6' long and 1" square, which weighs 19 lbs. I wish to support this at one end so that it can vibrate freely, and at the same time have a definite point of suspension. I have here two small prisms of steel e (Fig. 88) fastened to a brass frame; the faces of the prisms meet at about an angle of 60° and form the edges about which the oscillation takes place: this frame and the edges can be placed on the end of the bar, and can be fixed there by tightening two nuts. The object of [Pg 301]having the edges on a sliding frame is that they may be applicable to different parts of the bar with facility. In some instruments used in experiments requiring extreme delicacy, the edges which are attached to the pendulum are supported upon plates of agate; they are to be adjusted on the same horizontal line, and the pendulum really vibrates about this line, as about an axis. For our purpose it will be sufficient to support the edges upon small pieces of steel. a b, Fig. 88, represents one side of the top of the iron bar; e is the edge projecting from it, with its edge perpendicular to the bar. c d is a steel plate bearing a knife edge on its upper surface; this piece of steel is firmly secured to the framework. There is of course a similar piece on the other side, supporting the other edge. The bar, thus delicately poised, will, when once started, vibrate backwards and forwards for an hour, as there is very little friction between the edges and the pieces which support them.

Fig. 88.

626. The general appearance of the apparatus, when mounted, is shown in Fig. 89. a b is the bar: at a the two edges are shown, and also the pieces of steel which support them. The whole is carried by a horizontal beam bolted to two uprights; and a glance at the figure will explain the arrangements made to secure the steadiness of the apparatus; the second pair of edges shown at b will be referred to presently (Art. 635).

627. This bar, as you see, vibrates to and fro; and we shall determine the length of a simple pendulum which would vibrate in the same period [Pg 302]of time. The length might be deduced by finding the time of vibration, and then calculating from Art. 606. This would be the most accurate mode of proceeding, but I have preferred to adopt a direct method which does not require calculation. A simple pendulum, consisting of a fine cord and a small iron sphere c, is mounted behind the edge, Fig. 89. The point from which the cord is suspended lies exactly in the line of the two edges, and there is an adjustment for lengthening or shortening the cord at pleasure.

Fig. 89.

[Pg 303]628. We first try with 6' of cord, so that the simple pendulum shall have the same length as the bar. Taking the ball in one hand and the bar in the other, I draw them aside, and you see, when they are released, that the bar performs two vibrations and returns to my hand before the ball. Hence the length of the isochronous simple pendulum is certainly less than the length of the bar; for a pendulum of that length is too slow.

629. I now shorten the cord until it is only half the length of the bar; and, repeating the experiment, we find that the ball returns before the bar, and therefore the simple pendulum is too short. Hence we learn that the isochronous pendulum is greater than half the length of the bar, and less than the whole length.

630. Let us finally try a simple pendulum two-thirds of the length of the bar. I make the experiment, and find that the ball and the bar return to my hand precisely at the same instant. Therefore two-thirds of the length of the bar is the length of the isochronous simple pendulum.

We may state generally that the time of vibration of a uniform bar about one end equals that of a simple pendulum whose length is two-thirds of the bar; no doubt the bar we have used is not strictly uniform, because of the edges; but in the positions they occupy, their influence on the time of vibrations is imperceptible.

632. For this rule to be verified, it is essentially necessary that the edges be properly situated on the bar; to illustrate this we may examine the oscillations of the small rod, shown at d (Fig. 89). This rod is also of iron 24" × 0"·5 × 0"·5, and it is suspended from a point near the centre by a pair of edges; if the edges could be placed so that the centre of gravity of the whole lay in the line of the edges, it is evident that the bar would rest indifferently however [Pg 304]it were placed, and would not oscillate. If then the edges be very near the centre of gravity, we can easily understand that the oscillations may be very slow, and this is actually the case in the bar d. By the aid of the stop-watch, I find that one hundred vibrations are performed in 248 seconds, and that therefore each vibration occupies 2·48 seconds. The length of the simple pendulum which has 2·48 seconds for its period of oscillation, is about 20'. Had the edges been at one end, the length of the simple pendulum would have been

A bar 72" long will vibrate in a shorter time when the edge is 15"·2 from one end than when it has any other position. The length of the corresponding simple pendulum is 41"·6.

