Huygens's Clocks
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Chapter 7. the Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methods vs. Energy Methods We have so far studied two distinct ways of analyzing physics problems: force methods, basically consisting of the application of Newton’s Laws, and energy methods, consisting of the application of the principle of conservation of energy (the conservations of linear and angular momenta can also be considered as part of this). Both have their advantages and disadvantages. More precisely, energy methods often involve scalar quantities (e.g., work, kinetic energy, and potential energy) and are thus easier to handle than forces, which are vectorial in nature. However, forces tell us more. The simple example of a particle subjected to the earth’s gravitation field will clearly illustrate this. We know that when a particle moves from position y0 to y in the earth’s gravitational field, the conservation of energy tells us that ΔK + ΔUgrav = 0 (7.1) 1 2 2 m v − v0 + mg(y − y0 ) = 0, 2 ( ) which implies that the final speed of the particle, of mass m , is 2 v = v0 + 2g(y0 − y). (7.2) Although we know the final speed v of the particle, given its initial speed v0 and the initial and final positions y0 and y , we do not know how its position and velocity (and its speed) evolve with time. On the other hand, if we apply Newton’s Second Law, then we write −mgey = maey dv (7.3) = m e . dt y This equation is easily manipulated to yield t v = dv ∫t0 t = − gdτ + v0 (7.4) ∫t0 = −g(t − t0 ) + v0 , - 138 - where v0 is the initial velocity (it appears in equation (7.4) as a constant of integration). -
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. -
Exomars Schiaparelli Direct-To-Earth Observation Using GMRT
TECHNICAL ExoMars Schiaparelli Direct-to-Earth Observation REPORTS: METHODS 10.1029/2018RS006707 using GMRT S. Esterhuizen1, S. W. Asmar1 ,K.De2, Y. Gupta3, S. N. Katore3, and B. Ajithkumar3 Key Point: • During ExoMars Landing, GMRT 1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, 2Cahill Center for Astrophysics, observed UHF transmissions and California Institute of Technology, Pasadena, CA, USA, 3National Centre for Radio Astrophysics, Pune, India Doppler shift used to identify key events as only real-time aliveness indicator Abstract During the ExoMars Schiaparelli separation event on 16 October 2016 and Entry, Descent, and Landing (EDL) events 3 days later, the Giant Metrewave Radio Telescope (GMRT) near Pune, India, Correspondence to: S. W. Asmar, was used to directly observe UHF transmissions from the Schiaparelli lander as they arrive at Earth. The [email protected] Doppler shift of the carrier frequency was measured and used as a diagnostic to identify key events during EDL. This signal detection at GMRT was the only real-time aliveness indicator to European Space Agency Citation: mission operations during the critical EDL stage of the mission. Esterhuizen, S., Asmar, S. W., De, K., Gupta, Y., Katore, S. N., & Plain Language Summary When planetary missions, such as landers on the surface of Mars, Ajithkumar, B. (2019). ExoMars undergo critical and risky events, communications to ground controllers is very important as close to real Schiaparelli Direct-to-Earth observation using GMRT. time as possible. The Schiaparelli spacecraft attempted landing in 2016 was supported in an innovative way. Radio Science, 54, 314–325. A large radio telescope on Earth was able to eavesdrop on information being sent from the lander to other https://doi.org/10.1029/2018RS006707 spacecraft in orbit around Mars. -
Aerothermodynamic Analysis of a Mars Sample Return Earth-Entry Vehicle" (2018)
Old Dominion University ODU Digital Commons Mechanical & Aerospace Engineering Theses & Dissertations Mechanical & Aerospace Engineering Summer 2018 Aerothermodynamic Analysis of a Mars Sample Return Earth- Entry Vehicle Daniel A. Boyd Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/mae_etds Part of the Aerodynamics and Fluid Mechanics Commons, Space Vehicles Commons, and the Thermodynamics Commons Recommended Citation Boyd, Daniel A.. "Aerothermodynamic Analysis of a Mars Sample Return Earth-Entry Vehicle" (2018). Master of Science (MS), Thesis, Mechanical & Aerospace Engineering, Old Dominion University, DOI: 10.25777/xhmz-ax21 https://digitalcommons.odu.edu/mae_etds/43 This Thesis is brought to you for free and open access by the Mechanical & Aerospace Engineering at ODU Digital Commons. It has been accepted for inclusion in Mechanical & Aerospace Engineering Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. AEROTHERMODYNAMIC ANALYSIS OF A MARS SAMPLE RETURN EARTH-ENTRY VEHICLE by Daniel A. Boyd B.S. May 2008, Virginia Military Institute M.A. August 2015, Webster University A Thesis Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE AEROSPACE ENGINEERING OLD DOMINION UNIVERSITY August 2018 Approved by: __________________________ Robert L. Ash (Director) __________________________ Oktay Baysal (Member) __________________________ Jamshid A. Samareh (Member) __________________________ Shizhi Qian (Member) ABSTRACT AEROTHERMODYNAMIC ANALYSIS OF A MARS SAMPLE RETURN EARTH-ENTRY VEHICLE Daniel A. Boyd Old Dominion University, 2018 Director: Dr. Robert L. Ash Because of the severe quarantine constraints that must be imposed on any returned extraterrestrial samples, the Mars sample return Earth-entry vehicle must remain intact through sample recovery. -
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 -
Huygens Probe Set to Detach from Cassini Orbiter Tonight 24 December 2004
