SETTING up and MOVING a PENDULUM CLOCK by Brian Loomes, UK
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The Swinging Spring: Regular and Chaotic Motion
References The Swinging Spring: Regular and Chaotic Motion Leah Ganis May 30th, 2013 Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Outline of Talk I Introduction to Problem I The Basics: Hamiltonian, Equations of Motion, Fixed Points, Stability I Linear Modes I The Progressing Ellipse and Other Regular Motions I Chaotic Motion I References Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Introduction The swinging spring, or elastic pendulum, is a simple mechanical system in which many different types of motion can occur. The system is comprised of a heavy mass, attached to an essentially massless spring which does not deform. The system moves under the force of gravity and in accordance with Hooke's Law. z y r φ x k m Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics We can write down the equations of motion by finding the Lagrangian of the system and using the Euler-Lagrange equations. The Lagrangian, L is given by L = T − V where T is the kinetic energy of the system and V is the potential energy. Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics In Cartesian coordinates, the kinetic energy is given by the following: 1 T = m(_x2 +y _ 2 +z _2) 2 and the potential is given by the sum of gravitational potential and the spring potential: 1 V = mgz + k(r − l )2 2 0 where m is the mass, g is the gravitational constant, k the spring constant, r the stretched length of the spring (px2 + y 2 + z2), and l0 the unstretched length of the spring. -
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda • Introduction to the elastic pendulum problem • Derivations of the equations of motion • Real-life examples of an elastic pendulum • Trivial cases & equilibrium states • MATLAB models The Elastic Problem (Simple Harmonic Motion) 푑2푥 푑2푥 푘 • 퐹 = 푚 = −푘푥 = − 푥 푛푒푡 푑푡2 푑푡2 푚 • Solve this differential equation to find 푥 푡 = 푐1 cos 휔푡 + 푐2 sin 휔푡 = 퐴푐표푠(휔푡 − 휑) • With velocity and acceleration 푣 푡 = −퐴휔 sin 휔푡 + 휑 푎 푡 = −퐴휔2cos(휔푡 + 휑) • Total energy of the system 퐸 = 퐾 푡 + 푈 푡 1 1 1 = 푚푣푡2 + 푘푥2 = 푘퐴2 2 2 2 The Pendulum Problem (with some assumptions) • With position vector of point mass 푥 = 푙 푠푖푛휃푖 − 푐표푠휃푗 , define 푟 such that 푥 = 푙푟 and 휃 = 푐표푠휃푖 + 푠푖푛휃푗 • Find the first and second derivatives of the position vector: 푑푥 푑휃 = 푙 휃 푑푡 푑푡 2 푑2푥 푑2휃 푑휃 = 푙 휃 − 푙 푟 푑푡2 푑푡2 푑푡 • From Newton’s Law, (neglecting frictional force) 푑2푥 푚 = 퐹 + 퐹 푑푡2 푔 푡 The Pendulum Problem (with some assumptions) Defining force of gravity as 퐹푔 = −푚푔푗 = 푚푔푐표푠휃푟 − 푚푔푠푖푛휃휃 and tension of the string as 퐹푡 = −푇푟 : 2 푑휃 −푚푙 = 푚푔푐표푠휃 − 푇 푑푡 푑2휃 푚푙 = −푚푔푠푖푛휃 푑푡2 Define 휔0 = 푔/푙 to find the solution: 푑2휃 푔 = − 푠푖푛휃 = −휔2푠푖푛휃 푑푡2 푙 0 Derivation of Equations of Motion • m = pendulum mass • mspring = spring mass • l = unstreatched spring length • k = spring constant • g = acceleration due to gravity • Ft = pre-tension of spring 푚푔−퐹 • r = static spring stretch, 푟 = 푡 s 푠 푘 • rd = dynamic spring stretch • r = total spring stretch 푟푠 + 푟푑 Derivation of Equations of Motion -
Pendulum Time" Lesson Explores How the Pendulum Has Been a Reliable Way to Keep Time for Centuries
IEEE Lesson Plan: P endulum Time Explore other TryEngineering lessons at www.tryengineering.org L e s s o n F o c u s Lesson focuses on how pendulums have been used to measure time and how mechanical mechanism pendulum clocks operate. Students work in teams to develop a pendulum out of everyday objects that can reliably measure time and operate at two different speeds. They will determine the materials, the optimal length of swing or size of weight to adjust speed, and then develop their designs on paper. Next, they will build and test their mechanism, compare their results with other student teams, and share observations with their class. Lesson Synopsis The "Pendulum Time" lesson explores how the pendulum has been a reliable way to keep time for centuries. Students work in teams to build their own working clock using a pendulum out of every day materials. They will need to be able to speed up and slow down the motion of the pendulum clock. They sketch their plans, consider what materials they will need, build the clock, test it, reflect on the assignment, and present to their class. A g e L e v e l s 8-18. Objectives Learn about timekeeping and engineering. Learn about engineering design and redesign. Learn how engineering can help solve society's challenges. Learn about teamwork and problem solving. Anticipated Learner Outcomes As a result of this activity, students should develop an understanding of: timekeeping engineering design teamwork Pendulum Time Provided by IEEE as part of TryEngineering www.tryengineering.org © 2018 IEEE – All rights reserved. -