THE CENTRE OF OSCILLATION.

633. It appears that corresponding to each compound pendulum we have a specific length equal to that of the isochronous simple pendulum. To take as example the 6' bar already described (Art. 625), this length is 4'. If I measure off from the edges a distance of 4', and mark this point upon the bar, the point is called the centre of oscillation. More generally the centre of oscillation is found by drawing a line equal to the isochronous simple pendulum from the centre of oscillation through the centre of gravity.

634. In the bar d the centre of oscillation would be at a distance of 20' below the edges; and in general the position will vary with the position of the edges.

  635. In the 6' bar b is the centre of oscillation. I take another pair of edges and place them on the bar, so that the line of [Pg 305]the edges passes through b. I now lift the bar carefully and turn it upside down, so that the edges b rest upon the steel plates. In this position one-third of the bar is above the axis of suspension, and the remaining two-thirds below. a is of course now at the bottom of the bar, and is on a level with the ball, c: the pendulum is made to oscillate about the edges b, and the time of its vibration may be approximately determined by direct comparison with c, as already explained. I find that, when I allow c and the bar to swing together, they both vibrate precisely in the same time. You will remember, that when the ball was suspended by a string 4' long, its vibrations were isochronous with those of the bar when suspended from the edges a. Without having altered c, but having made the bar to vibrate about b, I find that the time of oscillation of the bar is still equal to that of c. Therefore, the period of oscillation about a is equal to that about b. Hence, when the bar is vibrating about b, its centre of oscillation must be 4' from b, that is, it must be at a: so that when the bar is suspended from a, the centre of oscillation is b; while, when the bar is suspended from b, the centre of oscillation is a. This is an interesting dynamical theorem. It may be more concisely expressed by saying that the centre of oscillation and the centre of suspension are reciprocal.

636. Though the proof that we have given of this curious law applies only to a uniform bar, yet the law is itself true in general, whatever be the nature of the compound pendulum.

  637. We alluded in the last lecture (Art. 610) to the difficulty of measuring with accuracy the length of a simple pendulum; but the reciprocity of the centres of oscillation and suspension, suggested to the ingenious Captain Kater a method by which this difficulty could be [Pg 306]evaded. We shall explain the principle. Let one pair of edges be at a. Let the other pair of edges, b, be moved as near as possible to the centre of oscillation. We can test whether b has been placed correctly: for the time taken by the pendulum to perform 100 vibrations about a should be equal to the time taken to perform 100 vibrations about b. If the times are not quite equal, b must be moved slightly until the times are properly brought to equality. The length of the isochronous simple pendulum is then equal to the distance between the edges a and b; and this distance, from one edge to the other edge, presents none of the difficulties in its exact measurement which we had before to contend with: it can be found with precision. Hence, knowing the length of the pendulum and its time of oscillation, gravity can be found in the manner already explained (Art. 608).

638. I have adjusted the two edges of the 6' bar as nearly as I could at the centres of oscillation and suspension, and we shall proceed to test the correctness of the positions. Mounting the bar first by the edges at a, I set it vibrating. I take the stop-watch already referred to (Art. 612), and record the position of its hands. I then place my finger on the stud, and, just at the moment when the bar is at the middle of one of its vibrations, I start the watch. I count a hundred vibrations; and when the pendulum is again at the middle of its stroke, I stop the watch, and find it records an interval of 110·4 seconds. Thus the time of one vibration is 1·104 seconds. Reversing the bar, so that it vibrates about its centre of oscillation b, I now find that 110·0 is the time occupied by one hundred vibrations counted in the same manner as before; hence 1·100 seconds is the time of one vibration about b: thus, the periods of the vibrations are very nearly equal, as they differ only by ¹/₂₅₀th part of a second.[Pg 307]

