General References for Pendulums and Timekeeping

Michael R. Matthews

School of Education, University of New South Wales

m.matthews@unsw.edu.au

Adas, M. 1989, Machines as the Measure of Men Technology and Ideologies of Western Dominance……., Ithaca, NY.

Ariotti, P.E. 1968, ‘Galileo on the Isochrony of the Pendulum’, Isis 59, 414-426.

Ariotti, P.E. 1972, ‘Aspects of the Conception and Development of the Pendulum in the 17th Century’, Archive for History of the Exact Sciences 8, 329-410.

Ariotti, P.E. 1973, ‘Toward Absolute Time The Undermining and Refutation of the Aristotelian Conception of Time in the 16th and 17th Centuries’, Annals of Science 30, 31-50.

Ariotti, P.E. 1975, ‘The Concept of Time in Western Antiquity’. In J.T. Fraser & N. Lawrence (eds.), The Study of Time II Proceedings of the Second Conference of the International Society for the Study of Time, Springer-Verlag, New York, 1975, pp. 69-80.

Atwood, S.G. 1975, ‘The Development of the Pendulum as a Device for Regulating Clocks Prior to the 18th Century’. In J.T. Fraser & N. Lawrence (eds.), The Study of Time II Proceedings of the Second Conference of the International Society for the Study of Time, Springer-Verlag, New York, 1975, pp. 417-450.

Aveni, A.F. 1989, Empires of Time Calendars, Clocks, and Cultures, Basic Books, New York.

Barnett, J.E. 1998, Time’s Pendulum From Sundials to Atomic Clocks, the Fascinating History of Timekeeping and How Our Discoveries Changed the WorldHarcourt Brace & Co., New York.

Bedini, S.A. & Reti, L. 1974, ‘Horology’. In L. Reti (ed.), The Unknown LeonardoMcGraw-Hill Book Company, New York.

Bedini, S.A. 1963, ‘Galileo Galilei and Time Measurement’, Physis 5, 145-165.

Bedini, S.A. 1963, ‘The Scent of Time’, Transactions of the American Philosophical Society liii.

Bedini, S.A. 1980, ‘The Mechanical Clock and the Industrial Revolution’. In K. Maurice & O. Mayr (eds.), Clockwork Universe. German Clocks and Automata 1550-1650Neale Watson Academic Publications, New York.

Bedini, S.A. 1986, `Galileo and Scientific Instrumentation'. In W.A. Wallace (ed.) Reinterpreting GalileoCatholic University of America Press, Washington, pp.127-154.

Bedini, S.A. 1991, The Pulse of Time Galileo Galilei, the Determination of Longitude, and the Pendulum ClockOlschki, Florence.

Bell, A.E. 1947, Christiaan Huygens and the Development of Science in the Seventeenth Century….., London.

Bell, A.E. 1947, Christiaan Huygens and the Development of Science in the Seventeenth CenturyEdward Arnold, London.

Bell, T.H. & Bell, C. 1963, The Riddle of Time, Viking Press, New York.

Bochner, S. 1966, The Role of Mathematics in the Rise of Science, Princeton University Press, Princeton, NJ.

Bos, H.J.M. et al. (eds.) 1980, Studies on Christiaan Huygens, Swets & Zeitlinger, Lisse.

Bos, H.J.M. 1980, ‘Christiaan Huygens – A Biographical Sketch’. In H.J.M. Bos et al. (eds.), Studies on Christiaan Huygens, Swets & Zeitlinger, Lisse, pp.7-18.

Bos, H.J.M. 1986, ‘Introduction’. In R.J. Blackwell (trans.), Christiaan Huygens’ The Pendulum Clock, Iowa State University Press, Ames, IA.

Boxer, C.R. 1965, The Dutch Seaborne Empire, 1600-1800, ….., London.

Boxer, C.R. 1969, The Portuguese Seaborne Empire, 1415-1825, …., London.

Boxer, C.R. 1984, From Lisbon to Goa, 1500-1750 Studies in Portuguese Maritime Enterprise…., London.

Boxer, C.R. 1988, Dutch Merchants and Mariners in Asia, 1602-1795, …., London.

Brearley, H.C. 1919, Time Telling Through the Ages, Doubleday, Page & Co., New York.

Burke, J.G. (ed.) 1984, The Uses of Science in the Age of Newton, ……, Berkeley, CA.

Burton, E. 1968, Clocks and WatchesHamlyn, Feltham.

Callahan, J.F. 1948, Four Views of Time in Ancient Philosophy, Harvard University Press, Cambridge.

Cambell, J. (ed.) 1957, Man and Time Papers from the Eranos Yearbooks, …., New York.

Capek, M. (ed.) 1976, The Concepts of Space and Time Their Structure and Development, (Boston Studies in the Philosophy of Science xxii), Reidel, Dordrecht.

Capek, M. 1973, ‘Time’, Dictionary of the History of Ideas iv389-398.

Cipolla, C. 1967, Clocks and Culture 1300-1700, Collins, London.

Cousins, F.W. 1969, Sundials A Simplified Approach by Means of the Equatorial Dial, John Baker,

Czudková, L. & Musilová, J. 2000, ‘The Pendulum A Stumbling Block of Secondary School Mechanics’, Physics Education 35(6), 428-435.

Daumas, M. 1972, Scientific Instruments of the Seventeenth and Eighteenth Centuries and their MakersB.T. Batsford, London.

Davies, P. 1994, About Time Einstein’s Unfinished Revolution, Penguin, London.

Dobson, R.D. 1985, ‘Galileo Galilei and Christiaan Huygens’, Antiquarian Horology 15, 261-270.

Dougherty, J.P. 1991, ‘Abstraction and Imagination in Human Understanding’. In D. O. Dahlstrom (ed.), Nature and Scientific Method, Catholic University of America Press, Washington, DC., 51-62.

Drake, S. 1974, ‘Mathematics and Discovery in Galileo’s Physics’, Historia Mathematica 1, 129-150.

Drake, S. 1990, ‘The Laws of Pendulum and Fall’. In his Galileo Pioneer ScientistUniversity of Toronto Press, Toronto, pp.9-31.

Edwardes, E.L. 1977, The Story of the Pendulum Clock, J. Sherratt, Altrincham.

Fraser, J.T. & Lawrence, N. (eds.) 1975, The Study of Time II, New York.

Fraser, J.T. (ed.) 1966, The Voices of Time, ….., New York.

Fraser, J.T. (ed.) 1981, The Voices of Time A Cooperative Survey of Man’s Views of Time as Expressed by the Sciences and the Humanities, 2nd edit., University of Massachusetts Press, Amherst.

Fraser, J.T. (ed.) 1981, The Voices of Time A Cooperative Survey of Man’s Views of Time as Expressed by the Sciences and the Humanities, 2nd edit., University of Massachusetts Press, Amherst.

Fraser, J.T., Lawrence, N., & Park, D. (eds.) 1978, The Study of Time III, …, New York.

Fraser, J.T., Lawrence, N., & Park, D. (eds.) 1981, The Study of Time IV, ….., New York.

Fraser, J.T. 1982, The Genesis and Evolution of Time, The Harvester Press, Brighton.

Fraser, J.T. 1987, Time the Familiar Stranger, Microsoft Press, Redmond, WA

Galison, P. 2003, Einstein’s Clocks, Poincaré’s Maps Empires of TimeW.W. Norton & Co., New York.

Gaukroger, S. 1981, ‘Aristotle on the Function of Sense Perception’, Studies in History and Philosophy of Science 1275-89.

Gauld, C. 1993, ‘The Historical Context of Newton's Third Law and the Teaching of Mechanics, Research in Science Education, 23, 95-103.

Gauld, C. 1994, Newton's Third Law after Newton, Research in Science Education, 24, 93-101.

Gauld, C.F. 1991, `History of Science, Individual Development and Science Teaching', Research in Science Education 21, 133-140.

Gauld, C.F. 1998, ‘Colliding Pendulums, Conservation of Momentum and Newton's Third Law’, Australian Science Teachers Journal 44(3), 37-38.

Gauld, C.F. 1998, Making more plausible what is hard to believe Historical justifications and illustrations of Newton's third law, Science & Education, 7(2), 159-172.

Gauld, C.F. 1998, Solutions to the problem of impact in the 17th and 18th centuries and teaching Newton's third law today. Science & Education, 7(1), 49-67.

Gauld, C.F. 1999, ‘Using Colliding Pendulums to Teach Newton's Third Law’, The Physics Teacher, 37, 25-28.

Gimbel, J. 1992, The Medieval Machine The Industrial Revolution of the Middle Ages, Pimplico, London.

Goudsmit, S.A. & Clariborne, R. (eds.) 1966, Time, Time-Life Books,

Gould, R. T. 1923, The Marine Chronometer, Its History and Development, J.D. Potter, London.

Hackmann, W.D. 1979, ‘The Relationship Between Concept and Instrument Design in Eighteenth-Century Experimental Science’, Annals of Science 36, 205-224.

Hall, A.R. 1950, ‘Robert Hooke and Horology’, Notes and Records of the Royal Society of London 8, 167-177.

Hall, A.R. 1957, ‘Newton on the Calculation of Central Forces’, Annals of Science 13, 62-71.

Hall, A.R. 1963, From Galileo to Newton 1630-1720, Harper & Row, New York. (derivation of pendulum laws from false premises)

Hall, B.S. 1978, ‘The Scholastic Pendulum’, Annals of Science 35, 441-462.

Hawking, S.W. 1988, A Brief History of Time, Bantam Books, London.

Heard, P.F., Divall, S.A. & Johnson, S.D. 2000, ‘Can “Ears-On” Help Hands-On Science Learning – for Girls and Boys?’, International Journal of Science Education 22(11), 1133-1146.

Howse, D. 1980, Greenwich Time and the Discovery of Longitude, Oxford University Press, Oxford.

Huygens, C. 1673/1986, Horologium Oscillatorium. The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Fpendula as Applied to Clocks, R.J. Blackwell, trans., Iowa State University Press, Ames.

Janich, P. 1985, Protophysics of Time Constructivie Foundation and History of Time MeasurementReidel, Boston.

Jespersen, J. & Fitz-Randolph, J. 1977, From Sundials to Atomic Clocks, National Bureau of Standards, Washington. (Reprinted 1982, Dover Publications, New York.)

Kassler, J.C. 1982, ‘Music as Model in Early Science’, History of Science XX, 103-131.

Kline, M. 1959, Mathematics and the Physical World, Dover Publications, New York.

Koyré, A. 1953, ‘An Experiment in Measurement’, Proceedings of the American Philosophical Society 7222-237. Reproduced in his Metaphysics and Measurement (1968).

Landes, D.S. 1983, Revolution in Time. Clocks and the Making of the Modern World, Harvard University Press, Cambridge, MA.

Laudan, L. 1981, ‘The Clock Metaphor and Hypotheses The Impact of Descartes on English Methodological Thought’. In his Science and Hypothesis, Reidel, Dordrecht, pp.27-58.

