Search Results for Pendulum*

Biographies

1. Huygens biography
   
   • In 1656 he patented the first pendulum clock, which greatly increased the accuracy of time measurement.
     
   • His work on the pendulum was related to other mathematical work which he had been doing on the cycloid as a result of the challenge by Pascal.
     
   • Huygens believed that a pendulum swinging in a large are would be more useful at sea and he invented the cycloidal pendulum with this in mind.
     
   • He built several pendulum clocks to determine longitude at sea and they underwent sea trials in 1662 and again in 1686.
     
   • In the Horologium Oscillatorium sive de motu pendulorum (1673) he described the theory of pendulum motion.
     
   • At this time Huygens patented his design of pendulum clock with the solution of the longitude problem in mind.
     
   • Despite this Huygens did begin to experiment with clocks regulated by springs, but their accuracy was poorer than his pendulum clocks.
     
   • Horologium Oscillatorium contains work on the pendulum.
     
   • In it Huygens proves that the cycloid is tautochronous, an important theoretical result but one which had little practical application to the pendulum.
     
   • He also solves the problem of the compound pendulum.
     
   • However there is much more than work on pendulums.
     
   • He continued to work on improving lenses and on a spring regulated clock and on new pendulum clocks.
     
   • the ideas of mass, weight, momentum, force, and work were finally clarified in Huygens' treatment of the phenomena of impact, centripetal force and the first dynamical system ever studied - the compound pendulum.
     
   • The theory of the pendulum .
     

2. Foucault biography
   
   • Foucault's next idea was that if he was able to design the support for a pendulum which allowed it to freely move in any direction without any resistance, then once set in motion it would retain its plane of swing in space while the Earth rotated beneath it.
     
   • In January 1851 he succeeded in constructing such a pendulum in the basement of his house.
     
   • Every scientist in Paris received an invitation to view the pendulum in the Paris Observatory on 3 February 1851.
     
   • A paper by Foucault on his pendulum was read by Arago to the Academy of Sciences on the same day as the experiment was carried out in the Observatory.
     
   • where T is the time in hours taken for the pendulum to return to its original position and q is the latitude at which the experiment is carried out.
     
   • Plana presented a paper to the Academy of Sciences of Turin on Foucault's pendulum in March.
     

3. Mouton biography
   
   • His most famous work Observationes diametrorum solis et lunae apparentium published in 1670 studied interpolation and a standard of measurement based on the pendulum.
     
   • This volume contains interesting memoirs on interpolations and on the project of a universal standard of measurement based on the pendulum.
     
   • Certainly one could not measure the circumference of the earth, so he proposed a standard based on the length of a pendulum.
     
   • He conducted experiments which led him to the conclusion that a simple pendulum of length one virgula would oscillate 3959.2 times in 30 minutes.
     
   • Mouton also produced 10 place tables of logarithmic sines and cosines and an astronomical pendulum of remarkable precision.
     

4. Hooke biography
   
   • Realising the weakness of the pendulum clock in keeping time on a ship which was pitching and tossing, he wondered about the:- .
     
   • Rather than the balance wheel being controlled by a pendulum which in turn operated through gravity, he reasoned that controlling the balance wheel with a spring would have huge advantages for a portable timekeeper that one might carry around or one which would have to continue to keep the correct time on a ship.
     
   • Hooke invented the conical pendulum and was the first person to build a Gregorian reflecting telescope.
     
   • In 1666 he proposed that gravity could be measured using a pendulum.
     
   • And particularly that of the oval figure of the Earth which was read by me to this Society about 27 years since upon the occasion of the carrying the pendulum clocks to sea and at two other times since, though I have had the ill fortune not to be heard, and I conceive there are some present that may very well remember and do know that Mr Newton did not send up that addition to his book till some weeks after I had read and showed the experiments and demonstration thereof in this place and had answered the reproachful letter of Dr Wallis from Oxford.
     

5. Galileo biography
   
   • In fact he had returned to work on the theory of motion in 1602 and over the following two years, through his study of inclined planes and the pendulum, he had formulated the correct law of falling bodies and had worked out that a projectile follows a parabolic path.
     