639. It would be difficult to render the times of oscillation exactly equal by merely altering the position of B. In Kater’s pendulum the two knife-edges are first placed so that the periods are as nearly equal as possible. The final adjustments are given by moving a small sliding-piece on the bar until it is found that the times of vibration about the two edges are identical. We shall not, however, use this refinement in a lecture experiment; I shall adopt the mean value of 1·102 seconds. The distance of the knife-edges is about 3'·992; hence gravity may be found from the expression (Art. 608)

The value thus deduced is 32'·4, which is within a small fraction of the true value.

640. With suitable precautions Kater’s pendulum can be made to give a very accurate result. It is to be adjusted so that there shall be no perceptible difference in the number of vibrations in twenty-four hours, whichever edge be the axis of suspension: the distance between the edges is then to be measured with the last degree of precision by comparison with a proper standard.

THE CENTRE OF PERCUSSION.

Fig. 90.

641. The centre of oscillation in a body free to rotate about a fixed axis is identical with another remarkable point, called the centre of percussion. We proceed to examine some of the properties of a body thus suspended with reference to the effects of a blow. For the purposes of these experiments the method of suspension by edges is however quite unsuited.[Pg 308]

642. We shall first use a rod suspended from a pin about which the rod can rotate. a b, Fig. 90, is a pine rod 48" × 1" × 1", free to turn round b. Suppose this rod be hanging vertically at rest. I take a stick in my hand, and, giving the rod a blow, an impulsive shock will instantly be communicated to the pin at b; but the actual effect upon b will be very different according to the position at which the blow is given. If I strike the upper part of the rod at d, the action of a b upon the pin is a pressure to the left. If I strike the lower part at a, the pressure is to the right. But if I strike the point c, which is distant from b by two-thirds of the length of the rod, there is no pressure upon the pin. Concisely, for a blow below c, the pressure is to the right; for one above c, it is to the left; for one at c it is nothing.

643. We can easily verify this by holding one extremity of a rod between the finger and thumb of the left hand, and striking it in different places with a stick held in the right hand; the pressure of the rod, when struck, will be so felt that the circumstances already stated can be verified.

Fig. 91.

644. A more visible way of investigating the subject is shown in Fig. 91. f b is a rod of wood, suspended from a beam by the string f g. A piece of paper is fastened to the rod at f by means of a small slip of wood clamped firmly to the rod; the other ends of this piece of paper are similarly clamped at p and q.

645. When the rod receives a blow on the right-hand side at a, we find that the piece of paper is broken across at e, because the end f has been driven by the blow towards q, and consequently caused the fracture of the paper at a place, e, [Pg 309]where it had been specially narrowed. I remove the pieces of paper, and replace them by a new piece precisely similar. I now strike the rod at b,—a smart tap is all that is necessary,—and the piece of paper breaks at d. Finally replacing the pieces of paper by a third piece, I find that when I give the rod a tap (not a violent blow) at c, neither d nor e are broken.

646. This point c, where the rod can receive a blow without producing a shock upon the axis of suspension is the centre of percussion. We see, from its being two-thirds of the length of the rod distant from f, that it is identical with the centre of oscillation of the rod, if vibrating about knife-edges at f. It is true in general, whatever be the shape of the body, that the centre of oscillation is identical with the centre of percussion.

647. The principle embodied in the property of the centre of percussion has many practical applications. Every cricketer well knows that there is a part of his bat from which the ball flies without giving his hands any unpleasant feeling. The explanation is simple. The bat is a body suspended from the hands of the batman; and if the ball be struck with the centre of percussion of the bat, there is no shock experienced. The [Pg 310]centre of percussion in a hammer lies in its head, consequently a nail can receive a violent blow with perfect comfort to the hand which holds the handle.

