# Understanding Pendulums: A Brief Introduction

Foucault pendulum

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Join Our Free Trial Now! In the third, the impulse is applied while the point P is moving away from the rest position Fig. In general the impulses change the pendulum period, and this change is the escapement error. In the first, an impulse is applied while the pendulum is moving towards the rest position and the effect is to reduce the pendulum period. In the second, an impulse is applied while the pendulum is at the rest position and the pendulum period is unchanged.

In the third, an impulse is applied while the pendulum is moving away from the rest position and the pendulum period is increased. Reviews of various types of escapement are given by Bateman e , Britten , Rawlings and Roberts Bateman d suggested that escapement error could be reduced by increasing the pendulum quality, Q, since this decreased the level of impulses needed to keep the pendulum in motion. The effect of impulses on a pendulum can be represented graphically by a phase diagram in which angular acceleration is plotted against angular velocity.

The phase diagram for a real pendulum driven by an escapement with impulses at the ends of the swing is shown schematically in Fig. The impulses are assumed to be short enough to be regarded as instantaneous. The effect of the impulses is exaggerated for clarity. One complete circuit of the phase diagram, in the direction shown by the arrow, represents one complete oscillation of the pendulum.

Each impulse, represented by a horizontal line, causes an instantaneous increase in the angular velocity of the pendulum. Energy losses due to friction etc. If the impulses exactly replace the energy lost, then the circuit repeats exactly. In other words the pendulum amplitude is constant. Arrows show the sense of time. In its stable state the phase diagram repeats exactly and is the corresponding limit circuit.

If the pendulum amplitude is disturbed by an extraneous impulse then the phase diagram spirals back to the limit circuit. A numerical example is given by Denny When a clock is launched by pushing the pendulum it automatically settles into the stable state.

The dead beat escapement Fig. It is sometimes called the Graham escapement, and is used in precision pendulum clocks Roberts , The teeth are of a different shape to those of an anchor escapement Fig. Impulses applied by a dead beat escapement are fairly close approximations to instantaneous impulses.

If the clock is fitted with a seconds hand then the seconds hand visibly recoils for a short distance after each tick. The anchor escapement Figs. The invention of the anchor escapement is sometimes attributed to Robert Hooke Britten ; Bennet et al. It has also been attributed to William Clement Edwardes The phase diagram for a real pendulum driven by a recoil escapement is shown schematically in Fig. The impulse is of finite length and extends around the end of a swing. The force due to the impulse is always in the same direction, but the direction of motion of the pendulum reverses at the end of a swing.

Hence, before the end of the swing the force due to the impulse is in the same direction as 58 4 Pendulum Clocks the motion of the pendulum and the angular acceleration of the pendulum increases, but after the end of the swing the force due to the impulse is in the opposite direction to the motion of the pendulum and the angular acceleration decreases.

The net effect can be represented as an instantaneous impulse at the end of a swing, as shown in Fig. The numerous variables make it impossible to derive precise theoretical values of the effect of escapement error on the pendulum period, and a trial and error approach still has to be used. Modern precision electronic timing equipment makes it possible to measure pendulum periods very accurately.

This makes a trial and error approach easier since the effects of changes made to an escapement can found quickly and easily Tekippe The amplitude of a clock pendulum cannot be kept precisely constant so minimising changes in pendulum period due to inevitable changes in amplitude is an important aspect of clock design.

Unwanted changes in amplitude are more significant for a pendulum swinging with a large amplitude, so precision clocks usually have small amplitude pendulums Roberts , For large pendulum amplitude there are two ways to reduce changes in pendulum period due to circular error. The other is to increase the restoring force as the pendulum angle increases.

Huygens Huygens showed that if the path of the point mass. To achieve this he proposed the use of cycloidal cheeks to make a pendulum isochronous. This is shown in Fig.

### A Brief Introduction

The arc ACB in the figure corresponds to one complete revolution of the circle. Rolling the circle past A or B results in a cusp, such as that at D in Fig.