Lloyd, H.A. 1957, ‘Mechanical Timekeepers’. In C.J. Singer, E.J. Holmyard, A.R. Hall & T.I. Williams (eds.), History of Technology (5 vols.) 3, 648-675.

Lloyd, H.A. 1970, Old ClocksDover Publications, New York.

Macey, S.L. (ed.) 1994, Encyclopedia of Time, ….., Hamden, CT.

Macey, S.L. 1980, Clocks and Cosmos Time in Western Life and Thought, …., Hamden, CT.

Macey, S.L. 1991, Time A Bibliographical Guide,….., Hamden, CT.

MacLachlan, J. 1973, `A Test of an "Imaginery Experiment" of Galileo's', Isis 64, 374-379.

MacLachlan, J. 1976, ‘Galileo’s Experiments with Pendulums Real and Imaginary’, Annals of Science 33173-185.

Martins, R. de A.1993, ‘Huygens’s Reaction to Newton’s Gravitational Theory’. In J.V. Field & F.A.J.L. James (ed.), Renaissance and Revolution Humanists, Scholars, Craftsmen and Natural Philosophers in Early Modern EuropeCambridge University Press, Cambridge, pp. 203-214.

Mathew, K.M. 1988, History of the Portuguese Navigation in India (1497-1600), Mittal Publications, Delhi.

Matthews, M.R. (ed.) 1989b, The Scientific Background to Modern Philosophy, Hackett Publishing Company, Indianapolis.

Matthews, M.R. 1987a, `Galileo's Pendulum & the Objects of Science', in B.& D.Arnstine (eds.) Philosophy of Education, pp. 309-319, Philosophy of Education Society, PLACE???.

Matthews, M.R. 1987b, `Experiment as the Objectification of Theory Galileo's Revolution'. In J. Novak (ed.) Misconceptions and Educational Strategies in Science and MathematicsCornell University, Ithaca, pp.289-299.

Matthews, M.R. 1989a, ‘Galileo and Pendulum Motion A Case for History and Philosophy in the Science Classroom’, Research in Science Education 19, 187-197.

Matthews, M.R. 1992, ‘History, Philosophy and Science Teaching The Present Rapprochement’, Science & Education 1(1), 11-48.

Matthews, M.R. 1994a, Science Teaching The Role of History and Philosophy of Science, Routledge, New York.

May, W.E. 1973, A History of Marine Navigation, Norton, New York.

May, W.E. 1976, ‘How the Chronometer Went to Sea’, Antiquarian HorologyMarch, 638-663.

McCloskey, M. & Kargon, R. 1988, ‘The Meaning and Use of Historical Models in the Study of Intuitive Physics’. In S. Strauss (ed.), Ontology, Phylogeny and Historical Development, Ablex Publishing Corporation, Norwood, NJ., pp. 49-67.

McMullin, E. 1985, ‘Galilean Idealization’, Studies in the History and Philosophy of Science 16, 347-373.

Mecke, G. & Mecke, V. 1971, ‘The Development of Formal Thought as Shown by Explanations of the Oscillations of a Pendulum A Replication Study’, Adolescence 6(22), 219-228.

Milham, W.I. 1945, Time & Timekeepers. Including the History, Construction, Care, and Accuracy of Clocks and Watches, Macmillan, London.

Modrak, D.K.W. 1987, Aristotle The Power of Perception, …., Chicago.

Molland, A.G. 1989, ‘Aristotelian Holism and Medieval Mathematical Physics’. In S. Caroti (ed.), Studies in Medieval Natural Philosophy, ….., Florence, pp.227-235.

Morison, S.E. 1942, Admiral of the Ocean Sea A Life of Christopher Columbus, Little Brown, Boston.

Naylor, R.H. 1974a, ‘Galileo's Simple Pendulum’, Physis 16, 23-46.

Naylor, R.H. 1974b, `The Evolution of an Experiment Guidobaldo del Monte and Galileo's Discourse Demonstration of the Parabolic Trajectory', Physis 16323-348.

Naylor, R.H. 1974c, `Galileo and the Problem of Free Fall', British Journal for the History of Science 7, 105-134.

Naylor, R.H. 1976, ‘Galileo Real Experiment and Didactic Experiment’, Isis 67(238), 398-419.

Naylor, R.H. 1976, ‘Galileo Search for the Parabolic Trajectory’, Annals of Science 33, 153-174.

Naylor, R.H. 1977, `Galileo's Theory of Motion Processes of Conceptual Change in the Period 1604-10', Annals of Science 34365-392.

Naylor, R.H. 1980a, `The Role of Experiment in Galileo's Early Work on the Law of Fall', Annals of Science 37, 363-378.

Naylor, R.H. 1980b, `Galileo's Theory of Projectile Motion', Isis 71550-570.

Naylor, R.H. 1989, `Galileo's Experimental Discourse', in D. Gooding et al (eds.) The Uses of ExperimentCambridge University Press, Cambridge.

Naylor, R.H. 1990, ‘Galileo’s Method of Analysis and Synthesis’, Isis 81, 695-707.

Needham, J., Ling, W., & Solla Price, D.J. de 1960, Heavenly Clockwork, Cambridge University Press, Cambridge.

Nersessian, N.J. 1989, ‘Conceptual Change in Science and in Science Education’, Synthese 80(1), 163-184. In M.R. Matthews (ed.) History, Philosophy and Science Teaching Selected Readings, OISE Press, Toronto.

Newton, I. 1729/1934, Mathematical Principles of Mathematical Philosophy, (translated A. Motte, revised F. Cajori), University of California Press, Berkeley.

Newton, R.G. 2004, Galileo’s Pendulum From the Rhythm of Time to the Making of MatterHarvard University Press, Cambridge MA.

North, J.D. 1975, ‘Monasticism and the First Mechanical Clocks’. In Fraser, J.T. & Lawrence, N. (eds.) 1975, The Study of Time, New York.

Palisca, C.V. 1961, ‘Scientific Empiricism in Musical Thought’. In Rhys, H.H. (ed.) 1961, Seventeenth Century Science and the Arts, Princeton University Press, Princeton, pp.91-137.

Parker, J. (ed.) 1965, Merchants and Scholars Essays in the History of Exploration and Trade, …., Minneapolis, MN.

Patterson, L.D. 1952, ‘Pendulums of Wren and Hooke’, Osiris X, 277-321.

Petry, M.J. 1993, ‘Classifying the Motion Hegel on the Pendulum’. In M.J. Petry (ed.) Hegel and Newtonism, Kluwer Academic Publishers, Dordrecht, pp.291-316.

Piaget, J. 1970, The Child’s Conception of Time, Basic Books, New York. (orig. 1946)

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Price, D.J. de S. 1975, ‘Clockwork before the Clock and Timekeepers before Timekeeping’. In J.T. Fraser & N. Lawrence (eds.), The Study of Time II, ….., Berlin, pp. 367-380.

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Quinones, R.J. 1972, The Renaissance Discovery of Time, Harvard University Press, Cambridge.

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Robertson, J. Drummond 1931, The Evolution of Clockwork, Cassell, London. [chap. 5 on Galileo vs Huygens, chaps.6, 7 on Galileo and pendulum clock]

Rossum, G. D-V. 1996, History of the Hour Clocks and Modern Temporal Orders, Chicago University Press, Chicago.

Sarlemijn, A. 1993, ‘Pendulums in Newtonian Mechanics’. In M.J. Petry (ed.) Hegel and NewtonismKluwer Academic Publishers, Dordrecht, pp.267-290.

Schwarz, C. 1995, ‘The Not-So-Simple Pendulum’, The Physics Teacher 33(4), 225-228.

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Pendulum References in Physics Education

Colin Gauld *

The simplest type of pendulum (the simple pendulum) consists of a spherical ball suspended from a thin string so that the ball can move backwards and forwards in one plane along a path which is a portion of a circle. As Matthews (2000) has pointed out this device has been central to the early beginnings of modern science. The physical pendulum (also known as the compound pendulum) consists of a solid object pivoted about a fixed point and the motions of both the simple and the physical pendulum are described by the same relationship, namely,

where τ is the torque acting on the system, I is the moment of inertia of the object about the pivot point, m is the mass of the object, d is the length of the line between the centre of mass of the object and the pivot point and θ is the angle between this line and the vertical. In an ideal pendulum, for which the amplitude, θ₀, is constant, the periodic time is constant and depends the length, d, the value of g and the size of θ₀. In a real pendulum the amplitude decreases with time as energy is expended in the system so that the motion is damped. For small values of θ₀, sinθ » θ, and the pendulum executes simple harmonic motion with a period, T = 2πÖ(I/mgd), that is independent of θ₀. For a simple pendulum, d = l and I = ml², so that T = 2πÖ(l/g).

Kater’s pendulum is a physical pendulum which has two points of support allowing it to be suspended upside down. When the position of a mass is changed on the pendulum rod or the position of one of the points is adjusted so that the period in both positions is the same, the length of the equivalent simple pendulum is equal to the distance between the supports.

If a pendulum is constructed so that the line between the point of support and the centre of mass when at rest is not vertical but at an angle φ to the vertical the effective value of g is reduced and the period of oscillation is T = 2πÖ(l/gcosφ). Escriche’s pendulum is one example of a variable gravity pendulum.

Huygens (Matthews, 2000) and Newton (1729/1960, Propositions 48-52) showed that if a pendulum moved along a cycloidal rather than a circular path its period was independent of its amplitude for all values of θ₀. The cycloidal pendulum was thus shown to be an ideally suited device to regulate the mechanism of a time-keeping instrument.

If the supporting string of a simple pendulum can stretch elastically the pendulum is called in the literature an elastic pendulum

The bifilar pendulum is a simple or physical pendulum which is suspended from two points so that it is constrained to oscillate about a fixed horizontal axis through the line joining the two points of suspension rather than having the freedom to move in any direction. In “Newton’s cradle” a number of bifilar pendulums hang in a row so that their steel bobs just touch one another. An extended bob, supported in the above manner by two strings attached to different points on the bob, as well as oscillating about a fixed horizontal axis, can also oscillate about a vertical axis.

In the most common version of the ballistic pendulum a large block of wood is suspended by four strings so that it remains horizontal when it swings. It is used to determine the speed of a projectile fired into the block by measuring the height the block rises and using the law of conservation of momentum during the collision and the law of conservation of mechanical energy after the block has begun to move (Resnick & Halliday, 1966, pp.219-220; Taylor, 1941, pp.205-206). A more recent version, the Blackwood pendulum, consists of a solid rod which is suspended at its top (Blackwood, 1973, p.104). The bob, at the bottom end of the rod, is a cage into which a spherical metal ball is fired and captured causing the pendulum to swing upwards. When it stops, a ratchet holds the bob in its highest position so that its height can be measured. This information provides the data from which the initial speed of the ball can be calculated.

The ring pendulum is a physical pendulum in which the object is a ring which is suspended from a point on the inside surface.