THE CONICAL PENDULUM.

  648. I have here a tripod (Fig. 92) which supports a heavy ball of cast iron by a string 6' long. If I withdraw the ball from its position of rest, and merely release it, the ball vibrates to and fro, the string continues in the same plane, and the motion is that already discussed in the circular pendulum. If at the same instant that I release the ball, I impart to it a slight push in a direction not passing through the position of rest, the ball describes a curved path, returning to the point from which it started. This motion is that of the conical pendulum, because the string supporting the ball describes a cone.

649. In order to examine the nature of the motion, we can make the ball depict its own path. At the opposite point of the ball to that from which it is suspended, a hole is drilled, and in this I have fitted a camel’s hair paint-brush filled with ink. I bring a sheet of paper on a drawing-board under the vibrating ball; and you see the brush traces an ellipse upon the paper, which I quickly withdraw.

650. By starting the ball in different ways, I can make it describe very different ellipses: here is one that is extremely long and narrow, and here another almost circular. When the magnitude of the initial velocity is properly adjusted, and its direction is perpendicular to the radius, I can make the string describe a right cone, and the ball a horizontal circle, but it requires some care and several trials in order to succeed in this. The ellipse may also become very narrow, so [Pg 311]that we pass by insensible gradations to the circular pendulum, in which the brush traces a straight line.

Fig. 92.

651. When the ball is moving in a circle, its velocity is uniform; when moving in an ellipse, its velocity is greatest at the extremities of the least axes of this ellipse, and least at the extremities of the greatest axes; but, when the ball is vibrating to and fro, as in the [Pg 312]ordinary circular pendulum, the velocity is greatest at the middle of each vibration, and vanishes of course each time the pendulum attains the extremity of its swing. It is worthy of notice that under all circumstances the brush traces an ellipse upon the paper; for the circle and the straight line are only extreme cases, the one being a very round ellipse and the other a very thin one. If, however, the arc of vibration be large the movement is by no means so simple.

652. How are we to account for the elliptic movement? To do so fully would require more calculation than can be admitted here, but we may give a general account of the phenomenon.

Let us suppose that the ellipse a c b d, Fig. 93, is the path described by a particle when suspended by a string from a point vertically above o, the centre of the ellipse. To produce this motion I withdraw the particle from its position of rest at o to a. If merely released, the particle would swing over to b, and back again to a; but I do not simply release it, I impart a velocity impelling it in the direction a t. Through o draw c d parallel to a t. If I had taken the particle at o, and, without withdrawing it from its position of rest, had started it off in the direction o d, the particle would continue for ever to vibrate backwards and forwards from c to d. Hence, when I release the particle at a, and give it a velocity in the direction a t, the particle commences to move under the action of two distinct vibrations, one parallel to a b, the other parallel to c d, and we have to find the effect of these two vibrations impressed simultaneously upon the same particle. They are performed in the same time, since all vibrations are isochronous. We must conceive one motion starting from a a towards o at the same moment that the other commences to start from o towards d. [Pg 313]After the lapse of a short time, the body has moved through a y in its oscillation towards o, and in the same time through o z in its oscillation towards d; it is therefore found at x. Now, when the particle has moved through a distance equal and parallel to a o, it must be found at the point d, because the motion from o to d takes the same time as from a to o. Similarly the body must pass through b, because the time occupied by going from a to b, would have been sufficient for the journey from o to d, and back again. The particle is found at p, because, after the vibration returning from b has arrived at q, the movement from d to o has travelled on to r. In this way the particle may be traced completely round its path by the composition of the two motions. It can be proved that for small motions the path is an ellipse, by reasoning founded upon the fact that the vibrations are isochronous.

Fig. 93.