   • He then described an experiment using a pendulum to verify his property of inclined planes and used these ideas to give a theorem on acceleration of bodies in free fall:- .
     
   • One would expect that Galileo's understanding of the pendulum, which he had since he was a young man, would have led him to design a pendulum clock.
     
   • In fact he only seems to have thought of this possibility near the end of his life and around 1640 he did design the first pendulum clock.
     

6. Richer biography
   
   • Richer's second important work was to examine the periods of pendulums at different points on the Earth.
     
   • He examined the period of a pendulum while on the expedition to Cayenne, French Guyana and found that the pendulum beat more slowly than in Paris.
     

7. Gregory biography
   
   • Gregory hung his pendulum clock on the wall beside the same window.
     
   • Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock.
     

8. Taylor biography
   
   • Taylor, in his studies of vibrating strings was not attempting to establish equations of motion, but was considering the oscillation of a flexible string in terms of the isochrony of the pendulum.
     
   • He tried to find the shape of the vibrating string and the length of the isochronous pendulum rather than to find its equations of motion.
     

9. Saurin biography
   
   • Saurin made contributions to the calculus, wrote on Jacob Bernoulli's problem of quickest descent and Huygens' theory of the pendulum.
     
   • He defended Huygens' theory of the pendulum after it was attacked by le Chevalier de Liouville in eclaircissement sur une difficulte propose aux mathematiciens par M le Chevalier de Liouville (1722).
     

10. Arago biography
    
    • In fact they were asked to carry out a second scientific task at the same time for they were asked to use a pendulum to measure the force of gravity at the various locations on their travels so that the data might be used for a more accurate estimate of the exact shape of the Earth.
      
    • We described above the adventures he had in taking readings in the south, but later, in 1821, the two extended their results to the north making measurements of the force of gravity using a pendulum in Scotland at Leith, near Edinburgh, and in the Shetland Islands.
      

11. Mathieu Claude biography
    
    • In the same year, together with Biot, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.
      
    • The fact that the project had been so major and unrepeatable was its great virtue, argued Mathieu, for the metre was now fixed for all time and even if the Archive Metre were damaged, its length was known in relation to the pendulum so could be reconstructed.
      

12. Stokes biography
    
    • He also used his work on the motion of pendulums in fluids to consider the variation of gravity at different points on the earth, publishing a work on geodesy of major importance On the variation of gravity at the surface of the earth in 1849.
      
    • Stokes' work on the motion of pendulums in fluids led to a fundamental paper on hydrodynamics in 1851 when he published his law of viscosity, describing the velocity of a small sphere through a viscous fluid.
      

13. Robins biography
    
    • Robins invented the ballistic pendulum which allowed precise measurements of the velocity of projectiles fired from guns.
      
    • When a bullet is fired its momentum is transferred to the bob and can be determined from the amplitude of the pendulum's swing.
      

14. Borda biography
    
    • It considered a proposal which had already been made to the French government to base the metre on the length of a pendulum which beat at the rate of one second.
      
    • His argument against the pendulum standard was that it based one unit on another, which might itself change, and also that the second itself was an arbitrary unit based on the division of a day by 12 × 60 × 60.
      

15. Zhang Heng biography
    
    • It was in the shape of an egg with eight dragon heads around the top, each with a copper ball in its mouth, and a pendulum in the centre.
      

16. Airy biography
    
    • He improved the orbital theory of Venus and the Moon, studied interference fringes in optics, made a mathematical study of the rainbow and computed the density of the Earth by swinging a pendulum at the top and bottom of a deep mine.
      

17. Appell biography
    
    • In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity.
      

18. Bowditch biography
    
    • More scientific publications followed such as one on a meteor explosion in 1807, three papers on orbits of comets (1815, 1818, 1820), and in 1815 he studied Lissajous figures while studying the motion of a pendulum suspended from two points.
      

19. Molyneux William biography
    
    • is most easily perceived through an ordinary telescope of 18 inches long: insomuch that we may determine to the greatest niceity and exactness, when a star passes just over the cross-hairs, even to the single beat of a second-pendulum.
      