When a point P moves along a curve the centre of curvature of the curve moves along another curve called the evolute which can be expressed parametrically in terms of its arc length Coxeter The evolute of a cycloid is an identical cycloid, which can be expressed graphically as shown in Fig. An inextensible string, length equal to the arc length BD, is clamped at one at D, and held tight against the cycloid at the other end P.

P then traces the desired identical evolute cycloid. The same idea can be used for a real pendulum fitted with a spring suspension Sect. This is because of practical difficulties in persuading a real pendulum to follow a cycloidal path Emmerson The use of auxiliary springs to increase the restoring force as the pendulum angle increases was suggested by Edouard Phillips in Roberts Restoring springs are used in Bulle clocks Miles ; Bavister , which were produced in the first quarter of the twentieth Century Rawlings Bulle clocks are battery powered and usually have a pivot suspension Sect.

Impulses are given to the pendulum by a coil mounted on the pendulum which swings over a permanent magnet mounted on the clock case Fig. The coil acts as a bob and there is a rating nut below it. A contact near the top of the pendulum completes the circuit to provide the impulses Robinson Pendulums in Bulle clocks have a large amplitude, and an auxiliary spring is used to compensate for changes in pendulum period due circular error.

In this context an auxiliary spring is called an isochronous spring. The 60 4 Pendulum Clocks Fig. In the rest position Fig. As the pendulum swings away from the rest position Fig. The synchronous spring is very light so its pivots introduce negligible additional friction. Edouard Phillips analysed the synchronous spring using elliptic integrals, and his analysis was published posthumously in Bavister Numerical calculations by Bavister, using values taken from a Bulle clock, showed that with careful adjustment of the adjustable pivot, a synchronous spring could be very effective in reducing changes in pendulum period due to circular error.

This was confirmed by measurements of the pendulum period on the same Bulle clock Ridout and Thackery Another approach is to compensate for changes in pendulum period due to circular error by ensuring that changes in pendulum period due to escapement error previous section are of opposite sign Tekippe He states that in Ferdinand Berthoud knew that he could make a clock that kept better time by modifying the escapement Tekippe Berthoud carried out an experiment References 61 with three different escapements driving the same pendulum. An escapement with a large recoil decreased the pendulum period when the driving weight, and hence the pendulum amplitude, was increased.

A dead beat escapement increased the pendulum period when the driving weight was increased. An escapement with a little recoil caused no change in the pendulum period when the driving weight was increased. In other words it was isochronous for changes in amplitude. This concept of compensating circular error by changes in escapement error of opposite sign became part of traditional knowledge among French clockmakers.

Experiments reported by Tekippe using modern precision electronic timing equipment and an escapement with a slight recoil showed clearly what happened when the driving weight was doubled. The pendulum amplitude increased until it reached the new stable value limit cycle corresponding to the increased weight.

There was an initial step decrease in the pendulum period due the change in escapement error. The pendulum period then increased as the pendulum amplitude increased, initially rapidly, until it settled to a value corresponding to the increased driving weight. In this particular experiment the stable value of the pendulum period was unchanged by increasing the driving weight. In other words there is an isochronous combination of pendulum and escapement.

Eur J Phys 36 3 — Anonymous Clock. Horological J 1 : 3—8 Bateman D b Vibration theory and clocks. Part 2. Forced harmonic motion. Horological J 2 —52 Bateman D c Vibration theory and clocks. Part 3. Q and the practical performance of clocks. Horological J 3 —55 Bateman D d Vibration theory and clocks. Part 4. Q and classical escapement theory. Horological J 4 —70 Bateman D e Vibration theory and clocks. Part 6. Errors in escapements. Horological Sci Newsl —1 :5 Bavister R A study into the effectiveness of the isochronous spring.