653. Close examination reveals a very interesting circumstance connected with this experiment. It may be observed that the ellipse described by the body is not quite fixed in position, but that it [Pg 314]gradually moves round in its plane. Thus, in Fig. 92, the ellipse which is being traced out by the brush will gradually change its position to the dotted line shown on the board. The axis of the ellipse revolves in the same direction as that in which the ball is moving. This phenomenon is more marked with an ellipse whose dimensions are considerable in proportion to the length of the string. In fact, if the ellipse be very small, the change of position is imperceptible. The cause of this change is to be found in the fact already mentioned (Art. 598), that though the vibrations of a pendulum are very nearly isochronous, yet they are not absolutely so; the vibrations through a long arc taking a minute portion of time longer than those through a short arc.

This difference only becomes appreciable when the larger arc is of considerable magnitude with reference to the length of the pendulum.

Fig. 94.

654. How this causes displacement of the ellipse may be explained by Fig. 94. The particle is describing the figure a d c b in the direction shown by the arrows. This motion may be conceived to be compounded of vibrations a c and b d, if we imagine the particle to have been started from a with the right velocity perpendicular to o a. At the point a, the entire motion is for the instant perpendicular to o a; in fact, the motion is then exclusively due to the vibration b d, and there is no movement parallel to o a. We may define the extremity of the major axis of the ellipse to be the position of the particle, when the motion parallel to that axis vanishes. Of course this applies equally to the other extremity of the axis c, and similarly at the points b or d there is no motion of the particle parallel to b d.

  655. Let us follow the particle, starting from a until it [Pg 315]returns there again. The movement is compounded of two vibrations, one from a to c and back again, the other along b d; from o to d, then from d to b, then from b to o, taking exactly double the time of one vibration from d to b. If the time of vibration along a c were exactly equal to that along b d, these two vibrations would bring the particle back to a precisely under the original circumstances. But they do not take place in the same time; the motion along a c takes a shade longer, so that, when the motion parallel to a c has ceased, the motion along d b has gone past o to a point q, very near o. Let a p = o q, and when the motion parallel to a c has vanished, the particle will be found at p; hence p must be the extremity of the major axis of the ellipse. In the next revolution, the extremity of the axis will advance a little more, and thus the ellipse moves round gradually.

THE COMPOSITION OF VIBRATIONS.

656. We have learned to regard the elliptic motion in the conical pendulum as compounded of two vibrations. The importance of the composition of vibrations justifies us in examining this subject experimentally in another way. The apparatus which we shall employ is represented in Fig. 95.

a is a ball of cast iron weighing 25 lbs., suspended from the tripod by a cord: this ball itself forms the support of another pendulum, b. The second pendulum is very light, being merely a [Pg 316]globe of glass filled with sand. Through a hole at the bottom of the glass the sand runs out upon a drawing-board placed underneath to receive it.

Fig. 95.

Thus the little stream of sand depicts its own journey upon the drawing-board, and the curves traced out thus indicate the path in which the bob of the second pendulum has moved.

Fig. 96.

657. If the lengths of the two pendulums be equal, and their vibrations be in different planes, the curve described is an ellipse, passing at one extreme into a circle, and at the other into a straight line. This is what we might have expected, for the two vibrations are each performed in the same time, and therefore the case is analogous to that of the conical pendulum of Art. 648.

658. But the curve is of a very different character when the cords are unequal. Let us study in particular the case in which the second pendulum is only one-fourth the length of the cord supporting the iron ball. This is the experiment represented in Fig. 95. The form of the path delineated by the sand is shown in Fig. 96. The arrowheads placed upon the curve show the direction in which it is traced. Let us suppose that the formation of the figure commences at a; it then goes on to b, to o, to c, to d, and back to a: this shows us that the bob of the lower pendulum must have performed two vibrations up and down, while that of the upper has made one right and left. The motion is thus compounded of two vibrations at right angles, and the time of one is half that of the other.

The time of vibration is proportional to the square root of the length; and, since the lower pendulum is one-fourth the length of the upper, its time of vibration is one-half that of the upper. In this experiment, therefore, we have a confirmation of the law of Art 605.

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