20. Atwood biography
    
    • He demonstrated elementary mechanics and hydrostatics with pulleys, pendulums, and air-pumps, as well as electricity, magnetism, and optics, including Leonhard Euler's principles of achromatic lenses.
      

21. Puiseux biography
    
    • He wrote on geometry, where he discovered new properties of evolutes and involutes and mechanics where he studied the conical pendulum, the tautochrone and similar topics.
      

22. Monge biography
    
    • This paper, in which Monge generalised the results obtained by Huygens on space curves (as part of Huygens's investigation of the pendulum) and added many important new discoveries, is described in detail in [Rev.
      

23. Bessel biography
    
    • a correction in 1826 to the seconds pendulum, the length of which is precisely calculated so that it requires exactly one second for a swing.
      

24. Riccati biography
    
    • He also worked on cycloidal pendulums, the laws of resistance in a fluid and differential geometry.
      

25. Poisson biography
    
    • In particular he studied applications to a number of physical problems such as the pendulum in a resisting medium and the theory of sound.
      

26. Lepaute biography
    
    • This work contains the first mathematical work by Nicole-Reine Lepaute who calculated the tables of oscillations of a pendulum which were contained in her husband's work.
      

27. Lissajous biography
    
    • He had produced them in 1815 with a compound pendulum and, because of this, sometimes the figures are referred to as Bowditch figures or Lissajous-Bowditch figures.
      

28. Newton biography
    
    • The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth.
      

29. Auzout biography
    
    • The pendulum clock, used in this calibration, found its principal application in enabling the times at which selected stars crossed the meridian to be accurately noted.
      

30. Mercator Nicolaus biography
    
    • Mercator invented such a marine chronometer, a pendulum clock, and on the strength of this invention he was elected a Fellow of the Royal Society in November 1666.
      

31. Ollerenshaw biography
    
    • Every true mathematician sees mathematics everywhere -- in a child's swing or a pendulum, in the outline shape of a tree and that of its leaves, in the clouds, in the way a circular tube is made from straight strips of paper.
      

32. Mandelbrot biography
    
    • They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition.
      

33. Ceva Giovanni biography
    
    • Although he wrongly concluded that the periods of oscillation of two pendulums were in the same ratio as their lengths, he later corrected the error.
      

34. Picard Jean biography
    
    • Picard was also involved with the measurement of the length of the second pendulum.
      

35. Gompertz biography
    
    • Gompertz also wrote about scientific instruments, writing Theory of astronomical instruments (1822), A new instrument called the differential sextant (1825) and On the converted pendulum (1829).
      

36. Schlafli biography
    
    • Other papers which he published investigate a variety of topics such as partial differential equations, the motion of a pendulum, the general quintic equation, elliptic modular functions, orthogonal systems of surfaces, Riemannian geometry, the general cubic surface, multiply periodic functions, and the conformal mapping of a polygon on a half-plane.
      

37. Greenhill biography
    
    • There, with his books around him, his tables covered in neat disorder with innumerable scraps of material and apparatus to be used as dynamical models, his walls festooned with every variety of pendulum, simple or compound, contrived from articles purchased below a prescribed limit of cost at the local stores, upon his floor the treasured roll of Turkish carpet from his room of long ago at St John's, and above the mantelpiece the portrait of his beloved teacher, Clerk Maxwell, smiling approval - with all these and the precious memories they recalled, the scholar was content.
      

38. Halley biography
    
    • he improved the sextant, collected a number of valuable facts relative to the ocean and atmosphere, noted the equatorial retardation of the pendulum, and made on St Helena, on 7 November 1677, the first complete observation of a transit of Mercury.
      

39. Bossut biography
    
    • The theory of evolutes for plane curves had been presented by Huygens about a century earlier in connection with the study of pendulum clocks.
      

40. Boulliau biography
    
    • Boulliau was a close associate of Huygens who turned to him first with his discovery of the rings of Saturn and sent him pendulum clocks.
      

41. Yunus biography
    
    • Perhaps it is worth mentioning that, contrary to claims which are often made, there is no evidence to suggest that ibn Yunus used a pendulum for time measurements.
      

