Skip to main content

Types of Two-Dimensional Pendulums and Their Uses in Education

Science & Education volume 13, pages 401–415 (2004)Cite this article

Abstract

Pendulums which swing in two dimensions simultaneously and aredesigned to leave a record of their motion are termed `harmonographs'.The curves which they draw are known, alternatively, as `Bowditchcurves' or `Lissajous curves'. A variety of designs of harmonographshave been invented over the years. These may be a `Y-suspended'`simple' pendulum, or they may be a complex `physical' pendulumsystem. Harmonographs have been built as demonstration apparatusin physics (or mathematics) or as `art' machines for enjoying theaesthetics of the curves produced.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (Canada)

References

  • Annett, C.H.: 1979, 'Observation of the First Overtone Vibrational Mode in an Automobile Whip Antenna', American Journal of Physics 47(9), 820–822.

    Google Scholar 

  • Ashton, A.: 1999, Harmonograph: A Visual Introduction to Harmony, Wooden Books, Walkmill, Wales.

    Google Scholar 

  • Bowditch, N.: 1815, 'On the Motion of a Pendulum Suspended from Two Points', Memoirs of the American Academy of Arts and Sciences 3(Part II), 416–436.

    Google Scholar 

  • Bulman, A.D.: 1968a, Model-Making for Physicists, Thomas Y. Crowell, New York.

    Google Scholar 

  • Bulman, A.D.: 1968b, Models for Experiments in Physics, Thomas Y. Crowell, New York.

    Google Scholar 

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

    Google Scholar 

  • Collins, A.F.: 1931, Experimental Mechanics, D. Appleton, New York and London.

    Google Scholar 

  • 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.

    Google Scholar 

  • Cundy, H.M. & Rollett, A.P.: 1961, Mathematical Models, Oxford University Press, New York.

    Google Scholar 

  • Dean, J.: 1815, 'Of the Apparent Motion of the Earth Viewed from the Moon, Arising from the Moon's Librations', Memoirs of the American Academy of Arts and Sciences 3(Part II), 241–245.

    Google Scholar 

  • Eaton, V.E. et al.: 1940, 'Kater's Reversible Pendulum', Selective Experiments in Physics (No. 71990-M52b), Central Scientific Company, Chicago.

    Google Scholar 

  • Eaton, V.E. et al.: 1941, 'The Compound Pendulum', Selective Experiments in Physics (No. 71990-M54b), Central Scientific Company, Chicago.

    Google Scholar 

  • Engineering: 1874, 'Tisley's Compound Pendulum', Engineering 17, 101–102.

    Google Scholar 

  • Feather, N.: 1959, An Introduction to the Physics of Mass, Length and Time, Edinburgh University Press, Edinburgh.

    Google Scholar 

  • French, A.P.: 1971, Vibrations and Waves, Norton, New York.

    Google Scholar 

  • Garland, G.D.: 1965, The Earth's Shape and Gravity, Pergamon Press, Oxford.

    Google Scholar 

  • Goold, J., Benham, C.E., Kerr, R. & Wilberforce, L.R.: 1909, in H.C. Newton (ed.), Harmonic Vibrations and Vibration Figures, Newton & Co., Scientific Instrument Makers, London.

    Google Scholar 

  • Greenslade, T.B., Jr.: 1979, '19th Century Textbook Illustrations. XXVII. Harmonographs', The Physics Teacher 17(4), 256–258.

    Google Scholar 

  • Greenslade, T.B., Jr.: 1992, '19th Century Textbook Illustrations. LI. The Kaleidophone', The Physics Teacher 30(1), 38–39.

    Google Scholar 

  • Greenslade, T.B., Jr.: 1993, 'All about Lissajous Figures', The Physics Teacher 31(6), 364–370.

    Google Scholar 

  • Greenslade, T.B., Jr.: 1998, 'The Double-Elliptic Harmonograph', The Physics Teacher 36(2), 90–91.

    Google Scholar 

  • Hagen, J.G.: 1879, 'Ueber die Verwendung des Pendels zur graphischen Darstellung der Stimmga-belcurven', Zeitschrift für Mathematik und Physik 24, 285–303.

    Google Scholar 

  • Halladay, D., Resnick, R. & Walker, J.: 1993, Fundamentals of Physics: Extended with Modern Physics, John Wiley & Sons, New York.

    Google Scholar 

  • Ingersoll, L.R., Martin, M.J. & Rouse, T.A.: 1953, A Laboratory Manual of Experiments in Physics, 6th edn, McGraw-Hill, New York.

    Google Scholar 

  • Jesse, K.E.: 1980, 'Kater Pendulum Modification', American Journal of Physics 48(9), 785–786.

    Google Scholar 

  • Kater, H.: 1818, 'An Account of Experiments for Determining the Length of the Pendulum Vibrating Seconds in the Latitude of London', Philosophical Transactions of the Royal Society of London 8, 33–102.

    Google Scholar 

  • Knight, D.M.: 1998, Science in the Romantic Era, Ashgate (Variorum Collected Studies Series), Aldershot, Hampshire and Brookfield, VT.

  • Lenzen, V.F. & Multhauf, R.P.: 1965, 'Development of Gravity Pendulums in the 19th Century', United States National Museum Bulletin 240: Contributions from the Museum of History and Technology, Paper 44, Smithsonian Institution, Washington, DC, pp. 301–348.

    Google Scholar 

  • Lovering, J.: 1881, 'Anticipation of the Lissajous Curves', Proceedings of the American Academy of Arts and Sciences 16(new series 8), 292–298.