Pioneers in Optics: Christiaan Huygens
Downloaded from Microscopy Pioneers https://www.cambridge.org/core Pioneers in Optics: Christiaan Huygens Eric Clark From the website Molecular Expressions created by the late Michael Davidson and now maintained by Eric Clark, National Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 . IP address: [email protected] 170.106.33.22 Christiaan Huygens reliability and accuracy. The first watch using this principle (1629–1695) was finished in 1675, whereupon it was promptly presented , on Christiaan Huygens was a to his sponsor, King Louis XIV. 29 Sep 2021 at 16:11:10 brilliant Dutch mathematician, In 1681, Huygens returned to Holland where he began physicist, and astronomer who lived to construct optical lenses with extremely large focal lengths, during the seventeenth century, a which were eventually presented to the Royal Society of period sometimes referred to as the London, where they remain today. Continuing along this line Scientific Revolution. Huygens, a of work, Huygens perfected his skills in lens grinding and highly gifted theoretical and experi- subsequently invented the achromatic eyepiece that bears his , subject to the Cambridge Core terms of use, available at mental scientist, is best known name and is still in widespread use today. for his work on the theories of Huygens left Holland in 1689, and ventured to London centrifugal force, the wave theory of where he became acquainted with Sir Isaac Newton and began light, and the pendulum clock. to study Newton’s theories on classical physics. Although it At an early age, Huygens began seems Huygens was duly impressed with Newton’s work, he work in advanced mathematics was still very skeptical about any theory that did not explain by attempting to disprove several theories established by gravitation by mechanical means. -
The Case of Switzerland and the World Watch Industry *
469 Technological discontinuities and flexible production networks: The case of Switzerland and the world watch industry * Amy Glasmeier tain and augment their competitiveness in a global Unrr~rs~t~of Texas at Austin, Texas, USA economy. On the eve of the electronics revolution, the Swiss watch production system, centered in the mountainous Jura region, was flexible, cost The twentieth-century history of the Swiss watch industry effective, and extremely profitable. Both horizon- illustrates how cultures and industrial production systems ex- tally and vertically disintegrated, the Swiss system perience great difficulty adapting to external change at differ- offered enormous variety while maintaining qual- ent points in time. The current emphasis on production net- ity and timeliness of delivery. “The multiplicity of works - unique reservoirs of potential technological innovation realized through cooperation rather than competition among enterprises, and the competition and emulation firms - lacks a detailed appreciation of historic networks, and that characterized the industry, yielded a product in particular their fragile character in times of economic of superior quality known the world over for high turmoil. While networks can and do promote innovation within fashion, design, and precision” [21, p. 481. an existing technological framework, historical experience sug- Beginning in the 1970s when foreign competi- gests their fragmented, atomistic structure is subject to dis- organization and disintegration during periods of technological tion hurdled technological frontiers in watch change. An exclusive focus on “production” ignores other movements, advancing from mechanical to elec- constraints that are powerful forces governing the reaction tric, electronic, digital and finally quartz technol- abilities of regions. Previous research has largely relied on a ogy, the Jura’s undisputed dominance ended. -
A History of the Citizen Watch Company, from the Pages of Watchtime Magazine