42. Peirce Charles biography
    
    • Coast Survey being put in charge of pendulum experiments in November 1872, then promoted to assistant in December of that year.
      

43. Cassini biography
    
    • Another measurement made by Jean Richer, namely that a pendulum with a period of one second is shorter in Cayenne than Paris, led him to explain this by suggesting that the Earth was flattened at the poles.
      

44. Brouncker biography
    
    • In particular he carried out many experiments including some on ballistics, some on the pendulum and a study of the variation of the magnetic needle.
      

45. Horrocks biography
    
    • Not content with this theory without evidence, he supported it by analogy with the conical pendulum.
      

46. Anstice biography
    
    • The first paper was On the motion of a free pendulum but the next two are the ones of real interest.
      

47. Gauss biography
    
    • He discussed a modified Foucault pendulum in 1854.
      

48. Wien biography
    
    • In 1912 [Einstein] turned by letter to W Wien with the request to measure the difference between the periods of oscillation of pendulums made of uranium and lead, as well as the proportionality of inertial and gravitational masses of a uranium and a lead weight, respectively, namely with a torsion balance.
      

49. Blackburn biography
    
    • In Blackburn's rooms they swung on 'Blackburn's pendulum', an ingenious device with a double suspension, which could oscillate in planes at right angles to each other.
      

History Topics

1. Longitude1
   
   • Huygens was particularly important to the Academie Royale des Sciences as he had patented the pendulum clock in 1656 and several of his clocks had been tried, although not very successfully, at sea in an attempt to find the longitude.
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   • Having invented the pendulum clock in 1657, Huygens turned his attention to the longitude problem, convinced that the horological approach - to produce a marine timekeeper that would keep accurate and regular time for months on end in any climate, regardless of the motion of the ship - would soon make it possible to discover the longitude.
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   • After beginning work for the Academie Royale in Paris, Huygens tried to perfect the operation of his pendulum clocks.
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   • His early timekeepers were controlled by pendulums but, in anything but a flat calm, their going was most erratic.
     
   • Observations of Jupiter's moons were made with three telescopes and Picard used two pendulum clocks to measure time, one with a pendulum beating once per second, the other clock beating every half second.
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   • However Picard took with him a pendulum clock which had been carefully calibrated in the Paris Observatory before he left.
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   • Picard had to shorten its pendulum by about 0.2 cm to get it to keep correct time.
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   • Other expeditions which set out from Paris on longitude measurements were all told to watch out for any unexpected variations in the performance of their pendulum clocks.
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   • Varin and des Hayes found that, like those of Picard, their clocks did not run correctly and they had to shorten the length of the pendulums.
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2. 20th century time
   
   • There had been remarkable progress towards more and more accurate measurement and at the beginning of the century pendulum clocks had been perfected to the extent that they recorded time to an accuracy of less than 1/100 of a second error in a day.
     
   • This was achieved with a nearly free pendulum clock.
     
   • The free pendulum is one which is completely free from mechanical tasks, such as being part of the driving mechanism of the clock, that would stop it from being completely regular.
     
   • R J Rudd introduced a genuine free pendulum clock in 1898, then W H Shortt introduced a clock with two pendulums in 1921.
     
   • One pendulum was a true free pendulum, the other was part of the driving mechanism of the clock.
     

3. Longitude2
   
   • Moore purchased two clocks for Flamsteed with pendulums 4 m long which were hung above the dials of the clocks.
     
   • There are the two dials on the left with the pendulum bobs visible above the dials.
     
   • By 1727 he had made a very fine clock with a 'gridiron' pendulum which consisted of nine alternating steel and brass rods to eliminate the effects of temperature changes.
     
   • In 1730 Harrison visited London, taking with him his gridiron pendulum and the grasshopper escapement which he had developed, and there he learnt exactly what was required to win the longitude prize.
     

4. Measurement
   
   • Scientists had long seen the benefits of rationalising measures and those such as Wren had proposed a new system based on the yard defined as the length of a pendulum beating at the rate of one second in the Tower of London.
     
   • The basic length should be that of a pendulum which beat at the rate of one second.
     
   • An immediate problem was that the pendulum length depended on the latitude at which the experiment was performed so a latitude had to be chosen.
     