Electrical Horology Group Paper No. The life and work of Robert Hooke. Antiquarian Horology, 12 6 : —, Emmerson A Some mathematics of the cantilever pendulum. Horological Sci Newsl, —3 , 11—19 Errata. Christiaan Huygens, the pendulum and the cycloid. Horological Sci Newsl ——32 Emmerson A Equations of motion for the spring suspended pendulum.

Horological Sci Newsl ——14 Haine J Synchronous oscillations of asymmetric coupled pendulums. In: Edwardes EL The story of the pendulum clock. Antiquarian Horology 31 6 — Matthys R Accurate clock pendulums. Practical manual for the use of clockmakers and jewellers. Mayfield, Ashbourne Penney D The earliest pendulum clocks: a re-evaluation. In: Bavister RA study into the effectiveness of the isochronous spring. Antiquarian Horological Society, Ticehurst Roberts D Precision pendulum clocks: year quest for accurate timekeeping in England. Their repair and maintenance, 2nd edn.

N A G Press Ltd. It does this by impulses whose timing is determined by the pendulum Sect. By contrast, in a driven pendulum, energy is transferred to a pendulum by periodic forces whose period is not controlled by the pendulum. A driven pendulum is sometimes called a forced pendulum. There are three distinct ways in which a simple rod pendulum Sect. In rotary driving the pendulum is subjected to prescribed varying torques about the suspension point. Experimental results, using a specially designed real pendulum with an electromagnetic drive have confirmed the existence of both periodic and chaotic behaviour Baker and Gollub A simple rod pendulum can be started from the rest position by using rotary driving.

A simple string pendulum Sect.

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In horizontal driving the suspension point of a simple rod pendulum is subjected to prescribed varying horizontal displacements Lamb ; Tritton , In vertical driving, sometimes called lifting, the suspension point is subjected to prescribed varying vertical displacements Pippard ; Bishop et al. In both cases there are two forces on the simple rod pendulum. One is that due to gravity on the point mass, m, Fig.

The other is that needed to impose the prescribed force or displacement, and this force varies with time. A simple rod pendulum can be started from the rest position by using horizontal driving, but it cannot be started from the rest position by using vertical driving. A simple string pendulum can be driven by horizontal driving, and by vertical driving. In the latter case random process theory, described in this chapter, is needed for understanding the effects of driving.

Driven damped simple harmonic motion provides a useful approximation to some aspects of the behaviour of driven pendulums. Some aspects of horizontal driving are described in this chapter. Broadly, a random process can be defined as a process which has a random appearance, such as those shown in Figs.

Alternatively, a random process can be defined as a process that displays chaotic behaviour. In a Gaussian random processes instantaneous values follow the Gaussian distribution or Normal distribution. Many naturally occurring random processes, such as water waves, are approximately Gaussian. Random process theory Papoulis ; Bendat and Piersol ; Pook can be used to analyse Gaussian random processes Fig.

## The sympathy of two pendulum clocks: beyond Huygens’ observations

The random processes shown in Figs. They were computer generated so were precisely determined by the algorithms used, and can therefore be described as chaotic. In some ways random process theory is an alternative to chaos theory as described, for example by Baker and Gollub , that can be used to highlight different aspects of behaviour.

Figure 5. In the figure amplitude is plotted against time. A feature of a broad band random process is that individual cycles cannot be distinguished. In a narrow band random process Fig. In general, a random process may be described by the function S. Assume that S. Stationary means that statistical parameters characterising the process are independent of time.

Ergodic means, broadly, that different samples of the same process yield the same values for statistical parameters. Only stationary random processes can be ergodic, and in practice most are. The RMS can equally well be calculated for periodic processes such as a sine wave Fig. The use of RMS first became popular in electrical engineering because it can be used directly in calculations involving power.

Instantaneous values of S. These can be characterised by the exceedance, P. The cumulative probability, 1 — P. The probability density, p. The values shown in Fig. The obvious difference between them can be expressed numerically by measures of what is called their bandwidth. Measures of bandwidth include the irregularity factor, the spectral density function, and the spectral bandwidth. The irregularity factor, I , is used in metal fatigue Pook For a process with zero mean it is the ratio of upward going zero crossings to positive peaks, and lies in the range 0—1.

