Torsional Pendulum

5 Key excerpts on "Torsional Pendulum"

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  eBook - PDF

  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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  Mechanical Science

  • W. C. Bolton(Author)
  • 2013(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  When the pendulum is displaced through an angle θ then the restoring moment is mgh sin θ . For Free vibrations 395 Fig. 16.7 Compound pendulum. small angles this becomes approximately mgh θ . When released, an angular acceleration α is produced where mgh θ ≈ I O α with I O being the moment of inertia of the pendulum about an axis through O. If I G is the moment of inertia about the centre of gravity G then by the parallel axis theorem I O = I G + mh 2 The moment of inertia I G can be represented by mk 2 g , where k g is the radius of gyration of the pendulum about the centre of gravity. Thus mgh θ ≈ (mk 2 g + mh 2 ) α Thus θ / α = (k 2 g + h 2 )/gh and so equation [10] gives periodic time = 1 f = 2 π θ α = 2 π k 2 g + h 2 gh [15] Example A connecting rod has a mass 5.0 kg and radius of gyration about an axis through its centre of gravity of 170 mm. The connecting rod is supported on a knife edge inside one of the bearings so that it can oscillate like a pendulum about the knife edge, this being 200 mm above the centre of gravity. What is the periodic time of the oscillations? Using equation [15], periodic time = 2 π k 2 g + h 2 gh = 2 π 0 . 170 2 + 0 . 200 2 9 . 8 × 0 . 200 = 1 . 2 s 396 Mechanical Science 16.3 Torsional vibrations of an elastic system Consider an elastic system consisting of a rod attached at one end to a rigid support and at the free end to a rotor of moment of inertia I (figure 16.8). In the same way as with a spring being stretched we can write F = kx , for the relationship between the force F and the resulting extension x , then we can write for the twisting of the rod T = q θ for the relationship between the torque T and the angle of twist θ . With the spring k is referred to as the stiffness, with the twisted rod q is the torsional stiffness. Thus when the free end of the rod in figure 16.8 is twisted through angle θ then the restoring torque T = q θ . When released this torque gives the rotor an angular acceler-ation α where T = I α .

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  Simulations of Oscillatory Systems

  with Award-Winning Software, Physics of Oscillations

  • Eugene I. Butikov(Author)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  In our approach to the problem we try to rely primarily on the physics un-derlying the investigated phenomena. In this chapter we are concerned with free oscillations of a torsion spring pendulum, and with forced oscillations of the pen-dulum kinematically driven by an external sinusoidal force, including cases of damping caused by dry (Coulomb) friction, and both by viscous and dry friction. Mathematically, the pendulum driven by an external force is equivalent to the spring-mass system with the body residing on the horizontally oscillating base. The simple formulae of analytical solutions are confirmed by graphs obtained in computer simulations. New results cover quantitative description of the resonant growth of oscillations under sinusoidal forcing, and closed-form analytical solu-tions at sub-resonant frequencies. These solutions correspond to multiple asym-metric steady-state regimes coexisting at the same values of the system parame- 11.1. BASICS OF THE THEORY 309 ters. Characteristics of such regimes depend on the initial conditions. Our analyt-ical and numerical solutions are illustrated by a simplified version of the relevant simulation program (Java applet) available on the web [58]. 11.1.2 The Physical System The rotating component of the torsion spring oscillator investigated in the paper is a balanced flywheel whose center of mass lies on the axis of rotation (Figure 11.1), similar to devices used in mechanical watches. j q ( t ) ( t ) + -q 0 q 0 d d + -Figure 11.1: Schematic diagram of the driven torsion oscillator with dry friction. A spiral spring with one end attached to the flywheel flexes when the flywheel is turned. The other end of the spring is attached to the exciter — a driving rod, which can be turned by an external force about the axis common with the flywheel axis. The spring provides a restoring torque whose magnitude is proportional to the angular displacement of the flywheel relative to the driving rod.

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  Experiments and Demonstrations in Physics

  Bar-Ilan Physics Laboratory

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

    (Publisher)

  Fig. 1. Schematic of the experiment.

  Weighing a swinging pendulum allows determinations of the centripetal force on the pendulum bob. This force reaches a maximum when the bob passes its lower position, two times a period. A steel ball and a thread form a pendulum (Fig. 1 ). The thread is attached to the Force sensor, which measures the vertical component of the tension. For small-angle oscillations, the angular position of the pendulum is

  where ω2 = g/l, l is the length of the pendulum, θo is the maximum angular displacement; the initial phase of the oscillations is taken zero. The angular velocity of the pendulum is θ′ = ωθo cosωt. The centripetal force equals

  where m is the mass of the bob, and v is its linear velocity.

  The maximum kinetic energy equals the potential energy at the point of maximum deflection θo :

  The maximum centripetal force thus is

  For θo = 90°, the maximum centripetal force is twice the weight of the bob. The tension T in the pendulum thread is the sum of the bob weight component along the thread and of the centripetal force:

  The vertical component of the tension equals

  For small angles, cosθ ≈ 1 – θ2 /2. Neglecting terms involving powers of θ greater than two,

  At the lower position of the pendulum (θ = 0, ωt = 0),

  while at the upper position (θ = θ0 , ωt = π/2),

  For small-angle oscillations, the weight measured by the Force sensor oscillates with an amplitude ΔTv = mgθ0 2 ; the frequency of the oscillations is 2ω. In our setup, m = 45 g, and l = 30 cm. The maximum displacement of the pendulum is determined with a scale positioned behind the pendulum. A small permanent magnet attached to the bob and a coil positioned under the pendulum serve for detecting the instants when the pendulum passes its lower position. The voltage induced in the coil is measured with the Voltage sensor and displayed by the Graph tool, along with the force measured by the Force sensor (Fig. 2 ). It is seen that the maximum force is reached when the bob passes its lower position. The initial weight of the pendulum is excluded using the Tare button of the Force sensor. From the measurements, the maximum kinetic energy of the pendulum, mv0 2 /2, is determined through ΔTv = mv0 2 /l, so that mv0 2 /2 = lΔTv /2. This kinetic energy can be compared with the maximum potential energy of the pendulum, mglsinθ0

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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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