   • Diplomatic wording allowed an international agreement to be reached, but in March 1791 Borda, as chairman of the Commission of Weights and Measures, proposed using instead of the length of a pendulum, the length of 1/10,000,000 of the distance from the pole to the equator of the Earth.
     
   • Britain became hostile to the metre as did Germany which wanted a standard based on the pendulum.
     
   • Now Borda had argued against using the length of a pendulum which beats at the rate of one second to define the metre in 1791 on the reasonable grounds that the second was not a fixed unit but could change with time.
     

5. Classical time
   
   • Long before his discovery of Jupiter's moons he discovered the fundamental property of the pendulum in 1583.
     
   • In fact Galileo used the biological clock built into his body, for he used his own pulse to compare the time taken for the pendulum to swing.
     
   • Galileo does not seem to have realised that his discovery might be used to design an accurate clock until many years later, but around 1640 he did design the first pendulum clock.
     
   • The first to succeed in making a pendulum clock was Huygens in 1656, see [Acad.
     

6. U of St Andrews History
   
   • Gregory hung his pendulum clock on the wall beside the same window.
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   • Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock.
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7. Maxwell's House
   
   • Those by Maxwell are on 'The Conical Pendulum', Descartes' Ovals', 'Meloid and Apioid', and 'Trifocal curves'.
     
   • For example the manuscript on the Conical Pendulum is signed by Maxwell and dated 25 May 47.
     
   • This last manuscript is signed and dated by Tait - 27 May 47, this being two days after Maxwell's Conical Pendulum manuscript.
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8. Gravitation
   
   • Up until this time people had watched a pendulum swinging and seen the bob of the pendulum going back and forward.
     
   • Although the time taken for the bob to rise and fall depended on the length of the pendulum, it did not depend on the weight of the bob.
     

9. Elliptic functions
   
   • For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions.
     

10. Physical world
    
    • He could not conduct experiments to test objects falling in a vacuum, nor could he conduct experiments with a pendulum consisting of a point mass supported by a weightless string swinging without air resistance.
      

Famous Curves

1. Involute
   
   • It was studied by Huygens when he was considering clocks without pendulums that might be used on ships at sea.
     
   • He used the involute of a circle in his first pendulum clock in an attempt to force the pendulum to swing in the path of a cycloid.
     

2. Cycloid
   
   • He constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in a cycloid arc.
     

Societies etc

1. London Royal Society
   
   • It was ordered that Mr Wren be desired to prepare against the next meeting for the pendulum experiment.
     
   • Another related topic was the pendulum clock, and Huygens, who was the first foreign scientist to visit and work with the Society which he did in 1661, was the leading expert.
     

2. London Royal Society
   
   • It was ordered that Mr Wren be desired to prepare against the next meeting for the pendulum experiment.
     
   • Another related topic was the pendulum clock, and Huygens, who was the first foreign scientist to visit and work with the Society which he did in 1661, was the leading expert.
     

3. History of the Royal Society
   
   • It was ordered that Mr Wren be desired to prepare against the next meeting for the pendulum experiment.
     

References

1. References for Galileo
   
   • H Erlichson, Galileo's pendulums and planes, Ann.
     
   • D K Hill, Pendulums and planes : what Galileo didn't publish, Nuncius Ann.
     
   • J MacLachlan, Galileo's experiments with pendulums : real and imaginary, Ann.
     
   • R Naylor, Galileo's simple pendulum, Physis 16 (1974), 23-46.
     

2. References for Fizeau
   
   • A D Aczel, Pendulum (Washington Square Press, Washington, 2003).
     

3. References for Airy
   
   • A Chapman, The pit and the pendulum : George Biddell Airy and his measurements of gravity, Astronomy Now 5 (1991), 18-20.
     

4. References for Ivory
   
   • O Sheynin, Ivory's treatment of pendulum observations, Historia Math.
     

5. References for Yunus
   
   • D A King, Ibn Yunus and the pendulum : a history of errors, Arch.
     

6. References for Robins
   
   • B D Steele, Muskets and pendulums : Benjamin Robins, Leonhard Euler, and the ballistics revolution (1742-1753), Technology and culture 35 (1994), 348-382.
     

