Pendulum

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  Pendulum in Physics

  • (Author)
  • 2014(Publication Date)
  • The English Press

    (Publisher)

  Mathematically, for small swings the Pendulum approximates a harmonic oscillator, and its motion approximates to simple harmonic motion: Compound Pendulum The length L of the ideal simple Pendulum above, used for calculating the period, is the distance from the pivot point to the center of mass of the bob. For a real Pendulum consisting of a swinging rigid body, called a compound Pendulum , the length is more difficult to define. A real Pendulum swings with the same period as a simple Pendulum with a length equal to the distance from the pivot point to a point in the Pendulum called the center of oscillation . This is located under the center of mass, at a distance called the radius of gyration, that depends on the mass distribution along the Pendulum. However, for the usual sort of Pendulum in which most of the mass is concentrated in the bob, the center of oscillation is close to the center of mass. ________________________ WORLD TECHNOLOGIES ________________________ Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. This means if any Pendulum is turned upside down and swung from a pivot at the center of oscillation, it will have the same period as before, and the new center of oscillation will be the old pivot point. History One of the earliest known uses of a Pendulum was in the 1st century seismometer device of Han Dynasty Chinese scientist Zhang Heng. Its function was to sway and activate one of a series of levers after being disturbed by the tremor of an earthquake far away. Released by a lever, a small ball would fall out of the urn-shaped device into one of eight metal toad's mouths below, at the eight points of the compass, signifying the direction the earthquake was located. Many sources claim that the 10th century Egyptian astronomer Ibn Yunus used a Pendulum for time measurement, but this was an error that originated in 1684 with the British historian Edward Bernard.

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  Chaotic Pendulum, The

  • Moshe Gitterman(Author)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 1 Pendulum Equations 1.1 Mathematical Pendulum The Pendulum is a massless rod of length l with a point mass (bob) m at its end (Fig. 1.1). When the bob performs an angular deflection φ from the equilibrium downward position, the force of gravity mg provides a restoring torque -mgl sin φ. The rotational form of Newton’s second law of motion states that this torque is equal to the product of the moment of inertia ml 2 times the angular acceleration d 2 φ/dt 2 , d 2 φ dt 2 + g l sin φ = 0 (1.1) Fig. 1.1 Mathematical Pendulum. For small angles, sin φ ≈ φ , Eq. (1.1) reduces to the equation of a har-monic oscillator. The influence of noise on an oscillator has been considered earlier [2]. The main difference between the oscillator and the Pendulum 1 2 The Chaotic Pendulum is that the former has a fixed frequency p g/l, whereas the Pendulum pe-riod decreases with increasing amplitude. Multiplying Eq. (1.1) by dφ/dt and integrating, one obtains the general expression for the energy of the Pendulum, E = l 2 2 dφ dt 2 + gl (1 -cos φ ) (1.2) where the constants were chosen to make the potential energy vanishes at the downward vertical position of the Pendulum. Systems with constant energy are called conservative systems. In the ( φ, dφ/dt ) plane, the trajec-tories are contours of constant energy. Depending on the magnitude of the energy E, there are three different types of phase trajectories in the ( φ, dφ/dt ) plane (Fig. 1.2): Fig. 1.2 Phase plane of a mathematical Pendulum. 1. E < 2 gl. The energy is less than the critical value 2 gl, which is the energy required for the bob to reach the upper position. Under these conditions, the angular velocity dφ/dt vanishes for some angles ± φ 1 , i.e., the Pendulum is trapped in one of the minima of the cosine potential well, performing simple oscillations (“librations”) around the position of the min-

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  Workshop Physics Activity Guide Module 2