Because a simple pendulum is free to move in any direction the two-dimensional pendulum can be used to study the nature of the motion of the bob in two dimensions. In the conical pendulum the pendulum bob moves in a horizontal circle so that the string is at a fixed angle to the vertical. The period of the conical pendulum is T = 2πÖ(h/g) where h is the distance of the point of suspension above the plane in which the bob moves.

The Blackburn pendulum is a bifilar pendulum in which the lower part is another pendulum supported by a single string attached to the V formed by the upper pair of strings. The pendulum as a whole is constrained to move in one plane while the lower portion can move at right angles to this plane. The motion of the bob thus consists of two perpendicular oscillatory motions.

If the material which is used to connect the pendulum bob to the point of suspension is solid and elastic the pendulum can be turned upside down so that the fixed point is at the bottom and the bob moves from side to side above it. This is called an inverted pendulum. If the connecting rod is rigid the inverted pendulum is unstable but stability can be achieved by driving it either vertically or horizontally from the bottom.

A double pendulum often consists of two physical pendulums with one usually but not always being suspended from the bottom of the other. It can also consist of two simple pendulums with one suspended from the bob of the other.

Pendulums can be arranged so that they are joined near the tops of the strings by a horizontal string. The motion of one pendulum is then communicated by this string to the pendulum to which it is linked. Such pendulums are called coupled pendulums.

The name “pendulum” is also used for other systems in which the potential energy is not always (or not totally) gravitational. In the spring-mass pendulum the bob is suspended by a spring so that it can move up and down. In this case the kinetic energy which the system has in its equilibrium position is converted into gravitational and elastic potential energy as the spring stretches or compresses. Ignoring the mass of the spring, the period of such a pendulum is T = 2πÖ(m/κ) where κ is the spring constant.

The torsion pendulum also consists of a mass suspended by a spring but in this case the potential energy is stored in the spring as it twists. Changes in gravitational potential energy are of minor consideration. The period of this pendulum is given by T = 2πÖ(I/k) where k is the torsion constant. However, as the spring twists so that the coils close up the length of the spring decreases a little and when the spring unwinds its length becomes a little greater. If the spring constant and the moment of inertia of the object are carefully chosen the period of the torsional motion and the period of the natural vertical motion (as a spring-mass pendulum) can be made equal. When this is the case the torsional motion slowly decreases and the vertical oscillations gradually increase until all the original rotational kinetic energy appears as kinetic energy of vertical motion and torsional motion ceases. In this mode, as the spring stretches it also unwinds a little and when it compresses it winds up a little. Gradually, the vertical motion decreases and the torsional motion increases again. This type of pendulum is called a Wilberforce pendulum and demonstrates the conservation of energy as the mode of vibration changes.

When it is pulled aside and released a pendulum will fall to its lowest position along a plane which contains the initial line of the string and the line of the string when it is vertical. If no other external influences than gravity act the pendulum will continue to swing in this plane until it stops. However, if the frame to which the point of support is attached is rotated a torque can be exerted on the string and the plane of oscillation may rotate about a vertical axis with the rotation of the frame. On the other hand, if the point of support is carefully designed to prevent this torque from acting the initial plane of oscillation can be maintained even as the frame rotates. This property was used by Foucault to show that the Earth itself was rotating.

PENDULUM CONTEXTS

In teaching physics/science the pendulum has been used both as a device to be studied and as a tool for finding out other things. For example, in the Principia Newton (1729/1960) presented the theory which lies behind the motion of the pendulum and also used it as a means of measuring the velocities of balls before and after they collided.

The simple pendulum is a physical system which is easy to make and to study and it is often used to teach investigative skills and skills of measurement. Its role in timekeeping is also something which students can explore.

Two everyday systems which can be modelled by pendulum motion are walking and swinging

and both have been extensively discussed in the physics education literature. If one considers the leg as a simple or double pendulum (with a second pivot at the knee) then the most comfortable leg movement is related to the natural period of this pendulum. One of the implications of the law of conservation of momentum is that forces within a system are unable to change the total momentum of the system. This raises the question of how the kinetic energy of a child’s swing can be increased by a person sitting on the swing and the subsequent discussion of this issue is most illuminating (see also Walker, 1977, pp.37-38).

In the 17^th and 18^th centuries “laws of motion” referred to the laws which governed elastic and inelastic collisions between two bodies and the laws enabled predictions to be made about the outcomes of different types of collisions (Gauld, 1998). Colliding pendulums were widely used to measure the velocities before and after the collisions to check the predictions. Today they can be very effective in demonstrating the law of conservation of momentum in a dynamic rather than a static context. “Newton’s cradle” consists of a series of colliding bifilar pendulums.

Coupled pendulums in which the motion of one pendulum influences the motion of a nearby pendulum can be used to demonstrate resonance. If one pendulum in a pair of equal-length, coupled pendulums is set in motion the second will begin to move and the first will begin to slow down. This continues until the second is moving with the same amplitude with which the first began and the first one has stopped. The total energy is transferred from one pendulum to the other and back again. Driven pendulums also demonstrate resonance at particular frequencies.

Chaotic motion can be demonstrated using a multiply-connected pendulum or a pendulum in which the point of suspension is driven backwards and forwards at different frequencies.

The importance of the pendulum in Galileo’s thinking has also been discussed in the physics education literature.

Since Piaget’s famous studies of adolescent thinking published in 1958 student conceptions of the pendulum and their explanations for its motion have been of interest to physics teachers and others. More recently error analysis of student responses to questions about the pendulum provide some idea of the pre-conceptions which students have when they first begin to learn physics.

MISCELLANEOUS

In the miscellaneous category are articles in which other phenomena than pendulum motion is the focus of attention. There are also articles which cover a wide variety of pendulum types.

REFERENCES

Blackwood, O.H., Kelly, W.C. & Bell, R.M.: 1973, General Physics, 4^th edition, Wiley, New York.

Gauld, C.: 1998, ‘Solutions to the Problem of Impact in the 17th and 18th Centuries and Teaching Newton’s Third Law Today’, Science & Education 7(1), 49-67.

Matthews, M.:2000, Time for Science Education, Kluwer/Plenum, New York.

Newton, I.: 1729/1960, The Mathematical Principles of Natural Philosophy, (translated A. Motte, 1729; revised F. Cajori, 1934), University of California Press, Berkeley, CA.

Piaget, J.: 1958, The Growth of Logical Thinking, Routledge & Kegan Paul, London.

Resnick, R. & Halliday, D.: 1966, Physics, Wiley, New York.

Taylor, L.: 1941, Physics: The Pioneer Science, Houghton Mifflin, Boston.

Walker, J.: 1977, The Flying Circus of Physics with Answers, Wiley, New York.

ENDNOTE

*  Paul McColl, a physics teacher from Bundoora Secondary College in Victoria and a doctoral student in science education at Monash University, assisted with the collection of the bibliographical material below and I owe a debt of gratitude to him for his contribution to this project.

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BIBLIOGRAPHY

Types of pendulums

Ballistic pendulum

Alt, R.L.: 1940, ‘A Corrupted Ballistic Pendulum’, American Journal of Physics 40(11), 1688-1689.

Barnes, G.: 1957, ‘Addition to the Ballistic Pendulum Experiment’, American Journal of Physics 25(7), 452-453.

Barton, R.W.: 1964, ‘A Versatile Ballistic Pendulum’, American Journal of Physics 32(3), 229-232.

Bayliss, L.T. & Ffolliott, C.F.: 1968, ‘Using a Blowgun with the Ballistic Pendulum’, American Journal of Physics 36(6), 558-559.

Christensen, F.E.: 1968, ‘Beck Ball Pendulum’, American Journal of Physics 36(9), 851.

Gupta, P.: 1985, ‘Blackwood Pendulum Experiment and the Conservation of Linear Momentum’, American Journal of Physics 53(3), 267-269.

Ivey, D.G.: 1956, ‘Modification of the Ballistic Pendulum Experiment’, American Journal of Physics 24(6), 459-460.

McCaslin, J.G.: 1984, ‘A Different Blackwood Pendulum Experiment’, The Physics Teacher 22(3), 184-186.

Peterson, F.C.: 1983, ‘Timing the Flight of the Projectile in the Classical Ballistic Pendulum Experiment’, American Journal of Physics 51(7), 602-604.

Sachs, A.: 1976, ‘Blackwood Pendulum Experiment Revisited’, American Journal of Physics 44(2), 182-183.

Sandin, T.R.: 1941, ‘Nonconservation of Linear Momentum in Ballistic Pendulums’, American Journal of Physics 41(3), 426-427.

Scheie, C.: 1973, ‘Ballistic Pendulum’, The Physics Teacher 11(7), 426-427.

Schramm, R.W.: 1962, ‘An Improved Ballistic Pendulum’, American Journal of Physics 30(5), 386-387.

Stoylov, S.P., Nsanzabera, J.C. & Karenzi, P.C.: 1972, ‘A Demonstration of Momentum Conservation Using Bow, Arrow and Ballistic Pendulum’, American Journal of Physics 40(3), 430-432.

Strnad, J.: 1970, ‘Trouble with the Ballistic Pendulum’, American Journal of Physics 38(4), 532-534.

Wagner, W.: 1985, ‘The Spring Gun Ballistic Pendulum: An Alternative Method for Finding the Initial Velocity’, American Journal of Physics 53(11), 1114-1115.

Weltin, H.: 1963, ‘Vertical Ballistic Pendulum Apparatus’, American Journal of Physics 31(9), 719-722.

Wicher, E.: 1977, ‘Ballistics Pendulum’, American Journal of Physics 45(7), 681-682.

Bifilar pendulum

Cromer, A.: 1995, ‘Many Oscillations of a Rigid Rod’, American Journal of Physics 63(1), 112-121.

Quist, G.M.: 1983, ‘The PET and the Pendulum: An Application of Microcomputers in the Undergraduate Laboratory’, American Journal of Physics 51(2), 145-149.

Schery, S.D.: 1976, ‘Design of an Inexpensive Pendulum for Study of Large-angle Motion’, American Journal of Physics 44(7), 666-670.

Sutton, R.S.: 1953, ‘An Experimental Encounter with Bifilar Pendula’, American Journal of Physics 21(2), 408.

Then, J.W.: 1965, ‘Bifilar Pendulum - An Experimental Study for the Advanced Laboratory’, American Journal of Physics 33(7), 545-547.

Then, J.W. & Chiang, K-R.: 1970, ‘Experimental Determination of Moments of Inertia by the Bifilar Pendulum Method’, American Journal of Physics 38(4), 537-539.

Blackburn pendulum

Bayman, B.F. & Thayer, D.: 1969, ‘A Rotating Two-dimensional Harmonic Oscillator’, American Journal of Physics 37(8), 841-842.

Case, W.: 1980, ‘Parametric Instability: An Elementary Demonstration and Discussion’, American Journal of Physics 48(3), 218-221.

Crowell, A.D.: 1981, ‘Motion of the Earth as Viewed from the Moon, and the Y-suspended Pendulum, American Journal of Physics, 49(5), 452-454.

Fox, J.W.: 1958, ‘Experiments with Modified Form of Simple Pendulum’, American Journal of Physics 26(8), 559-560.