The irregularity factor has the advantages that is easily understood, and is not restricted to Gaussian processes. The irregularity factor is 0. A Gaussian random process which is statistically stationary and ergodic can be described more precisely by its spectral density function SDF which describes the frequency content of the process.

This describes the relationship between the values of the random process S. The SDF, G. The SDF is sometimes plotted on a logarithmic scale and sometimes on a linear scale. In practice it is usually calculated using an algorithm known as the Fast Fourier Transform Bendat and Piersol It lies in the range 0—1 and is given by s m22 5. Values of spectral bandwidth are shown in the captions to Figs. These have a slowly varying random amplitude. There is no generally accepted definition of precisely what is meant by a narrow band random process, partly because of the physical difficulties of measuring bandwidth as "!

The probability density for the occurrence of a positive peak of amplitude S Fig. As the process is statistically symmetrical about zero, corresponding negative peaks also appear. The normalised exceedance, P. Conventionally, in discussion of the Rayleigh and related distributions, only positive peaks are described and shown in diagrams such as Fig.

Negative peaks are sometimes called troughs. Theoretically the Rayleigh distribution extends to 5. Clipping implies that higher peaks are reduced to the level given by the clipping ratio; truncation that they are omitted altogether. Physical limitations mean that the clipping ratio does not usually exceed four or five. In a narrow band random process the spectral density function is sharply peaked at the centre frequency, as shown in Fig. The figure is for water surface elevation of surface waves in the North Sea, significant wave height 4. The significant wave height is the average height, measured peak to trough, of the highest one third portion of the waves Pook so these are large waves.

In general, large surface waves are a narrow band random process. In damped simple harmonic motion Sect. In driven damped simple harmonic motion there is, in addition, a prescribed force on the point mass, m, along the straight line. This prescribed force can be either periodic or random.

If this periodic force, frequency fp , is represented by Fp cos! The phase, ", between the driving force and the oscillation is given by tan " D k! One of these is Eq. Another can be obtained from Eq. This is illustrated schematically by the resonance curve for driven damped simple harmonic motion shown in Fig. Driving damped simple harmonic motion by a broad band random process Fig. For this to occur the resonant frequency Fig. In other words, the spectral density function and resonance curve Fig. The spectral density function for a broad band random process Fig.

Using the broad band random process to drive a damped simple harmonic motion within this range of frequencies would result in a narrow band random process. In other words, a resonance is excited. For a narrow band random process the range of frequencies over which a resonance is exited is much more limited. In Fig. This is an example of periodic driving. If the imposed displacement is a sine wave Fig.

If the period of the imposed displacement is greater than the natural period of a simple rod pendulum, rod length, l, then the motion of the point mass, m, and the imposed displacement are in phase, and the pendulum oscillates about a virtual frictionless pivot which is above the actual frictionless pivot Fig. If the period of the imposed displacement is less than 72 5 Driven Pendulums Fig. Pendulum angle and point mass moving in phase. Pendulum angle and point mass moving out of phase that of a simple rod pendulum, rod length, l, then the motion of the point mass, m, and the imposed displacement are out of phase, and the pendulum oscillates about a virtual frictionless pivot which is below the actual frictionless pivot Fig.

The two modes of oscillation are similar to those of the lower rod of a double rod pendulum, shown in Fig. If the driving period is equal to the natural period then the virtual rod length, lV , is indeterminate, and pendulum behaviour is chaotic. A damped simple rod pendulum is a simple rod pendulum in which a resistance torque, opposed to the motion of the pendulum, is proportional to the angular velocity of the pendulum Sect.

If the horizontal driving, described above, is applied to a damped simple rod pendulum then, by analogy with driven damped simple harmonic motion Sect. For small amplitudes chaotic behaviour occurs when the driving period is approximately or exactly equal to the natural period.