  Mechanics II

  • Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  Name Section Date UNIT 14: SIMPLE HARMONIC MOTION Alaettin YILDIRIM/Shutterstock.com The swing of this antique clock Pendulum is one of many examples of what physicists call simple harmonic motion. Due to the Pendulum motion, each tick of the clock takes the same amount of time, and so it serves as a reliable timekeeper. And if the clock is running slow or fast, one can adjust the Pendulum to compensate. By the end of this unit, you should have an understanding of why a Pendulum undergoes simple harmonic motion. 460 WORKSHOP PHYSICS ACTIVITY GUIDE UNIT 14: SIMPLE HARMONIC MOTION OBJECTIVES 1. To learn about the quantities used to describe periodic (and harmonic) motion, such as period, frequency, amplitude, and phase. 2. To understand the basic properties of simple harmonic motion, in which the displacement of an object varies sinusoidally in time, and to experi- mentally measure such motion in different systems. 3. To use Newton’s laws to theoretically explore the factors that influence the motion of both a mass-spring system and a simple Pendulum. 14.1 OVERVIEW Any motion that repeats itself regularly is known as periodic motion. The Pendulum in a grandfather clock, molecules in a crystal, the vibrations of a car after it encounters a pothole on the road, and the rotation of Earth around the sun are examples of periodic motion. In this unit we will be especially interested in a type of periodic motion known as simple harmonic motion, which is often abbreviated as SHM. Simple harmonic motion involves a displacement that changes sinusoidally in time. We will study the behavior of two systems that undergo SHM—an object hanging from a spring (a mass-spring system) and a simple Pendulum that oscillates at small angles. Pendula and masses on springs are two common examples of periodic systems that oscillate with SHM (or at least approximately SHM).

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  Science Teaching

  The Contribution of History and Philosophy of Science, 20th Anniversary Revised and Expanded Edition

  • Michael R. Matthews(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  An isochronic Pendulum is one in which the period of the first swing is equal to that of all subsequent swings: this implies perpetual motion. We know that any Pendulum, when let swing, will very soon come to a halt: the period of the last swing will be by no means the same as the first. Furthermore, it was plain to see that cork and lead Pendulums have a slightly different frequency, and that large-amplitude swings do take somewhat longer than small-amplitude swings for the same Pendulum length. All of this was pointed out to Galileo, and he was reminded of Aristotle’s basic methodological claim that the evidence of the senses is to be preferred over other evidence in developing an understanding of the world. The fundamental laws of classical mechanics are not verified in experience; further, their direct verification is fundamentally impossible. Herbert Butterfield (1900–1979) conveys something of the problem that Galileo and Newton had in forging their new science: 11 They were discussing not real bodies as we actually observe them in the real world, but geometrical bodies moving in a world without resistance and without gravity History and Philosophy: Pendulum Motion 227 – moving in that boundless emptiness of Euclidean space which Aristotle had regarded as unthinkable. In the long run, therefore, we have to recognise that here was a problem of a fundamental nature, and it could not be solved by close observation within the framework of the older system of ideas – it required a transposition in the mind. (Butterfield 1949/1957, p. 5) An objectivist, non-empiricist account of science stresses that the transposi- tion in the mind is really the creation of a new theoretical object or system. Even for Galileo, the Pendulum seemed to stop at the top of its swing; it was only in his theory, not his perceptual mind, that it continued in smooth motion.

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  Assume the Pendulum consists of a massless string and a bob of mass M and volume V . In this problem, we consider the effect of the buoyancy of air which can be treated as an additional force acting opposite to gravity. (a) Show that for a Pendulum in air of density D, the equation of motion is I 0 ¨ θ = −Mgh sin θ + V Dgh sin θ, where I 0 = Mk 2 is the moment of inertia of the Pendulum with respect to the point of support, h is the distance from the point of support to the center of mass of the bob, and V is the volume of the bob. (b) Show that if l is the length of an equivalent simple Pendulum, x y s m l Cycloid Figure 12.16 An isochronous Pendulum can be constructed such that the length of the string varies with angle. However, the analysis requested in Problem 12.21 is easier if one assumes the bob is a bead on a cycloidal wire. Computational Projects 347 l = k 2 (1 − DV/M)h . (c) Show that if the density of air is given by D + δD, where D is a standard density, the length of the equivalent Pendulum changes by δl , where δl l 2 = h k 2 V M δD. Computational Projects Computational Project 12.1 Write a program to evaluate elliptic integrals of the first kind. Computational Project 12.2 Use the results of Computational Project 12.1 to write a program that evaluates the period of a simple Pendulum with arbitrary amplitude. Vary the amplitude and obtain a plot of period as a function of amplitude. Computational Project 12.3 Plot position vs. time for several points on a rod undergoing an impulse, as illustrated in Figure 12.4. Assume the impulse is J = 10 Ns, the rod has mass M = 2 kg and length L = 0.5 m. 13 Waves A wave is an oscillation of a medium, such as a water wave in the ocean or a wave propagating down a stretched slinky or a taut string. In general, the medium as a whole does not translate. For example, a wave in the ocean can be thought of as water molecules moving vertically up and down while the wave itself moves horizontally.