Update: Huygens Probe Set to Detach From Cassini Orbiter Tonight 24 December 2004 mission is approximately $3 billion. Many of these sophisticated instruments are capable of multiple functions, and the data that they gather will be studied by scientists worldwide. Aerosol Collector and Pyrolyser (ACP) will collect aerosols for chemical-composition analysis. After extension of the sampling device, a pump will draw the atmosphere through filters which capture aerosols. Each sampling device can collect about 30 micrograms of material. Descent Imager/Spectral Radiometer (DISR) can take images and make spectral measurements using sensors covering a wide spectral range. A few hundred metres before impact, the instrument will switch on its lamp in order to acquire spectra of the surface material. The highlights of the first year of the Cassini- Doppler Wind Experiment (DWE) uses radio Huygens mission to Saturn can be broken into two signals to deduce atmospheric properties. The chapters: first, the arrival of the Cassini orbiter at probe drift caused by winds in Titan's atmosphere Saturn in June, and second, the release of the will induce a measurable Doppler shift in the carrier Huygens probe on Dec. 24, 2004, on a path signal. The swinging motion of the probe beneath toward Titan. (read PhysOrg story) its parachute and other radio-signal-perturbing effects, such as atmospheric attenuation, may also The Huygens probe, built and managed by the be detectable from the signal. European Space Agency (ESA), is bolted to Cassini and fed electrical power through an Gas Chromatograph and Mass Spectrometer umbilical cable. It has been riding along during the (GCMS) is a versatile gas chemical analyser nearly seven-year journey to Saturn largely in a designed to identify and quantify various "sleep" mode, awakened every six months for atmospheric constituents. -
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. -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
Vleeuwen on Van Call
© 2013 Antiquarian Horological Society. Reproduction prohibited without permission. MARCH 2013 Jan van Call and the age of the pendulum clock in the Netherlands Pier van Leeuwen* The signature of Jan van Call, best known for his work on turret clocks, appears on a controversial wall clock that was the subject of a symposium held at the British Museum in 2011. At this symposium, the author sketched van Call’s documented biographical data against the colourful background of Anglo-Dutch history. This article, a much reduced version of that presentation, focuses on Van Call’s specific achievements and preserved heritage. From Kall to Nijmegen Brabant. The brass drum had twenty-four holes on one Jan van Call was born in Kall in the German Eifel and rule of about 45 cm. On 1 January 1647 Jan Becker moved to Batenburg near Nijmegen in the Dutch duchy Call was granted citizenship in the town of Nijmegen, of Guelders (since 1814 Province of Guelderland). This by which time he must have moved from Batenburg to old capital of the duchy had a rich medieval history, the capital. Amongst other constructional work he renowned among art historians as the birthplace of the produced pumps, drains and waterworks.3 Limburg brethren, court illuminators of at least two Books of Hours for the Duke of Berry. Nijmegen had an Repairing a monumental clock impressive medieval castle known as the Valkhof, once In the year in which he officially became a citizen of inhabited by emperor Charles V who made the old Nijmegen, Jan van Call repaired, for the fee of 280 duchy another part of his Habsburg realm. -
Evolution of Clock Escapement Mechanisms
Miodrag Stoimenov Associate Professor Evolution of Clock Escapement University of Belgrade Faculty of Mechanical Engineering Mechanisms Branislav Popkonstantinović Associate Professor The paper presents and explains the evolution of details design of the clock University of Belgrade escapement mechanisms through the ages. As particularly significant, the Faculty of Mechanical Engineering following mechanisms are emphasized: the crown wheel (verge & foliot), Ljubomir Miladinović anchor recoil, deadbeat and detached escapements, and their variations – Associate Professor gravity and chronometer escapements, as well as the English and Swiss University of Belgrade lever watch escapements. All important geometrical, kinematical and Faculty of Mechanical Engineering dynamical properties and the influence of these properties on the clock Dragan Petrović accuracy are explained. Associate Professor University of Belgrade Keywords: clock, escapement, evolution, mechanism, lever. Faculty of Mechanical Engineering 1. INTRODUCTION oscillator with the restoring force and standardized pace. The crown wheel rate, acted upon by driving torque, is This work is aimed at a qualitative analysis of regulated only by the foliot inertia with a rather advancement in escapement mechanism structural unpredictable error. When the crown wheel is rotating, definition and development of a timepiece over the past the pallet of the verge is caught by one of the wheel’s seven centuries. This study cowers the following issues teeth, rotating the verge and a solid foliot in one a) the crown wheel, the verge and foliot, b) the recoil direction and leading to engagement of the opposite anchor escapement, c) deadbeat escapements, and d) tooth with the second pallet [4]. Due to the foliot inertia, detached escapements. Measure of the effectiveness in that next tooth stops the wheel’s motion, and the wheel the regulator horological properties could be specified with its eigen driving torque finally stops the rotation of as follows: a) the level of constructive and dynamic the foliot too. -
The Evolution of Tower Clock Movements and Their Design Over the Past 1000 Years
The Evolution Of Tower Clock Movements And Their Design Over The Past 1000 Years Mark Frank Copyright 2013 The Evolution Of Tower Clock Movements And Their Design Over The Past 1000 Years TABLE OF CONTENTS Introduction and General Overview Pre-History ............................................................................................... 1. 10th through 11th Centuries ........................................................................ 2. 12th through 15th Centuries ........................................................................ 4. 16th through 17th Centuries ........................................................................ 5. The catastrophic accident of Big Ben ........................................................ 6. 18th through 19th Centuries ........................................................................ 7. 20th Century .............................................................................................. 9. Tower Clock Frame Styles ................................................................................... 11. Doorframe and Field Gate ......................................................................... 11. Birdcage, End-To-End .............................................................................. 12. Birdcage, Side-By-Side ............................................................................. 12. Strap, Posted ............................................................................................ 13. Chair Frame .............................................................................................