7. References for Foucault
   
   • Pendulum : Leon Foucault and the Triumph of Science (Washington Square Press, New York, 2003).
     

Additional material

1. St Andrews Physics Examinations
   
   • Define a simple pendulum and a compound pendulum.
     
   • How may the length of a simple pendulum and its time of vibration be determined by observations made with a compound pendulum? (Kater's method.) .
     
   • The length of a seconds pendulum at a certain place is 39.15 inches: find the local value of g.
     

2. University of Glasgow Examinations
   
   • Investigate the relation between the length of a simple pendulum and the time of oscillation.
     
   • A "second's" pendulum is lengthened 1 per cent.
     
   • Write down the pendulum formula.
     
   • Explain how the pendulum may be used to determine the force of gravity at a given place.
     
   • According to experiments of Sabine, the length of the seconds pendulum at London is 39.139 inches.
     

3. The Tercentenary of the birth of James Gregory
   
   • We can picture him laying plans for his observatory, gaining the confidence and friendship of men of goodwill who provided means for gathering the necessary instruments; his occasional journey to Edinburgh or London or homewards to the north: the famous occasion when he persuaded his townsfolk to hold a church door collection throughout Aberdeen to supply instruments for the observatory at St Andrews! Here stands his own pendulum clock, made by Joseph Knibb of London, with its large dial curiously divided into 60 parts for the seconds, a reminder that, only a few years before, Huygens had discovered the secret of the pendulum.
     

4. Gregory tercentenary
   
   • "Here stands his own pendulum clock," said Professor Turnbull, pointing to the clock in the hall.
     
   • "It was made by Joseph Knibb, of London, with its large dial curiously divided into 60 parts for the seconds, a reminder that only a few years before Huygens had discovered the secret of the pendulum.
     

5. Gregory's Astronomical Clock
   
   • The larger circle is divided into sixty seconds, each of which is sub-divided into three, as the pendulum beats thirds of seconds.
     
   • The pendulum is supported at the back on a knife-edge.
     

6. D'Arcy Thompson on Plato and Planets
   
   • But the fourth sphere was not coaxial with the third, but was set somewhat obliquely to it (just as the first sphere was to the second), and thus introduced another component in the form of a simple harmonic motion, causing the planet to perform apparently a pendulum-like vibration in the plane of the ecliptic, while all the while it was being carried around that circle by the proper motion of the second sphere.
     

7. Edinburgh Physics Examinations
   
   • Upon what physical fact does the possibility of constructing a Compensation Pendulum depend? Show the application to a measuring rod, whose length is to be independent of temperature changes.
     

8. Nevil Maskelyne measures the Earth's density
   
   • But were that the case, the attraction of mountains, and even smaller inequalities in the earth's surface, would be very great, contrary to experiment, and would affect the measures of the degrees of the meridian much more than we find they do; and the variation of gravity in different latitudes, in going from the equator to the poles, as found by pendulums, would not he near so regular as it has been found by experiment to be.
     

9. Max Planck: 'The Nature of Light
   
   • Though the name has been retained for simplicity, the physical theory of light or optics, in its most general sense, has as little to do with the eye and light perceptions as the theory of the pendulum has to do with sound perception.
     

10. Nevil Maskelyne measures the Earth's density
    
    • But were that the case, the attraction of mountains, and even smaller inequalities in the earth's surface, would be very great, contrary to experiment, and would affect the measures of the degrees of the meridian much more than we find they do; and the variation of gravity in different latitudes, in going from the equator to the poles, as found by pendulums, would not he near so regular as it has been found by experiment to be.
      

Quotations

Chronology

1. Chronology for 1650 to 1675
   
   • Huygens patents the first pendulum clock.
     
   • As well as work on the pendulum he investigates evolutes and involutes of curves and finds the evolutes of the cycloid and of the parabola.
     

2. Mathematical Chronology
   
   • Huygens patents the first pendulum clock.
     
   • As well as work on the pendulum he investigates evolutes and involutes of curves and finds the evolutes of the cycloid and of the parabola.