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  Experiments and Demonstrations in Physics

  Bar-Ilan Physics Laboratory

  • Yaakov Kraftmakher(Author)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  et al (2002) observed oscillations of a Pendulum with one of three damping effects: friction not depending on the velocity (sliding friction), linear dependence on the velocity (eddy currents in a metal plate), and quadratic dependence on the velocity (air friction).

  For other experiments with Pendulums, see Lapidus (1970); Schery (1976); Hall and Shea (1977); Simon and Riesz (1979); Hall (1981); Yurke (1984); Eckstein and Fekete (1991); Ochoa and Kolp (1997); Peters (1999); Lewowski and Woźniak (2002); LoPresto and Holody (2003); Parwani (2004); Coullet et al (2005); Ng and Ang (2005); Jai and Boisgard (2007); Gintautas and Hübler (2009); Mungan and Lipscombe (2013).

  Theoretical background. For a mathematical Pendulum with friction, the motion equation is

  where m is the mass, a is the acceleration, x is the displacement, v is the velocity, k and λ are coefficients of proportionality, x′ = v = dx/dt, and x″ = d2 x/dt2 . The solution to this equation describing free oscillations is

  where A and φ1 depend on the initial conditions. The natural frequency of the system is ω0 = (k/m)½ = (g/l)½ , the decay of free oscillations is governed by the decay constant δ = λ/2m, and the angular frequency Ω is somewhat lower than ω0 : Ω2 = ω0 2 – δ2 .

  The transient process after application of a periodic driving force deserves special consideration. Whenever forced oscillations start, a process occurs leading to steady oscillations. This transient process is caused by the superposition of forced and free oscillations. Landau and Lifshitz (1982) and Pippard (1989) considered the problem in detail, and this analysis is partly reproduced here. For a driven mathematical Pendulum

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  Science Teaching

  The Contribution of History and Philosophy of Science, 20th Anniversary Revised and Expanded Edition

  • Michael R. Matthews(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  It remains to be seen how the Pendulum will feature in the final NGSS document, but the signs are not good. In the current (2012) draft, the Pendulum is mentioned four times, and each time it is in connection with the transformation of energy from potential to kinetic forms. This is a level of abstraction way beyond what is needed or called for, it is beyond the life experience of the students and it reifies the role played by the Pendulum in the history of physics and in its social utilisation. The draft document mentions Newton’s laws, his theory of gravitation and the conservation of momentum, but no mention of the Pendulum, which could so easily be used to make manifest and experiential each of these learning goals.

  Conclusion

  The Pendulum case has been introduced in this chapter as an example where the HPS can contribute to even routine science education. It provides an opportunity to learn about science at the same time as one is learning the subject matter of science. With good HPS-informed teaching the Pendulum-motion case enables students to appreciate the transition from common-sense and empirical descriptions characteristic of Aristotelian science, to the abstract, idealised and mathematical descriptions characteristic of the scientific revolution. The Pendulum provides a manageable, understandable and straightforward way into scientific thinking and away from everyday and empirical thinking; it shows, at the same time, how scientific, idealised thinking nevertheless is connected with the world through controlled experiment.

  The ‘contextual’ teaching of science, as suggested here, is not a retreat from serious, or hard, science, but the reverse. To understand what happened in the history of science takes effort. Further, it is appealing to students. A frequent refrain from intelligent students who do not go on with study in the sciences is that, ‘science is too boring; we only work out problems’.37

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