Whitaker, R.: 1991, ‘A Note on the Blackburn Pendulum’, American Journal of Physics 59(4), 330-333.

Conical pendulum

Anon: 1963, ‘The Conical Pendulum’, The Physics Teacher 1(5), 238-239.

Hilton, W.A.: 1963, ‘Another Version of the Conical Pendulum’, American Journal of Physics 31(1), 58-59.

Moses, T. & Adolphi, N.L.: 1998, ‘A New Twist for the Conical Pendulum’, The Physics Teacher 36(6), 372-373.

Richards, J.A.: 1956, ‘Conical Pendulum’, American Journal of Physics 24(9), 632.

Saitoh, A.: 1986, ‘Winding Motion’, Physics Education 21(2, 98-102.

Verwiche, F.: 1964, ‘The Conical Pendulum Paradox’, The Physics Teacher 3(5), 238.

Coupled Pendulums

Blair, J.M.: 1971, ‘Laboratory Experiments Involving Two-mode Analysis of Coupled Oscillations’, American Journal of Physics 39(5), 555-557.

McKibben, J.L.: 1977, ‘Triple Pendulum as an Analog to Three Coupled Stationary States’, American Journal of Physics 45(11), 1022-1026.

Moloney, M.J.: 1978, ‘String-coupled Pendulum Oscillators: Theory and Experiment’, American Journal of Physics 46(12), 1245-1246.

Priest, J. & Poth, J.: 1982, ‘Teaching Physics with Coupled Pendulums’, The Physics Teacher 20(2), 80-85.

Damped pendulum

Allen, M. & Saxl, E.J.: 1972, ‘The Period of Damped Simple Harmonic Motion’, American Journal of Physics 40(7), 942-944.

Basano, L. & Ottonello, P.: 1991, ‘Digital Damping: The Single-oscillation Approach’, American Journal of Physics 59(11), 1018-1023.

Benham, T.A.: 1947, ‘Bessel Functions in Physics: Theory’, American Journal of Physics 15(4), 285-294.

Boving, R., Hellemans, J. & de Wilde, R.: 1983, ‘Teaching Damped and Forced Oscillations in the Student Laboratory’, Physics Education 18(6), 275-276.

Crawford, F.S.: 1975, ‘Damping of a Simple Pendulum’, American Journal of Physics 43(3), 276-277.

McInerney, M.: 1985, ‘Computer-aided Experiments with the Damped Harmonic Oscillator’, American Journal of Physics 53(10), 991-996.

Permann, D. & Hamilton, I.: 1992, ‘Self-similar and Erratic Transient Dynamics for the Linearly Damped Simple Pendulum’, American Journal of Physics 60(5), 442-450.

Squire, P.: 1986, ‘Pendulum Damping’, American Journal of Physics 54(11), 984-991.

Zonetti, L., Camago, A., Sartori, J., de Sousa, D., and Nunes, L. 1999, ‘A Demonstration of Dry and Viscous Damping of an Oscillating Pendulum’, European Journal of Physics, 20(2), 85-88.

Double pendulum

Bender, P.: 1985, ‘A fascinating Resonant Double Pendulum’, American Journal of Physics 53(11), 1114.

Bueche, F. & Pavelka, C.: 1964, ‘An Undergraduate Laboratory Experiment for Studying the Motion of Coupled Mechanical Systems’, American Journal of Physics 32(3), 226-228.

Lee, S.M.: 1970, ‘The Double-Simple Pendulum Problem’, American Journal of Physics 38(4), 536-537.

Levien, R. & Tan, S.: 1993, ‘Double Pendulum: An Experiment in Chaos’, American Journal of Physics 61(11), 1038-1044.

Romer, R.H.: 1970, ‘A Double Pendulum “Art Machine”’, American Journal of Physics 38(9), 1116-1121.

Satterley, J.: 1950, ‘Some Experiments in Dynamics, Chiefly on Vibrations’, American Journal of Physics 18(7), 405-416.

Shinbrot, T., Grebogi, C., Wisdom, J. & Yorke, J.: 1992, ‘Chaos in a Double Pendulum’, American Journal of Physics 60(6), 491-499.

Elastic pendulum

Anicin, B., Davidovic, D. and Babovic, V. 1993, ‘On the Linear Theory of the Elastic Pendulum’, European Journal of Physics, 14(3), 132-135.

Carretero-Gonzalez, R., Numez-Yepez, H.& Salas-Brito, A.: 1994, ‘Regular and Chaotic Behavior in an Extensible Pendulum’, European Journal of Physics, 15(3), 139-148.

Cayton, T.E.: 1975, ‘The Laboratory Spring-mass Oscillator: An Example of Parametric Instability’, American Journal of Physics 45(8), 723-732.

Cuerno, R., Rañada, A. & Ruiz-Lorenzo, J.: 1992, ‘Deterministic Chaos in the Elastic Pendulum: A Simple Laboratory for Nonlinear Dynamics’, American Journal of Physics 60(1), 73-79.

Davidovic, D., Anacin, B. & Babovic, V.: 1996, ‘The Libration Limits of the Elastic Pendulum’, American Journal of Physics 64(3), 338-342.

Dobrovolskis, A.: 1941, ‘Rubber Band Pendulum’, American Journal of Physics 41(9), 1103-1106.

Foucault pendulum

Brown, W.A.: 1961, ‘Suspension for Foucault Pendulum’, American Journal of Physics 29(9), 646.

Crane, H.R.: 1981, ‘Short Foucault Pendulum: A Way to Eliminate Precession Due to Ellipticity’, American Journal of Physics 49(11), 1004-1006.

Crane, H.R.: 1990, ‘The Foucault Pendulum as a Murder Weapon and a Physicist’s Delight’, The Physics Teacher 28(5), 264-269.

Curott, D.R.: 1972, ‘The Role of the Constraining Force in a Foucault Pendulum’, American Journal of Physics 40(7), 1007-1009.

French, A.P.: 1978, ‘The Foucault Pendulum’, The Physics Teacher 16(1), 61-62.

Hart, J., Miller, R. & Mills, R.: 1987, ‘A Simple Geometric Model for Visualizing the Motion of a Foucault Pendulum’, American Journal of Physics 55(1), 67-70.

Hecht, K.T.: 1983, ‘The Crane Foucault Pendulum: An Exercise in Action-angle Variable Perturbation Theory’, American Journal of Physics 51(2), 110-114.

Hilton, W.A.: 1978, ‘The Foucault Pendulum: A Corridor Demonstration’, American Journal of Physics 46(4), 436-438.

Horne, J.E.: 1996, ‘Classroom Foucault Pendulum’, The Physics Teacher 34(4), 238-239.

Kruglak, H.: 1983, ‘A Very Short, Portable Foucault Pendulum’, The Physics Teacher 21(7), 477-479.

Kruglak, H., Oppliger, L., Pittet, R. & Steele, S.: 1978, ‘A Short Foucault Pendulum for a Hallway Exhibit’, American Journal of Physics 46(4), 438-440.

Kruglak, H. & Pittet, R.: 1980, ‘Portable, Continuously Operating Foucault Pendulum’, American Journal of Physics 48(5), 419-420.

Kruglak, H. & Steele, S.: 1984, ‘A 25cm Continuously Operating Foucault Pendulum’, Physics Education 19(6), 294-296.

Kimball, W.S.: 1945, ‘Foucault Pendulum Starpath and the N-leaved Rose’, American Journal of Physics 13(5), 271-277.

Leonard, B.E.: 1981, ‘A Short Foucault Pendulum for Corridor Display’, The Physics Teacher 19(6), 421-423.

Mackay, R.S.: 1953, ‘Sustained Foucault Pendulums’, American Journal of Physics 21(3), 180-183.

Mattila, J.O.: 1991, ‘The Foucault Pendulum as a Teaching Aid’, Physics Education 26(2), 120-123.

McClatchey, S. & Flint, N.: 1981, ‘A Sustained Demonstration Foucault Pendulum’, The Physics Teacher 19(2), 134.

Miller, D. & Caudill, G.W.: 1966, ‘Driving Mechanism for a Foucault Pendulum’, American Journal of Physics 34(7), 615-616.

Noble, W.J.: 1952, ‘Direct Treatment of the Foucault Pendulum’, American Journal of Physics 20(6), 334-336.

Opat, G.: 1991, ‘The Precession of a Foucault Pendulum Viewed as a Beat Phenomenon of a Conical Pendulum Subject to a Coriolis Force’, American Journal of Physics 59(9), 822-823.

Reynhardt, E., van der Walt, T. & Soskolsky, L.: 1986, ‘A Modified Foucault Pendulum for a Corridor Exhibit’, American Journal of Physics 54(8), 759-761.

Romano, J.D.: 1997, ‘Foucault’s Pendulum as a Spirograph’, The Physics Teacher 35(3), 182-183.

Schulz-Dubois, E.O.: 1970, ‘Foucault Pendulum Experiment by Kammerlingh Onnes and Degenerate Perturbation Theory’, American Journal of Physics 38(2), 173-188.

Weltner, K.: 1979, ‘A New Model of the Foucault Pendulum’, American Journal of Physics 47(4), 365-366.

Inverted pendulum

Alessi, N. Fischer, C. & Gray, C.: 1992, ‘Measurement of Amplitude Jumps and Hysteresis in a Driven Inverted Pendulum’, American Journal of Physics 60(8), 755-756.

Blackburn, J., Smith, H. & Grønbich-Jensen, N.: 1992, ‘Stability amd Hopf Bifurcations in an Inverted Pendulum’, American Journal of Physics 60(10), 903-908.

Blitzer, L.: 1965, ‘Inverted Pendulum’, American Journal of Physics 33(12), 1076-1078.

Butikov, E.I.: 2001, ‘On the Dynamic Stabilization of an Inverted Pendulum’, American Journal of Physics 69(7), 755-768.

Duchesne, B., Fischer, C., Gray, C. & Jeffrey, K.: 1991, ‘Chaos in the Motion of an Inverted Pendulum: An Undergraduate Laboratory Experiment’, American Journal of Physics 59(11), 987-992.

Fenn, J., Bayne, D. & Sinclair, B.: 1998, ‘Experimental Investigation of the ‘Effective Potential’ of an Inverted Pendulum’, American Journal of Physics 66(11), 981-984.

Friedman, M.H., Campana, J.E., Kelner, L., Seeliger, E.H. & Yergeny, A.L.: 1982, ‘The Inverted Pendulum: A Mechanical Analog of the Quadrupole Mass Filter’, American Journal of Physics 50(10), 924-931.

Grandy, W. & Schöck, M.: 1997, ‘Simulations of Nonlinear Pivot-Driven Pendula’, American Journal of Physics 65(5), 376-381.

Jones, H.W.: 1969, ‘A Quick Demonstration of the Inverted Pendulum’, American Journal of Physics 37(9), 941.

Kalmus, H.P.: 1970, ‘The Inverted Pendulum’, American Journal of Physics 38(7), 874-878.