THE WORLD OF FINE WATCHES SPOTLIGHT www.watchtime.com A HISTORY OF THE CITIZEN WATCH COMPANY, FROM THE PAGES OF WATCHTIME MAGAZINE CCIITTIIZZEENN THe HisTory of ciTizen One of the original Citizen pocket watches that went on THE sale in December 1924 CITIZEN WATCH STORY How a Tokyo jeweler’s experiment in making pocket watches 84 years ago led to the creation of a global watch colossus n the 1920s, the young Emperor of Japan, than the imports. To that end, Yamazaki found - Goto. The mayor was a friend of Yamazaki’s. Hirohito, received a gift that reportedly de - ed in 1918 the Shokosha Watch Research Insti - When the fledgling watch manufacturer was I lighted him. The gift was from Kamekichi tute in Tokyo’s Totsuka district. Using Swiss ma - searching for a name for his product, he asked Yamazaki, a Tokyo jeweler, who had an ambi - chinery, Yamazaki and his team began experi - Goto for ideas. Goto suggested Citizen. A tion to manufacture pocket watches in Japan. menting in the production of pocket watches. watch is, to a great extent, a luxury item, he ex - The Japanese watch market at that time By the end of 1924, they began commercial plained, but Yamazaki was aiming to make af - was dominated by foreign makes, primarily production of their first product, the Caliber fordable watches. It was Goto’s hope that every Swiss brands, followed by Americans like 16 pocket watch, which they sold under the citizen would benefit from and enjoy the time - Waltham and Elgin. Yamazaki felt the time brand name Citizen. -
A Phenomenology of Galileo's Experiments with Pendulums
BJHS, Page 1 of 35. f British Society for the History of Science 2009 doi:10.1017/S0007087409990033 A phenomenology of Galileo’s experiments with pendulums PAOLO PALMIERI* Abstract. The paper reports new findings about Galileo’s experiments with pendulums and discusses their significance in the context of Galileo’s writings. The methodology is based on a phenomenological approach to Galileo’s experiments, supported by computer modelling and close analysis of extant textual evidence. This methodology has allowed the author to shed light on some puzzles that Galileo’s experiments have created for scholars. The pendulum was crucial throughout Galileo’s career. Its properties, with which he was fascinated from very early in his career, especially concern time. A 1602 letter is the earliest surviving document in which Galileo discusses the hypothesis of pendulum isochronism.1 In this letter Galileo claims that all pendulums are isochronous, and that he has long been trying to demonstrate isochronism mechanically, but that so far he has been unable to succeed. From 1602 onwards Galileo referred to pendulum isochronism as an admirable property but failed to demonstrate it. The pendulum is the most open-ended of Galileo’s artefacts. After working on my reconstructed pendulums for some time, I became convinced that the pendulum had the potential to allow Galileo to break new ground. But I also realized that its elusive nature sometimes threatened to undermine the progress Galileo was making on other fronts. It is this ambivalent nature that, I thought, might prove invaluable in trying to understand crucial aspects of Galileo’s innovative methodology. -
Sarlette Et Al
COMPARISON OF THE HUYGENS MISSION AND THE SM2 TEST FLIGHT FOR HUYGENS ATTITUDE RECONSTRUCTION(*) A. Sarlette(1), M. Pérez-Ayúcar, O. Witasse, J.-P. Lebreton Planetary Missions Division, Research and Scientific Support Department, ESTEC-ESA, Noordwijk, The Netherlands. Email: [email protected], [email protected], [email protected] (1) Stagiaire from February 1 to April 29; student at Liège University, Belgium. Email: [email protected] ABSTRACT 1. The SM2 probe characteristics The Huygens probe is the ESA’s main contribution to In agreement with its main purpose – performing a the Cassini/Huygens mission, carried out jointly by full system check of the Huygens descent sequence – NASA, ESA and ASI. It was designed to descend into the SM2 probe was a full scale model of the Huygens the atmosphere of Titan on January 14, 2005, probe, having the same inner and outer structure providing surface images and scientific data to study (except that the deploying booms of the HASI the ground and the atmosphere of Saturn’s largest instrument were not mounted on SM2), the same mass moon. and a similar balance. A complete description of the Huygens flight model system can be found in [1]. In the framework of the reconstruction of the probe’s motions during the descent based on the engineering All Descent Control SubSystem items (parachute data, additional information was needed to investigate system, mechanisms, pyro and command devices) were the attitude and an anomaly in the spin direction. provided according to expected flight standard; as the test flight was successful, only few differences actually Two years before the launch of the Cassini/Huygens exist at this level with respect to the Huygens probe. -
Mini Quartz Clock Movements