The above analysis for small amplitudes also applies to a simple string pendulum Fig. In this special case the modes of oscillation shown in Fig. The behaviour of a real string pendulum, driven at the clamp, is more complicated. For convenience call motion in a plane containing the horizontal driving left-right motion and motion perpendicular to this front-to-back motion. When the driving period is approximately equal to the natural period of the pendulum behaviour becomes chaotic and inevitable imperfections mean that front-to-back motion takes place, as well as left-right motion Tritton , Left-right motion oscillations are at the forced period of the pendulum, and front-to back motions are at the natural period of the pendulum.

In general, the combination of the two motions results in an elliptical motion which precesses Sect. The slightly different periods mean that the phase between the two oscillations changes so the ellipses change shape and, at times, become circles or arcs.

This driven damped simple rod pendulum is an example of random driving. By analogy with random driving of damped simple harmonic motion Sect. The amplitude of a narrow band random process varies with time, and the amplitude occasionally becomes relatively large. As for periodic driving previous section the analysis for small amplitudes also applies to a simple string pendulum Fig.

The pendulums used are hand held real versions of simple string pendulums such as that shown in Fig. Inevitable slight movements of the hand result in random driving. The resulting chaotic motion is similar to that observed in a real version of a simple string pendulum under periodic random driving Sect. This chaotic behaviour in hand held pendulums is described by Macbeth and Jurriaanse who ascribe it to occult influences. He then goes on to ascribe this to an occult influence whereas it is simply a consequence of the chaotic behaviour of a hand held pendulum. The random driving results in a two dimensional narrow band random process.

The amplitude varies with time and, in accordance with appropriately generalised versions of Eqs. Jurriaanse includes a number of pendulum charts, intended for occult predictions. A pendulum chart is semi circular, and divided into several sectors, as shown schematically in Fig. In use, a pendulum is held over the start position.

Its chaotic behaviour, similar to that described in Sect. One might just as well throw a die Chap. Clarendon, Oxford Jurriaanse D The practical pendulum book, with instructions for use and thirty-eight pendulum charts. Pendulum play. A scientific pastime needing no knowledge of science etc.

NEL Report In: Hall N ed The new scientist guide to chaos. Penguin, London, pp 22—32 Chapter 6 Scientific Instruments 6. Four of these uses are described in this chapter. If the effective length, l, of a pendulum and the pendulum period, P , are known then the value of the acceleration due to gravity, g, can be calculated by using Eq. The practical difficulty is the measurement of the effective length of a real pendulum. In the figure the vertical length of each pendulum, to the centre of the balls, is 18 cm.

If the ball at one end of the row is pulled back and released the impulse as it hits the next ball is transmitted along the row, and the ball at the other end flies off with the remaining balls remaining almost stationary. In Huygens pointed out that an explanation of its behaviour required conservation of both momentum and kinetic energy, although he did not use the latter term Hutzler et al.

The term is used because the foundations of mechanics, including conservation of momentum and kinetic energy were established by Isaac Newton in his Principia of Baker and Blackburn Pendulum length 18 cm. Apart from their use in clocks the Foucault pendulum is probably the best known use of pendulums. There are examples in 6. A Foucault pendulum usually consists of a spherical steel bob suspended from a long steel wire, with a pointer mounted below the bob so that its motion can be easily followed.

A Foucault pendulum can swing freely in any direction, and is a close approximation to a simple string pendulum Sect. Figure 6. In a Charpy impact testing machine a swinging pendulum is used to fracture notched steel test pieces. Interest in the effects of impacts on metals dates back to the early nineteenth century Siewert et al.

Impact testing of metals developed from the observation that metals are often more brittle under an impact than when loaded slowly Siewert et al. The earliest description of an impact test appears to be that given by Tredgold The pendulum impact testing machine developed by Georges Charpy Charpy is the basis of the Charpy impact testing machines that are now extensively used world wide. Methods 80 6 Scientific Instruments Fig. The general arrangement of a Charpy impact testing machine pendulum is shown in Fig. Despites its importance the Charpy impact test is not usually mentioned by writers on pendulums.