    Google Scholar 

  • Matthews, M.R.: 2000, Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion can Contribute to Science Literacy, Kluwer Academic/Plenum Publishers, New York.

    Google Scholar 

  • Matthews, M.R.: 2001, 'How Pendulum Studies can Promote Knowledge of the Nature of Science', Journal of Science Education and Technology 10(4), 359–368.

    Google Scholar 

  • Merivuori, H. & Sands J.A.: 1984, 'Standing Wave Pattern on Automobile Radio Antenna', The Physics Teacher 22(1), 33.

    Google Scholar 

  • Millikan, R.A., Roller, D. & Watson, E.C.: 1937, Mechanics, Molecular Physics, Heat, and Sound, Ginn and Company, Boston.

    Google Scholar 

  • Newburgh, R. & Newburgh, G.A.: 2000, 'Finding the Equation for a Vibrating Car Antenna', The Physics Teacher 38(1), 31–34.

    Google Scholar 

  • Pepper, J.H.: 1877, Cyclopaedic Science Simplified, 4th edn, Fredrick Warne, London.

    Google Scholar 

  • Peters, R.D.: 1999, 'Student-Friendly Precision Pendulum', The Physics Teacher 37(7), 390–393.

    Google Scholar 

  • Poynting, J.H. & Thomson, J.J.: 1900, A Text-Book of Physics: Sound, 2nd edn, Charles Griffin, London.

    Google Scholar 

  • Poynting, J.H. & Thomson, J.J.: 1909, A Text-Book of Physics: Properties of Matter, 5th edn, Charles Griffin, London.

    Google Scholar 

  • Rigge, W.F.: 1926, Harmonic Curves, The Creighton University, Omaha, NE.

    Google Scholar 

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

    Google Scholar 

  • Searle, G.F.C.: 1934, Experimental Physics: A Selection of Experiments, Cambridge University Press, Cambridge.

    Google Scholar 

  • Sommerfeld, A.: 1952, Mechanics: Lectures on Theoretical Physics, Vol.I, AcademicPress, New York and London.

    Google Scholar 

  • Stephenson, R.J.: 1969, Mechanics and Properties of Matter, 3rd edn, John Wiley & Sons, New York.

    Google Scholar 

  • Taylor, L.W.: 1941; 1959, Physics: The Pioneer Science, Houghton Mifflin, Boston; reprint, Dover, New York.

    Google Scholar 

  • Thomson, W. & Tait, P.G.: 1873, Elements of Natural Philosophy, Part I, The Clarendon Press, Oxford.

    Google Scholar 

  • Tisley, S.C.: 1873, 'On a Compound-Pendulum Apparatus', Report of the British Association for the Advancement of Science 43, 48.

    Google Scholar 

  • Tissandier, G.: 1883, Popular Scientific Recreations in Natural Philosophy, Astronomy, Geology, Chemistry, Etc., Ward, Lock, and Co., London..

    Google Scholar 

  • Tolansky, S.: 1969, 'Complex Curvilinear Designs from Pendulums', Leonardo 2, 267–274.

    Google Scholar 

  • Tuck, F.E.: 1913, 'How to Make a Wondergraph', in The Boy Mechanic, Book I: 700 Things for Boys to Do, Popular Mechanics Press, Chicago, pp. 436–439.

    Google Scholar 

  • Tyndall, J.: 1894, Sound, 3rd, edn, Appleton, New York.

    Google Scholar 

  • Watson, W.A.: 1902, A Text-Book of Physics, 3rd edn, Longmans, Green, London.

    Google Scholar 

  • Wheatstone, C.: 1827, 'Description of the Kaleidophone, or Phonic Kaleidoscope; A New Philosophical Toy, for the Illustration of Several Interesting and Amusing Acoustical and Optical Phenomena', Quarterly Journal of Science, Literature, and Art 23(new series), 344–351.

    Google Scholar 

  • Whitaker, R.J.: 1988a, 'L. R. Wilberforce and the Wilberforce Pendulum', The Physics Teacher 26(1), 37–39.

    Google Scholar 

  • Whitaker, R.J.: 1988b, 'Mathematics of the Spirograph', School Science and Mathematics 88(7), 554–564.

    Google Scholar 

  • Whitaker, R.J.: 1991, 'A Note on the Blackburn Pendulum', American Journal of Physics 59(4), 330–333.

    Google Scholar 

  • Whitaker, R.J.: 1993, 'The Wheatstone Kaleidophone', American Journal of Physics 61(8), 722–728.

    Google Scholar 

  • Whitaker, R.J.: 2001a, 'Harmonographs. I. Pendulum Design', American Journal of Physics 69(2), 162–173.

    Google Scholar 

  • Whitaker, R.J.: 2001b, 'Harmonographs. II. Circular Design', American Journal of Physics 69(2), 174–183.

    Google Scholar 

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

    Google Scholar 

  • Zahm, J.A.: 1892, Sound and Music, McClurg and Company, Chicago.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Astronomy, and Materials Science, Southwest Missouri State University, Springfield, MO 65804, U.S.A

    Robert J. Whitaker

Authors
  1. Robert J. Whitaker
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Whitaker, R.J. Types of Two-Dimensional Pendulums and Their Uses in Education. Science & Education 13, 401–415 (2004). https://doi.org/10.1023/B:SCED.0000041830.98845.5f

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:SCED.0000041830.98845.5f

Keywords

  • Lissajous Curve