McInerney, M.: 1985, ‘Computer-aided Experiments with the Damped Harmonic Oscillator’, American Journal of Physics 53(10), 991-996.

Michaelis, M.: 1985, ‘Stroboscopic Study of the Inverted Pendulum’, American Journal of Physics 53(11), 1079-1083.

Moloney, M.: 1996, ‘Inverted Pendulum Motion and the Principle of Equivalence’, American Journal of Physics 64(11), 1431.

Nelson, R. & Olsson, M.: 1986, ‘The Pendulum - Rich Physics from a Simple System’, American Journal of Physics 54(2), 112-121.

Ness, D.J.: 1967, ‘Small Oscillations of a Stabilized, Inverted Pendulum’, American Journal of Physics 35(10), 964-967.

Phelps, F.M. & Hunter, J.H.: 1965, ‘An Analytical Sopution of the Inverted Pendulum’, American Journal of Physics 33(4), 285-295.

Pippard, A. 1987, ‘The Inverted Pendulum’, European Journal of Physics, 8(3), 203-206.

Priest, J.: 1986, ‘Interfacing Pendulums to a Microcomputer’, American Journal of Physics 54(10), 953-955.

Scott, T.A.: 1983, ‘Resonance Demonstration’, The Physics Teacher 21(6), 409.

Smith, H. & Blackburn, J.: 1992, ‘Experimental Study of an Inverted Pendulum’, American Journal of Physics 60(10), 909-911.

Spencer, R.L. & Robertson, R.D.: 2001, ‘Mode Detuning in Systems of Weakly Coupled Oscillators’, American Journal of Physics 69(11), 1191-1197.

Kater pendulum

Candela, D., Martini, K.M., Krotkov, R.V. & Langley, K.H.: 2001, ‘Bessel’s Improved Kater Pendulum in the Teaching Laboratory’, American Journal of Physics 69(6), 714-720.

Crummett, W.: 1990, ‘Measurement of Acceleration due to Gravity’, The Physics Teacher 28(5), 291-295.

Jesse, K. & Born, H.: 1972, ‘Possible Sources of Error When Using the Kater Pendulum’, The Physics Teacher 10(8), 466.

Jesse, K.E.: 1980, ‘Kater Pendulum Modification’, American Journal of Physics 48(9), 785-786.

McCarthy, J.T.: 1950, ‘Use of WWV Signals to Time Pendulums’, American Journal of Physics 18(5), 306-307.

Peters, R.: 1997, ‘Automated Kater Pendulum’, European Journal of Physics, 18(3), 217-221.

Peters, R.D.: 1999, ‘Student-friendly Precision Pendulum’, The Physics Teacher, 37(7), 390-393.

Physical pendulum

Armstrong, H.L.: 1985, ‘An Experiment on the Inertial Properties of a Rigid Body’, Physics Education 20(3), 138-141.

Bartunek, P.: 1951, ‘A Driver for the Calthrop Resonance Pendulum’, American Journal of Physics 19(1), 57.

Basano, L. & Ottonello, P.: 1991, ‘Digital Pendulum Damping: The Single-oscillation Approach’, American Journal of Physics 59(11), 1018-1023.

Benenson, R. & Marsh, B.: 1988, ‘Coupled Oscillations of a Ball and a Curved-track Pendulum’, American Journal of Physics 56(4), 345-348.

Bulur, E., Aniltürk, S. & Mözer, A.M.: 1996, ‘Computer Analysis of Pendulum Motion: An Alternative Way of Collecting Experimental Data’, American Journal of Physics 64(10), 1333-1337.

Butikov, E.: 1999, ‘The Rigid Pendulum – an Antique but Evergreen Physical Model’, European Journal of Physics, 20(6), 429-441.

Cady, W.M.: 1942, ‘Remarkable Isochronous Pendulum’, American Journal of Physics 10(2), 114-116.

Corrado, L.: 1974, ‘The Meter Stick Pendulum’, The Physics Teacher 12(8), 494.

Cromer, A.: 1995, ‘Many Oscillations of a Rigid Rod’, American Journal of Physics 63(1), 112-121.

Freeman, I.M.: 1954, ‘Rectangular Plate Pendulum’, American Journal of Physics 22(4), 157-158.

Giltinan, D., Wagner, D. & Walkiewicz, T.: 1996, ‘The Physical Pendulum on a Cylindrical Support’, American Journal of Physics 64(2), 144-146.

Greenslade, T.B. & Owens, A.J.: 1980, ‘Reconstructed Nineteenth-century Experiment with Physical Pendula’, American Journal of Physics 48(6), 487-488.

Hinrichsen, P.F.: 1981, ‘Practical Applications of the Compound Pendulum’, The Physics Teacher 19(5), 286-292.

Horton, G.: 1966, ‘Some Laboratory Work with Physical Pendulums’, The Physics Teacher 4(2), 78-79.

Iona, M.: 1979, ‘The Physical Pendulum’, The Physics Teacher 17(4), 224, 276.

Irons, E.J.: 1947, ‘Graphical Treatment of the Physical Pendulum Problem’, American Journal of Physics 15(5), 426.

Kannewurf, C.R. & Jensen, H.C.: 1957, ‘Coupled Oscillations’, American Journal of Physics 25(7), 442-445.

Katz, E.: 1949, ‘Note on Pendulums’, American Journal of Physics 17(7), 439-441.

Kettler, J.: 1995, ‘The Variable Mass Physical Pendulum’, American Journal of Physics 63(11), 1049-1051.

Kolodiy, G.O.: 1979, ‘An Experiment with a Physical Pendulum’, The Physics Teacher 17(1), 52.

Levinson, D.A.: 1975, ‘Natural Frequencies of a Spherical Compound Pendulum’, American Journal of Physics 45(6), 579.

Marshall, J.: 1972, ‘Two Compound Pendulums with the Same Period of Oscillation’, School Science Review, 54(186), 130-131.

McInerney, M.: 1985, ‘Computer-aided Experiments with the Damped Harmonic Oscillator’, American Journal of Physics 53(10), 991-996.

Mills, D.S.: 1980, ‘The Physical Pendulum: A Computer-augmented Laboratory Exercise’, American Journal of Physics 48(4), 314-316.

Mires, R. & Peters, R.: 1994, ‘Motion of a Leaky Pendulum’, American Journal of Physics 62(2), 137-139.

Nicklin, R.C. & Rafert, J.B.: 1984, ‘The Digital Pendulum’, American Journal of Physics 52(7), 632-639.

Olssen, M.G.: 1981, ‘Spherical Pendulum Revisited’, American Journal of Physics 49(6), 531-534.

Pedersen, N.F. & Soerensen, O.H.: 1975, ‘The Compound Pendulum in Intermediate Laboratories and Demonstrations’, American Journal of Physics 45(10), 994-998.

Peters, R.: 1996, ‘Resonance Response of a Moderately Driven Rigid Planar Pendulum’, American Journal of Physics 64(2), 170-173.

Peters, R.D.: 1999, ‘Student-friendly Precision Pendulum’, The Physics Teacher 37(7), 390-393.

Peters, R. & Pritchett, T.: 1997, ‘The Not-so-simple Harmonic Oscillator’, American Journal of Physics 65(11), 1067-1073.

Peters, R. & Shepherd, J.: 1989, ‘A Pendulum with Adjustable Trends in the Period’, American Journal of Physics 57(6), 535-539.

Raychowdhury, P.N. & Boyd, J.N.: 1979, ‘Centre of Percussion’, American Journal of Physics 47(12), 1088-1089.

Reidl, C.J.: 1996, ‘Moment of Inertia of a Physical Pendulum’, The Physics Teacher 34(2), 114-115.

Sherfinski, J.: 1997, ‘A Counter-intuitive Physical Pendulum Lab’, The Physics Teacher 35(4), 252-253.

Spradley, J.L. : 1990, ‘Meter-stick Mechanics’, The Physics Teacher 28(5), 312-314.

Squire, P.: 1986, ‘Pendulum Damping’, American Journal of Physics 54(11), 984-991.

Sutton, R.S.: 1953, ‘An Experimental Encounter with Bifilar Pendula’, American Journal of Physics 21(5), 408.

Trilton, D.: 1986, Ordered and Chaotic Motion of a Forced Spherical Pendulum, European Journal of Physics, 7(3), 162-169.

Weltin, H.: 1964, ‘Inexpensive Physical Pendulum Experiment’, American Journal of Physics 32(4), 267-268.

Weltner, K., Esperodiao, A., Andrade, R. & Guedes, G.: 1994, ‘Demonstrating Different Forms of the Bent Tuning Curve with a Mechanical Oscillator’, American Journal of Physics 62(1), 56-59.

Worrell, F.T. & Correll, M.: 1958, ‘Elementary Experiment in Deriving an Empirical Relationship’, American Journal of Physics 26(9), 607-609.

Zilio, S.C.: 1982, ‘Measurement an Analysis of Large-angle Pendulum Motion’, American Journal of Physics 49(5), 450-452.

Ring pendulum

Jensen, H.C. & Haisley, W.E.: 1967, ‘On the Equivalence of Truncated Ring Pendula’, American Journal of Physics 35(10), 971-972.

Wagner, D., Walkiewicz, T. & Giltinan, D.: 1995, ‘The Partial Ring Pendulum’, American Journal of Physics 63(11), 1014-1017.

Walkiewicz, T.A. & Wagner, D.L.: 1994, ‘Symmetry Properties of a Ring Rendulum’, The Physics Teacher 32(3), 142-144.

Willey, D.G.: 1991, ‘Conservation of Mechanical Energy using a Pendulum’, The Physics Teacher 29(9), 567.

Simple pendulum

Abdel-Rahman, A.M.: 1983, ‘The Simple Pendulum in a Rotating Frame’, American Journal of Physics 51(8), 721-724.

Alford, W.L.: 1972, ‘Approximation for Horizontal Motion of a Plane Pendulum’, American Journal of Physics 42(5), 417-418.

Anderson, J.L.: 1959, ‘Approximations in Physics and the Simple Pendulum’, American Journal of Physics 27(3), 188-189.

Armstrong, H.L.: 1976, ‘Effect of the Mass of the Cord on the Period of a Simple Pendulum’, American Journal of Physics 44(6), 564-566.

Benham, T.A.: 1947, ‘Bessel Functions in Physics: Theory’, American Journal of Physics 15(4), 285-294.

Berg, R.: 1991, ‘Pendulum Waves: A Demonstration of Wave Motion Using Pendula’, American Journal of Physics 59(2), 186-187.

Blisard, T.J. & Duursema, C.H.: 1952, ‘A Demonstration of the Transformation of Mechanical Energy for Student Computation’, American Journal of Physics 20(9), 559-561.

Blitzer, L.: 1979, ‘Equilibrium and Stability of a Pendulum in an Orbiting Spaceship’, American Journal of Physics 47(3), 241-246.

Burns, G.P.: 1950, ‘Simple Pendulum’, American Journal of Physics 18(7), 468-469.

Cadwell, L. & Boyko, E.: 1991, ‘Linearization of the Simple Pendulum’, American Journal of Physics 59(11), 979-981.