A secondary objective was to check the reliability of pendulum clocks, which were to be used to determine the exact longitude of Cayenne. An unexpected finding was that clocks fitted with seconds pendulums ran more slowly in Cayenne latitude 4. To keep correct time the pendulums had to be shortened by about 2. The implication Eq. If the effective length of a real pendulum, l, and its period, P , are both known accurately, and the pendulum amplitude is small, then the value of g can be calculated using Eq.

The practical difficulty is the determination of the value 6. In other words, if the pendulum is suspended from a second frictionless pivot at the centre of oscillation corresponding to the original frictionless pivot then the pendulum period is the same. The distance between the pivots is therefore the effective length of the pendulum. Knife edge suspensions Sect. Kater made his knife edges as sharp as possible.

However, if they have a small radius it can be shown that the effective length is still the distance between the knife edges Rawlings The general arrangement is shown in Fig. The brass rod is of rectangular section with a heavy bob at the lower end and small adjustable weights near the upper end. The larger upper adjustable weight is clamped to the rod by a set screw and is used for coarse adjustments. The smaller weight is used for fine adjustments and is moved by a screw.

### Expertise. Insights. Illumination.

Despite their apparent simplicity, the behaviour of pendulums can be remarkably complicated. Historically, pendulums for specific purposes have been. Request PDF on ResearchGate | On Jan 1, , L. P. Pook and others published Understanding pendulums. A brief introduction.

Moving a weight towards a suspension decreases the pendulum period when that suspension is used, and increases it when the other suspension is used. The rod pendulum, length L, shown in Fig. The value of P obtained from Eq. As an example Lamb , let P1 D p 0. From Eq. Each pendulum is a real bifilar pendulum version of a bifilar pendulum Sect.

The Pendulum and Galileo

The bob is a hardened steel ball suspended by a pair of strings. The behaviour of a bifilar pendulum Fig. It then has one degree of freedom, and one mode of oscillation in which the point mass, m, moves in an arc in a vertical plane. This is also the primary mode of oscillation of a real bifilar pendulum. The centre of oscillation is close to the centre of mass, and the pendulum quality, Q, Sect. In the real bifilar pendulums the strings are attached at the top of the steel ball a short distance apart, so there are two additional degrees of freedom, and two secondary modes of oscillation are possible.

These are an oscillation in the plane of the strings and a torsional oscillation, The three degrees of freedom, and the three modes of oscillation are analogous to those of the dual string pendulum Sect. A closer analogy for the mode of oscillation in the plane of the strings is the trapezium pendulum Sect. The pendulum quality for the secondary modes of oscillation is low. In the analysis below it is therefore assumed that motions of the individual real bifilar pendulums correspond to motions in the primary mode of oscillation.

In these modes of oscillation interference between the real bifilar pendulums means that overall pendulum quality, Q, is low, even though the quality of individual pendulums is high, and oscillations decay rapidly. The time of swing is approximately the same for all five modes.

It is defined as the time interval for a given configuration to be repeated. To start the first primary mode of oscillation, from the rest position, one ball at an end of the row is pulled back and released. The impulse as it hits the next ball is transmitted along the row, and the ball at the other end of the row flies off, with the other four balls remaining almost stationary. This ball then swings back and hits the row. The oscillation continues with a ball flying off each end of the row alternately.

The central group of three balls remains almost stationary throughout the oscillation. The impulse as the pair of balls hits the next ball is transmitted along the row, and a pair of balls at the other end of the row flies off, still in contact, with the other three balls remaining almost stationary. The pair of balls then swings back and hits the row. The oscillation continues with a pair of balls flying off each end of the row alternately. The central ball remains almost stationary throughout the oscillation. To start the third primary mode of oscillation, from the rest position, a group of three balls at an end of the row, kept in contact, is pulled back and released.