Mini Quartz Clock Movements • 10 Year Warranty • Step Second Hand • Dimensions: 2-1/8"W x 2-1/8"H x 5/8"D • Runs on 1 "AA" Size Battery • American "I" Shaft - Diameter 5/16" • Free Set Of Hands and • Front Loading Hanger Mounting Hardware With • Accurate Within 2 Minutes A Year Each Clock Movement • Fits 3" Diameter Hole • Made in USA 1. Drill a 3/8" hole through the material you are working with and insert the movement. 2. Slide brass washer over shaft. 3. Attach dial mounting hex nut. 4. Gently press hour hand onto shaft at 12:00 position. 5. Place minute hand over shaft at 12:00 position. 6. Gently screw minute nut in place. 7. Press on second hand at 12:00 position. 8. Screw on cap nut if no second hand is used. Mini Movements Shaft Length Selecting the Proper Shaft Length Dials B A P r i c e E a c h P e r P k g O f Proper shaft length is important to ensure Stock# up to Thread Total 1 3 10 50 100 sufficient clearance when going through your dial Q-11 1/8" thick 3/16" 17/32" 4.95 4.23 4.45 3.80 4.25 3.63 3.95 3.38 3.75 3.21 board and when using a glass front on your clock. Q-12 1/4" thick 5/16" 5/8" 4.95 4.23 4.45 3.80 4.25 3.63 3.95 3.38 3.75 3.21 Overall Length (A) is measured from the tip of Q-13 3/8" thick 7/16" 3/4" 4.95 4.See23 4.4 website5 3.80 4.25 3for.63 3current.95 3.38 3.75 3.21 the hand shaft to the movement cast. -
Computer-Aided Design and Kinematic Simulation of Huygens's
applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock. -
Vertical Oscillations of Mass on Spring • Pendulum • Damped and Driven
Lecture 2 Outline • Vertical oscillations of mass on spring • Pendulum • Damped and Driven oscillations (more realistic) Vertical Oscillations (I) • At equilibrium (no net force), spring is stretched (cf. horizontal spring): spring force balances gravity Hooke’s law: (F ) = k∆y =+k∆L sp y − Newton’s law: (F ) =(F ) +(F ) = k∆L mg =0 net y sp y G y − ∆L = mg ⇒ k Vertical Oscillations (II) • Oscillation around equilibrium, y = 0 (spring stretched) block moves upward, spring still stretched (F ) =(F ) +(F ) = k (∆L y) mg net y sp y G y − − Using k∆L mg = 0 (equilibrium), (F ) = ky − net y − • gravity ``disappeared’’... as before: y(t)=A cos (ωt + φ0) Example • A 8 kg mass is attached to a spring and allowed to hang in the Earth’s gravitational field. The spring stretches 2.4 cm. before it reaches its equilibrium position. If allowed to oscillate, what would be its frequency? Pendulum (I) • Two forces: tension (along string) and gravity • Divide into tangential and radial... (F ) =(F ) = mg sin θ = ma net tangent G tangent − tangent acceleration around circle 2 d s = g sin θ dt2 − more complicated Pendulum (II) • Small-angle approximation sin θ θ (θ in radians) ≈ (F ) mg s net tangent ≈− L d2s g (same as mass on spring) dt2 = L s ⇒ s(t)=−A cos (ωt + φ ) or θ(t)=θ cos (ωt + φ ) ⇒ 0 max 0 (independent of m, cf. spring) Example • The period of a simple pendulum on another planet is 1.67 s. What is the acceleration due to gravity on this planet? Assume that the length of the pendulum is 1m. -
Relativity and Clocks
RELATIVITY AND CLOCKS Carroll 0. Alley Universityof Maryland CollegePark, Maryland Sumnary be mentioned. In this centennial year of the birth of Albert The internationaltimekeeping communityshould Einstein, it is fittingto review the revolutionary takegreat pride in the fact that the great stabil- andfundamental insights about time whichhe gave ity of cont-mporaryatomic clocks requires the us inthe Restricted Theory of Relativity (1905) first practical applications g Einstein's General and in the conseqences of the Principle of Equiv- Theory of Relativity.This circumstance can be ex- alence (l'. .The happiest thought of my life.. .") pected to produce a better understanding among whichhe developed (1907-1915) into histheory of physicists andengineers of the physical basis of gravityas curved space-time, the General Theory of gravity as curvedspace-time. For slow motions and Relativity. weak gravitational fields, such as we normally ex- perience on the earth, the primary curvature is It is of particularsignificance that the ex- thatof &, notspace. A body falls,according traordinary stability ofmodern atomicclocks has to Einstein's view, not because of the Newtonian recentlyallowed the experimental Study and accur- forcepulling it tothe earth, but because of the ate measurement of these basic effects of motion properties of time: clocksrun slower when moving and gravitationalpotential on time. Experiments andrun faster or slower, the higher or lower re- with aircraft flights and laser pulseremote time spectively they are in the earth's gravity field. comparison(Alley, Cutler, Reisse, Williams, et al, 1975)and an experiment with a rocketprobe (Vessot,Levine, et al, 1976) are brieflydes- Some Eventsin Einstein's cribed. Intellectual Development Properunderstanding and allowance for these Figure 1 shows Einstein in his study in Berlin remarkableeffects is now necessary for accurate several years after he hadbrought the General global time synchronization using ultrastable Theoryof Relativity to its complete form in 1915.