Cook, G. & Zaidens, C.: 1986, ‘The Quantum Point-mass Pendulum’, American Journal of Physics 54(3), 259-261.

Crawford, F.S.: 1975, ‘Damping of a Simple Pendulum’, American Journal of Physics 43(3), 276-277.

Curtis, R.K.: 1981, ‘The Simple Pendulum Experiment’, The Physics Teacher 19(1), 36.

Denman, H.H.: 1959, ‘Amplitude Dependence of Frequency in a Linear Approximation to the Simple Pendulum Equation’, American Journal of Physics 27(7), 524-525.

Di Lieto, A., Fenecia, S. and Mancini, P.: 1991, ‘A Computer-Assisted Pendulum for Didactics’, European Journal of Physics, 12(1), 51-52.

Epstein, S.T. & Olsson, M.G.: 1975, ‘Comment on “Effect of the Mass of the Cord on the period of a Simple Pendulum”’, American Journal of Physics 45(7), 671-672.

Erkal, C.: 2000, ‘The Simple Pendulum: a Relativistic Visit’, European Journal of Physics, 21(5), 377-384.

Fulcher, L.P. & Davis, B.F.: 1976, ‘Theoretical and Experimental Study of the Motion of the Simple Pendulum’, American Journal of Physics 44(1), 51-55.

Gleiser, R.J.: 1979, ‘Small Amplitude Oscillations of a Quasi-ideal Pendulum’, American Journal of Physics 47(7), 640-643.

Gough, W.: 1983, ‘The Period of a Simple Pendulum is not 2>pÖ(l/g)’, European Journal of Physics, 4(1), 53.

Grandy, W. & Schöck, M: 1997, ‘Simulations of Nonlinear Pivot-driven Pendula’, American Journal of Physics 65(5), 376-381.

Gupta, M.L.: 1972, ‘The Critical Points of a Simple Pendulum’, American Journal of Physics 40(3), 478-480.

Hall, D.E.: 1981, ‘Comments on Fourier Analysis of the Simple Pendulum’, American Journal of Physics 49(8), 792.

Haque-Copilah, S.: 1996, ‘Extremely Simple Demonstration of Forced Oscillations’, American Journal of Physics 64(4), 507-508.

Head, J.H.: 1995, ‘Building New Confidence with a Classic Pendulum Demonstration’, The Physics Teacher 33(1), 10-15.

Helrich, C. & Lehman, T.: 1979, ‘A Rolling Pendulum Bob: Conservation of Energy and Partition of Kinetic Energy’, American Journal of Physics 47(4), 367-368.

Hughes, J.V.: 1953, ‘Possible Motions of a Sphere Suspended on a String (the Simple Pendulum)’, American Journal of Physics 21(1), 47-50.

Jackson, D.P.: 1996, ‘Rendering the “Not-so-simple” Pendulum Experimentally Accessible’, The Physics Teacher 34(2), 86-89.

Jensen, H.C. & Monohan, J.R.: 1968, ‘Air Bearing Support for a Pendulum’, American Journal of Physics 36(5), 459-460.

Knauss, H.P. & Zilsel, P.R.: 1951, ‘Magnetically Maintained Pendulum’, American Journal of Physics 19(5), 318-320.

Mace, W.: 1972, ‘Isochronism and Hooke’s Law’, School Science Review, 53(185), 773-774.

Miller, B.J.: 1972, ‘More Realistic Treatment of the Simple Pendulum without Difficult Mathematics’, American Journal of Physics 42(4), 298-303.

Molina, M.I.: 1997, ‘Simple Linearization of the Simple Pendulum for any Amplitude’, The Physics Teacher 35(8), 489-490.

Montgomery, C.G.: 1978, ‘Pendulum on a Massive Cord’, American Journal of Physics 46(4), 411-412.

Pitucco, A.:1980, ‘An Approximation of a Simple Pendulum’, The Physics Teacher 18(9), 666.

Santarelli, V., Carolla, J. & Ferner, M.: 1993, ‘A New Look at the Simple Pendulum’, The Physics Teacher 31(4), 236-238.

Schwarz, C.: 1995, ‘The Not-so-simple Pendulum’, The Physics Teacher 33(4), 225-228.

Siddons, J.: 1976, ‘Bits and Pieces: A Physics Miscellany’, School Science Review, 57(200), 441-453 (esp. 451-453).

Simon, R. & Riesz, R.P.: 1979, ‘Large Amplitude Simple Pendulum: A Fourier Analysis’, American Journal of Physics 47(10), 898-899.

Zheng, T.F. et al.: 1994, ‘Teaching the Non-linear Pendulum’, The Physics Teacher 32(4), 248-251.

Spring-mass pendulum

Armstrong, H.L.: 1969, ‘The Oscillating Spring and Weight - An Experiment often Misinterpreted’, American Journal of Physics 37(4), 447-448.

Blair, J.M.: 1975, ‘Precision Timing Applied to a Driven Mechanical Oscillator’, American Journal of Physics 43(12), 1076-1078.

Cayton, T.E.: 1975, ‘The Laboratory Spring-mass Oscillator: An Example of Parametric Instability’, American Journal of Physics 45(8), 723-732.

Crawford, H.: 1964, ‘A Space Clock’, The Physics Teacher 2(6), 290.

Cushing, J.T.: 1984, ‘The Spring-mass Pendulum Revisited’, American Journal of Physics 52(10), 925-933.

Cushing, J.Y.: 1984, ‘The Method of Characteristics Applied to the Massive Spring Problem’, American Journal of Physics 52(10), 933-937.

Dewdney, J.W.: 1958, ‘Simple Pendulum Equivalent to Spring-Mass System’, American Journal of Physics 26(5), 340-341.

Edwards, T.W. & Hultsch, R.A.: 1972, ‘Mass Distribution and Frequencies of a Vertical Spring’, American Journal of Physics 40(3), 445-449.

Erlichson, H.:1976, The Vertical Spring-Mass System and its Equivalent’, The Physics Teacher 14(9), 573-574.

Fyfe, F.M., Stroink, G., March, R.H. & Calkin, M.G.: 1981, ‘Large-scale Spring Experiment’, American Journal of Physics 49(11), 1074-1075.

Galloni, E.E. & Kohen, M.: 1979, ‘Influence of the Mass of the Spring on its Static and Dynamic Effects’, American Journal of Physics 47(12), 1076-1078.

Glanz, P.K.: 1979, ‘Note on Energy Change in a Spring’, American Journal of Physics 47(12), 1091-1092.

Grant, F.: 1986, ‘Energy Analysis of the Conical-spring Oscillator’, American Journal of Physics 54(3), 227-233.

Greiner, M.: 1980, ‘Elliptical Motion from a Ball and Spring’, American Journal of Physics 48(6), 488-489.

Heard, T.C. & Newby, N.D.:1975, ‘Behavior of a Soft Spring’, American Journal of Physics 45(11), 1102-1106.

Holzworth, D.E. & Malone J.: 2000, ‘Pendulum Period versus Hanging-spring Period’, The Physics Teacher 38(1), 47.

Jalbert, R.: 1963, ‘On Springs and Simple Harmonic Motion’, The Physics Teacher 1(3), 124.

Karioris, F. & Mendelson, K.: 1992, ‘A Novel Coupled Oscillation Demonstration’, American Journal of Physics 60(6), 508-513.

Lai, H.M.: 1984, ‘On the Recurrence Phenomenon of a Resonant Spring Pendulum’, American Journal of Physics 52(3), 219-223.

Lipham, J.G. & Pollak, V.L.: 1978, ‘Constructing a “Misbehaving” Spring’, American Journal of Physics 46(1), 110-111.

McDonald, A.: 1980, ‘Deceptively Simple Harmonic Motion: A Mass on a Spiral Spring’, American Journal of Physics 48(3), 189-192.

Mills, D.S.: 1981, ‘The Spring and Mass Pendulum: An Exercise in Mathematical Modeling’, The Physics Teacher 19(6), 404-405.

Nunes da Silva, J.: 1994, ‘Renormalized Vibrations of a Loaded Spring’, American Journal of Physics 62(5), 423-426.

Olsson, M.G.: 1976, ‘Why Does a Mass on a Spring Sometimes Misbehave?’, American Journal of Physics 44(12), 1211-1212.

Ouseph, P. & Ouseph, J.: 1987, ‘Electromagnetically Driven Resonance Apparatus’, American Journal of Physics 55(12), 1126-1129.

Porta, A. & Sandoval, J.L.: 1982, ‘A Detection System for Mass-Spring Oscillations’, The Physics Teacher 20(3), 186.

Rusbridge, M.G.: 1980, ‘Motion of the Spring Pendulum’, American Journal of Physics 48(2), 146-151.

Scott, A.: 1985, ‘Transfer of Energy in a Spring-mass Pendulum’, The Physics Teacher 23(6), 356.

Sears, F.W.: 1969, ‘A Demonstration of the Spring-Mass Correction’, American Journal of Physics 37(6), 645-648.

Walker, J. & Soule, T.: 1996, ‘Chaos in a Simple Impact Oscillation: The Bender Bouncer’, American Journal of Physics 64(4), 397-409.

Torsion pendulum

Abbott, H. :1983, ‘Torsion Resonance Demonstrator’, The Physics Teacher 21(5), 333.

Allen, M. & Saxl, E.J.: 1972, ‘The Period of Damped Simple Harmonic Motion’, American Journal of Physics 40(7), 942-944.

Chapmen, S.: 1948, ‘Discovering the Torsion Pendulum Expression in the Freshman Laboratory’, American Journal of Physics 16(5), 308-309.

Cromer, A.: 1995, ‘Many Oscillations of a Rigid Rod’, American Journal of Physics 63(1), 112-121.

Green, R.E.: 1958, ‘Calibrated Torsion Pendulum for Moment of Inertia Measurements’, American Journal of Physics 26(7), 498-499.

Miller, J.S.: 1957, ‘Coupled Torsion Pendulums’, American Journal of Physics 25(9), 649-650.

Milotti, E.: 2001, ‘Non-linear Behavior in a Torsion Pendulum’, European Journal of Physics, 22(3), 239-248.

O’Connell, J.: 2000, ‘Magnetic Torsion Pendulum’, The Physics Teacher 38(6), 377-378.

Pollock, R.E.: 1963, ‘Resonant Detection of Light Pressure by a Torsion Pendulum in Air - An Experiment for Underclass Laboratories’, American Journal of Physics 31(12), 901-904.

Smedt, J. De & Bock, A. De: 1957, ‘Horizontal Pendulum with Variable Modulus of Torsion (Resonance Curve)’, American Journal of Physics 25(3), 155-156.

Taylor, K.N.: 1983, ‘Tinker Toys Have their Moments of Inertia’, The Physics Teacher 21(7), 456-458.

Tyagi, S. & Lord, A.E.: 1979, ‘An Inexpensive Torsional Pendulum Apparatus for Rigidity Modulus Determination’, American Journal of Physics 47(7), 632-633.