The impulse as the group of three balls hits the next ball is transmitted along the row, and a group of three balls at the other end of the row flies off, still in contact, with the other two balls remaining almost stationary. The group of three balls then swings back and hits the row. The oscillation continues with a group of three balls flying off each end of the row alternately. The central ball swings continuously throughout the oscillation. To start the fourth primary mode of oscillation, from the rest position, a group of four balls at an end of the row, kept in contact, is pulled back and released.

The impulse as the group of four balls hits the single ball is transmitted along the row, and a group of four balls at the other end of the row flies off, still in contact, with a single ball remaining almost stationary. The group of four balls then swings back and hits the single ball. The oscillation continues with a group of four balls flying off each end of the row alternately. The central group of three balls swings continuously throughout the oscillation. To start the fifth primary mode of oscillation all five balls, kept in contact, are pulled back and released.

The five balls then swing in unison in a uniform mode of oscillation in which all five balls stay in contact and move in phase. The first four primary modes of oscillation, in which balls fly off alternately from each end of the row, are called fly off modes of oscillation. These modes are unstable and degenerate steadily into the fifth primary mode of oscillation.

## The Simple Pendulum

The collisions between the balls means that the real bifilar pendulums are coupled pendulums in which energy is transferred between pendulums. Coupled pendulums, with identical or nearly identical periods, have a tendency to become synchronised, as first observed by Huygens in Baker and Blackburn This tendency to synchronisation is the reason why the fly off modes of oscillation degenerate into the fifth primary mode of oscillation.

There are N — 1 fly off modes of oscillation, and there is one uniform mode of oscillation. The first primary mode of oscillation is a fly off mode of oscillation in which one ball is pulled back from the rest position and released. The impulse as the first ball hits the second ball is transmitted to the second ball, which flies of with the first ball remaining almost stationary. The second ball swings back and the oscillation continues. The second primary mode of oscillation is a uniform mode of oscillation in which the two real bifilar pendulums 84 6 Scientific Instruments swing in unison.

Experiments with N up to five shows that degeneration of fly off modes of oscillation into a uniform mode of oscillation occurs increasingly rapidly as N increases. As simplifications it is assumed that the balls move along the same straight line in simple harmonic motion Sect. The assumption that the system is non dissipative includes the implicit assumption that collisions between the balls are perfectly elastic.

That is, all the kinetic energy absorbed during the elastic deformation as the balls collide is returned as kinetic energy as the balls return to their undeformed shape. In physics terminology the coefficient of restitution is one. The point of using hardened steel balls is that collisions are essentially elastic so negligible energy is lost during a collision. If immediately before a collision between the two balls the velocity of the right hand ball, mass m, is zero, then its momentum and kinetic energy are also zero. If at the same time the velocity of the left hand ball, mass M , is V , then its momentum, p, is given by Baker and Blackburn p D MV 6.

During the collision the left hand ball applies an impulse, over a very short period of time, to the right hand ball, and the right hand ball applies an impulse to the left hand ball. These impulses are of the same magnitude but opposite in sign, and have a much more drastic effect than do those of a clock escapement Sect. During an elastic collision some of the kinetic energy is stored as strain energy Gere and Timoshenko , and then returned. At either the North Pole or South Pole, the plane of oscillation of a pendulum remains pointing in the same direction while the Earth rotates underneath it, taking one sidereal day to complete a rotation.

When a Foucault pendulum is suspended somewhere on the equator, then the plane of oscillation of the Foucault pendulum is at all times co-rotating with the rotation of the Earth. What happens at other latitudes is a combination of these two effects. At the equator the equilibrium position of the pendulum is in a direction that is perpendicular to the Earth's axis of rotation.

Because of that, the plane of oscillation is co-rotating with the Earth. Away from the equator the co-rotating with the Earth is diminished. Between the poles and the equator the plane of oscillation is rotating both with respect to the stars and with respect to the Earth. Many people found the sine factor difficult to understand, which prompted Foucault to conceive the gyroscope in