Yu, Y-T.: 1942, ‘Double Torsion Pendulum in a Liquid’, American Journal of Physics 10(3), 152-153.

Two-dimensional pendulum

Livesey, D.: 1987, ‘The Precession of Simple Pendulum Orbits’, American Journal of Physics 55(7), 618-621.

Whitaker, R.J.: 2001, ‘Harmonographs. I. Pendulum Design’, American Journal of Physics 69(2), 162-173.

Worland, R.S. & Moelter, M.J.: 2000, ‘Two-dimensional Pendulum Experiments Using a Spark Generator’, The Physics Teacher 38(8), 489-492.

Variable gravity pendulums

Feliciano, J.: 1998, ‘The Variable Gravity Pendulum’, The Physics Teacher 36(1), 51-52.

Kwasnoski, J.B. & Murphy, R.S.: 1984, ‘The Classic Pendulum Experiment - on Jupiter or Saturn’, American Journal of Physics 52(1), 85.

Tufilaro, N.B., Abbott, T.A. & Griffiths, D.J.: 1984, ‘Swinging Attwood’s Machine’, American Journal of Physics 52(10), 895-903.

Vaquero, J.M. & Gallego, M.: 2000, ‘An Old Apparatus for Physics Teaching’ The Physics Teacher 38(7), 424-425.

Wilberforce pendulum

Berg, R. & Marshall, T.: 1991, ‘Wilberforce Pendulum Oscillations and Normal Modes’, American Journal of Physics 59(1), 32-38.

Debowska, E., Jakubowicz, S. and Mazur, Z.: 1999, ‘Computer Visualization of the Beating of a Wilberforce Pendulum’, European Journal of Physics, 20(2) 89-95.

Köpf, U.: 1990, ‘Wilberforce’s Pendulum Revisited’, American Journal of Physics 58(9), 833-837.

Whitaker, R.J.: 1988, ‘L.R. Wilberforce and the Wilberforce Pendulum’, The Physics Teacher 26(1), 37-39.

Williams, J. & Keil, R.: 1983, ‘A Wilberforce Pendulum’, The Physics Teacher 21(4), 257-258.

Pendulum Contexts

Chaos and the pendulum

Alessi, N., Fischer, C. & Gray, C.: 1992, Measurements of Amplitude Jumps and Hysteresis in a Driven Inverted Pendulum’, American Journal of Physics 60(8), 755-756.

Baker, G.: 1995, ‘Control of the Chaotic Driven Pendulum’, American Journal of Physics 63(9), 832-838.

Berdahl, J.P. & Lugt, K.V.: 2001, ‘Magnetically Driven Chaotic Pendulum’, American Journal of Physics 69(9), 1016-1019.

Blackburn, J. & Baker, G.: 1998, ‘A Comparison of Commercial Chaotic Pendulums’, American Journal of Physics 66(9), 821-830.

Blackburn, J., Smith, H. & Grønbich-Jensen, N.: 1992, ‘Stability and Hopf Bifurcations in an Inverted Pendulum’, American Journal of Physics 60(10), 903-908.

Carretero-Gonzalez, R., Numez-Yepez, H.& Salas-Brito, A. 1994, ‘Regular and Chaotic Behavior in an Extensible Pendulum’, European Journal of Physics, 15(3), 139-148.

Cohen, Y., Katz, S., Peres, A., Santo, E. & Yitzhaki, R.: 1988, ‘Stroboscopic Views of Regular and Chaotic Orbits’, American Journal of Physics 56(11), 1042.

Cuerno, R., Rañada, A. & Ruiz-Lorenzo, J.: 1992, ‘Deterministic Chaos in the Elastic Pendulum: A Simple Laboratory for Non-linear Dynamics’, American Journal of Physics 60(1), 73-79.

De Jong, M.L.: 1992, ‘Chaos and the Simple Pendulum’, The Physics Teacher 30(2), 115-121.

Duchesne, B., Fischer, C., Gray, C. & Jeffrey, K.: 1991, ‘Chaos in the Motion of an Inverted Pendulum: An Undergraduate Laboratory Experiment’, American Journal of Physics 59(11), 987-992.

Irons, F.: 1990, ‘Concerning the Non-Linear Behaviour of the Forced Spherical Pendulum including the Dowsing Pendulum’, European Journal of Physics, 11(2), 107-115.

Kautz, R.: 1993, ‘Chaos in a Computer-Animated Pendulum’, American Journal of Physics 61(5), 407-415.

Levien, R. & Tan, S.: 1993, ‘Double Pendulum: An Experiment in Chaos’, American Journal of Physics 61(11), 1038-1044.

Marega, E., Ioriatti, L. & Zilio, S.: 1991, ‘Harmonic Generation and Chaos in an Electromechanical Pendulum’, American Journal of Physics 59(9), 858-859.

Martin, S.J. & Ford, P.J.: 2001, ‘A Simple Experimental Demonstration of Chaos in a Driven Spherical Pendulum’, Physics Education 36(2), 108-114.

Mendelson, K. & Karioris, F.: 1991, ‘Chaoticlike Motion of a Linear Dynamical System’, American Journal of Physics 59(3), 221-224.

Oliver, D.: 1999, ‘A Chaotic Pendulum’, The Physics Teacher 37(3), 174.

Pemann, D. & Hamilton, I.: 1992, ‘Self-similar and Erratic Transient Dynamics for the Linearly Damped Simple Pendulum’, American Journal of Physics 60(5), 442-450.

Peters, R.: 1995, ‘Chaotic Pendulum Based on Torsion and Gravity in Opposition’, American Journal of Physics 63(12), 1128-1136.

Shew, W. Coy, H. & Lindner, J.: 1999, ‘Taming Chaos with Disorder in a Pendulum Array’, American Journal of Physics 67(8), 703-708.

Shinbrot, T., Grebogi, C., Wisdom, J. & Yorke, J.: 1992, ‘Chaos in a Double Pendulum’, American Journal of Physics 60(6), 491-499.

Taylor, M.: 2001, ‘Pendumonium’, Physics Education 36(5), 425.

Trilton, D.: 1986, Ordered and Chaotic Motion of a Forced Spherical Pendulum, European Journal of Physics, 7(3), 162-169.

Walker, J. & Soule, T.: 1996, ‘Chaos in a Simple Impact Oscillation: The Bender Bouncer’, American Journal of Physics 64(4), 397-409.

Galileo and the pendulum

Ball, M.: 1985, ‘Galileo Galilei and Christiaan Huygens: Addendum’, Antiquarian Horology, 15, 373-374.

Bjelic, D.: 1996, ‘Lebenswelt Structures of Galilean Physics: The Case of Galileo’s Pendulum’, Human Studies, 19, 409-432.

Dobson, R.: 1985, ‘Galileo Galilei and Christiaan Huygens’, Antiquarian Horology, 15, 261-270.

Drake, S.: 1986, ‘Galileo’s Physical Measurements’, American Journal of Physics 54(4), 302-306.

Erlichson, H.: 1997, ‘Galileo to Newton - a Liberal Arts Physics Course’, The Physics Teacher 35(9), 532-535.

Erlichson, H.: 1999, ‘Galileo’s Pendulum’, The Physics Teacher 37(8), 478-479.

Matthews, MR.: 1990, ‘Galileo and the Pendulum: A Case for History and Philosophy in the Classroom’, Australian Science Teachers Journal 36(1), 7-13.

Naylor, R.: 1974, ‘Galileo’s Simple Pendulum’, Physis: Rivista Internazionale di Storia della Scienza, 16, 23-46.

Naylor, R.: 1977, ‘Galileo’s Need for Precision: The “point” of the Fourth Day Pendulum Experiment’, Isis, 68, 97-103.

Wood, H.T.: 1994, ‘The Interrupted Pendulum’, The Physics Teacher 32(7), 422-423.

Yamazaki, M.: 1993, ‘Galileo and the Laws of Pendulum and Fall’, Journal of History of Science, 32, 12-18.

Historical Contexts

Conlin, M.: 1999, ‘The Popular and Scientific Reception of the Foucault Pendulum in the United States’,Isis, 90(2), 181-204.

Edwardes, E.: 1980, ‘The Suspended Foliot and New Light on Early Pendulum Clocks’, Antiquarian Horology, 12, 614-626.

Foley, V.: 1988, ‘Besson, da Vinci, and the Evolution of the Pendulum: Some Findings and Observations’’, History and Technology, 6, 1-43.

Garcia-Diego, J.: 1988, ‘On a Mechanical Problem of Lanz’, History and Technology, 5, 301-313.

Gauld, C.: 1998, ‘Solutions to the Problem of Impact in the 17th and 18th Centuries and Teaching Newton’s Third Law Today’, Science & Education 7(1), 49-67.

Hall, B.: 1978, ‘The Scholastic Pendulum’, Annals of Science, 35, 441-462.

King, D.: 1979, ‘Ibn Yunus and the Pendulum: A History of Errors’, Archives Internationale d’Histoire des Sciences, 29, 35-52.

Sheynin, O.: 1994, ‘Ivory’s Treatment of Pendulum Observations’, Historica Mathematica, 21, 174-184.

Investigating the motion of the simple pendulum

Araki, T.: 1994, ‘Measurement of Simple Pendulum Motion Using Flux-gate Magnetometer’, American Journal of Physics 62(6), 569-571.

Burris, J.A. & Hargrave, W.J.: 1944, ‘Simple Pendulum Energy Experiment’, American Journal of Physics 12(4), 215-217.

Chinn, L.: 1979, ‘Demonstration of the Conservation of Mechanical Energy’, The Physics Teacher 17(6), 385.

Crummett, W.: 1990, ‘Measurement of Acceleration due to Gravity’, The Physics Teacher 28(5), 291-295.

Curtis, R.K.: 1981, ‘The Simple Pendulum Experiment’, The Physics Teacher 19(1), 36.

Curzon, F., Locke, A., Lefrançois & Novick, K.: 1995, ‘Parametric Instability of a Pendulum’, American Journal of Physics 63(2), 132-136.

Denardo, B. & Masada, R.: 1990, ‘A Not-so-obvious Pendulum Experiment’, The Physics Teacher 28(1), 51-52.

Dix, F.: 1975, ‘A Pendulum Counter-timer Using a Photocell Gate’, American Journal of Physics 43(3), 280.

Hall, D.E. & Shea, M.J.: 1977, ‘Large-amplitude Pendulum Experiment: Another Approach’, American Journal of Physics 45(4), 355-357.

Lewowski, T. and Wozmiak K.: 2002, ‘Period of a Pendulum at Large Amplitudes: a Laboratory Experiment’, European Journal of Physics, 23(5), 461-464.

Li, S-P. & Feng, S-Y.: 1967, ‘Precision Measurement of the Period of a Pendulum Using an Oscilloscope’, American Journal of Physics 35(11), 1071-1073.

Matous, G. & Matolyak, J.: 1991, ‘Teaching Important Procedures with Simple Experiments’, The Physics Teacher 29(8), 541-542.

McCormick, W.W.: 1939, ‘A Pendulum Timer for the Elementary Laboratory’, American Journal of Physics 7(6), 260.

Santarelli, V., Carolla, J. & Ferner, M.: 1993, ‘A New Look at the Simple Pendulum’, The Physics Teacher 31(4), 236-238.

Smith, M.K.: 1964, ‘Precision Measurement of Period vs. Amplitude for a Pendulum’, American Journal of Physics 32(8), 632-633.

Pendulum collisions

Becchetti, F.D. & Cockerill, A.: 1984, ‘Collision Balls and Coupled Pendulums for the Overhead Projector’, The Physics Teacher 22(4), 258-259.

Chapman, S.: 1960, ‘Misconception Concerning the Dynamics of the Impact Ball Apparatus’, American Journal of Physics 28(8), 705-711.

Domenech, A. and Domenech, T.: 1988, ‘Relationships between Scattering Angles in Pendulum Collisions’, European Journal of Physics, 9(2), 116-122.

Erlich, R.: 1996, ‘Experiments with “Newton’s Cradle”’, The Physics Teacher 34(3), 181-183.

Erlichson, H.: 2001, ‘A Proposition Well Known to Geometers’, The Physics Teacher 39(3), 152-153.

Gauld, C.F.: 1998, ‘Colliding Pendulums, Conservation of Momentum and Newton’s Third Law’, Australian Science Teachers Journal, 44(3), 37-38.

Gauld, C.F.: 1999, ‘Using Colliding Pendulums to Teach Newton’s Third Law’, The Physics Teacher 37(2), 42-45.

Gavenda, J.D. & Edington, J.R.: 1997, ‘Newton’s Cradle and Scientific Explanation’, The Physics Teacher 35(7), 411-417.

Gupta, P.: 1985, ‘Blackwood Pendulum Experiment and the Conservation of Linear Momentum’, American Journal of Physics 53(3), 267-269.

Hecht, K.: 1961, ‘Collision Experiments in Shadow Projection’, American Journal of Physics 29(9), 636-639,

Herrmann, F. & Schmälzle, P.: 1981, ‘Simple Explanation of a Well-known Collision Experiment’, American Journal of Physics 49(8), 761-764.

McCaslin, J.G.: 1984, ‘A Different Blackwood Pendulum Experiment’, The Physics Teacher 22(3), 184-186.

Satterly, J.: 1945, ‘Ball Pendulum Impact Experiments’, American Journal of Physics 13, 170.

Siddons, J.: 1969, ‘Swinging Spheres’, School Science Review, 51(174), 152-153.

Pendulum resonance

Abbott, H. :1983, ‘Torsion Resonance Demonstrator’, The Physics Teacher 21(5), 333.

Bruns, D.G.: 1988, ‘Synchronized Swinging’, The Physics Teacher 26(4), 220-221.

Edge, R.D.: 1981, ‘Coupled and Forced Oscillations’, The Physics Teacher 19(7), 485-488.

Fajans, J. & Friedland, L.: 2001, Autoresonant (Nonstationary) Excitation of Pendulums, Plutinos, Plasmas and Other Nonlinear Oscillators’, American Journal of Physics 69(10), 1096-1102.

Grosu, T. & Ursu, D.: 1982, ‘Simple Apparatus for Obtaining Parametric Resonance’, American Journal of Physics 50(6), 561.

Lai, H.M.: 1984, ‘On the Recurrence Phenomenon of a Resonant Spring Pendulum’, American Journal of Physics 52(3), 219-223.

Olsen, L.O.: 1945, ‘Coupled Pendulums: An Advanced Laboratory Experiment’, American Journal of Physics 13(5), 321-324.

Ouseph, P. & Ouseph, J.: 1987, ‘Electromagnetically Driven Resonance Apparatus’, American Journal of Physics 55(12), 1126-1129.

Peters, R.: 1996, ‘Resonance Response of a Moderately Driven Rigid Planar Pendulum’, American Journal of Physics 64(2), 170-173.

Pinto, F.: 1993, ‘Parametric Resonance: an Introductory Experiment’, The Physics Teacher 31(6), 336-346.

Priest, J. & Poth, J.: 1982, ‘Teaching Physics with Coupled Pendulums’, The Physics Teacher 20(2), 80-85.

Scott, T.A.: 1983, ‘Resonance Demonstration’, The Physics Teacher 21(6), 409.

Stockman, H.E.: 1960, ‘Pendulum Parametric Amplifier’, American Journal of Physics 28(5), 506-507.

Student conceptions of the pendulum

Czudkova, L. & Musilova, J.: 2000, ‘The Pendulum: A Stumbling Block of Secondary School Mechanics’, Physics Education 35(6), 428-435.

Wolman, W.: 1984, ‘Models and Procedures: Teaching for Transfer of Pendulum Knowledge’, Journal of Research in Science Teaching, 21(4), 399-415.

Swinging and the pendulum

Anon.: 1966, ‘A Child’s Swing’, The Physics Teacher 4(7), 307.

Anon.: 1966, ‘A Child’s Swing’, The Physics Teacher 4(8), 374-375.

Burns, J.A.: 1970, ‘More on Pumping on a Swing’, American Journal of Physics 38(7), 920-922.

Case, W.: 1996, ‘The Pumping of a Swing from the Standing Position’, American Journal of Physics 64(3), 215-220.

Case, W. & Swanson, M.: 1990, ‘The Pumping of a Swing from the Seated Position’, American Journal of Physics 58(5), 463-467.

Curry, S.M.: 1976, ‘How Children Swing’, American Journal of Physics 44(10), 924-926.

Gore, B.F.: 1970, ‘The Child’s Swing’, American Journal of Physics 38(3), 378-379.

Gore, B.F.: 1971, ‘Starting a Swing from Rest’, American Journal of Physics 39(3), 347.

Hesketh, R.V.: 1975, ‘How to Make a Swing Go’, Physics Education 10(5), 367-369.

McMullan, J.: 1940, ‘On Initiating Motion in a Swing’, American Journal of Physics 40(5), 764-766.

Mellen, W.R.: 1994, ‘Spring String Swing Thing’, The Physics Teacher 32(2), 122-123.

Sanmartin, J.R.: 1984, ‘O Botafumeiro: Parametric Pumping in the Middle Ages’, American Journal of Physics 52(10), 937-945.

Siegman, A.F.: 1969, ‘Comments on Pumping on a Swing’, American Journal of Physics 37(8), 843-844.

Tea, P.L. & Falk, H.: 1968, ‘Pumping on a Swing’, American Journal of Physics 36(12), 1165-1166.

Time measurement and the pendulum

Aked, C.: 1994, ‘The First Free Pendulum Clock’, Bulletin of the Scientific Instrument Society, 41, 20-23.

Bazin, M. & Lucie, P.: 1981, ‘The Pendulum Reborn: Time Measurements in the Teaching Laboratory’, American Journal of Physics 49(8), 758-761.

Bensky, T.J.: 2001, ‘Measuring g with a Joystick Pendulum’, The Physics Teacher 39(2), 88-89.

Carlson, J.E.: 1991, ‘The Pendulum Clock’, The Physics Teacher 29(1), 8-11.

Crawford, H.: 1964, ‘A Space Clock’, The Physics Teacher 2(6), 290.

Crook, A.: 2001, ‘A Tale of a Clock’, European Journal of Physics, 22(5), 549-560.

Denny, M.: 2002, ‘The Pendulum Clock’, European Journal of Physics, 23(4), 449-458.

Dobson, R.: 1979, ‘Huygens, the Secret in the Coster-Fromanteel “Contract”, the Thirty-hour Clock’, Antiquarian Horology, 12, 193-196.

Edwardes, E.: 1983, ‘The Fromanteels and the Pendulum Clock’, Antiquarian Horology, 14, 250-265.

Kesteven, M.: 1978, ‘On the Mathematical Theory of Clock Escapements’, American Journal of Physics 46(2), 125-129.

Lee, R.: 1978, ‘Early Pendulum Clocks’, Antiquarian Horology, 11, 146-160.

Ments, M. v.: 1956, ‘Synchronization of Pendulum Clocks with the Help of Signals Taken from a Quartz-Crystal Clock’, American Journal of Physics 24(7), 489-495.

Mills, A.: 1993, ‘The Earl of Meath’s ‘Free Pendulum’ Water-driven Clock: An Incredible Scientific Instrument’, Bulletin of the Scientific Instrument Society, 39, 3-6.

Walking and the pendulum

Bachman, C.H.: 1976, ‘Some Observations on the Process of Walking’, The Physics Teacher 14, 360.

Dumont, A. & Waltham, C.: 1997, ‘Walking’, The Physics Teacher 35(6), 372-376.

Prigo, R.: 1976, ‘Walking Resonance’, The Physics Teacher 14, 360.

Sutton, R.M.: 1955, ‘Two Notes on the Physics of Walking’, American Journal of Physics 23(7), 490-491.

Miscellaneous

Asano, K.: 1975, ‘On the Theory of an Electrostatic Pendulum Oscillator’, American Journal of Physics 43(5), 423-427.

Bartunek, P.: 1956, ‘Some Interesting Cases of Vibrating Systems’, American Journal of Physics 24(5), 369-373.

Berg, R.: 1991, ‘Pendulum Waves: A Demonstration of Wave Motion Using Pendula’, American Journal of Physics 59(2), 186-187.

Chapman, A.: 1993, ‘The Pit and the Pendulum: G. B. Airy and the Determination of Gravity’, Antiquarian Horology, 21, 70-78.

Doyle, W.T. & Gibson, R.: 1979, ‘Demonstration of Eddy Current Forces’, American Journal of Physics 47(5), 470-471.

Dupré, A. & Janssen, J.: 2000, ‘An Accurate Determination of the Acceleration of Gravity in the Undergraduate Laboratory’, American Journal of Physics 68(8), 704-711.

Flaten, J.A. & Parendo, K.A.: 2001, ‘Pendulum Waves: A Lesson in Aliasing’, American Journal of Physics 69(7), 778-782.

Grøn, Ø.: 1983, ‘A Tidal Force Pendulum’, American Journal of Physics 51(5), 429-431.

Hageseth, G.T.: 1987, ‘The Liquid Pendulum’, The Physics Teacher 25(7), 427.

Morgan, B.H.: 1982, ‘Polarization Effects with Pendulums’, The Physics Teacher 20(8), 541-542.

Pollock, R.E.: 1963, ‘Resonant Detection of Light Pressure by a Torsion Pendulum in Air - An Experiment for Underclass Laboratories’, American Journal of Physics 31(12), 901-904.

Schmidt, V.H. & Childers, B.R.: 1984, ‘Magnetic Pendulum Apparatus for Analog Demonstration of First-order and Second-order Phase Transitions and Tricritical Points’, American Journal of Physics 52(1), 39-43.

Sheppard, D.M.: 1970, ‘Using One Pendulum and a Rotating Mass to Measure the Universal Gravitational Constant’, American Journal of Physics 38(3), 380.

Sleator, W.W.: 1948, ‘Demonstration Experiments with Pendulums’, American Journal of Physics 16(2), 93-496.

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