The Mighty Pendulum and the Measure of Heaven and Earth

  • Martin Beech is Full Professor of Astronomy at Campion College located at the University of Regina in Saskatchewan, Canada. His research interests include the study of stellar structure and evolution, the smaller bodies (asteroids, comets and meteoroids) of the solar system and the history of science.
  • ISBN: 978-1-938681-04-2


The world is in perpetual turmoil and motion – it is a vibrant extravaganza of interactions, oscillations and pulses. Movement surrounds us, and our lives dance to its ever-changing tune. Deep down, however, there is an underlying order, at least of sorts, and we can breakdown the complex motions of the everyday world into the summation of simple harmonic waves. Underpinning all this dynamical mêlée is the pendulum – perhaps the simplest experimental device ever invented: a weight and a piece of string. The pendulum for all its simplicity, however, has been at the historical center of humanities exploration of the Earth and the heavens, and this book is about some of that remarkable history. Based upon material used in a history of astronomy course, and written with the science undergraduate student in mind, the text takes the reader through a mathematical, physical and historical tour of the pendulums many contributions to fundamental science. From the study of Earth’s interior, to the measure of time and space; from the Victorian parlor room, to the boundaries of chaos; to the overarching domains of cosmology and the quantum world, the pendulum, indeed, the mighty pendulum has featured front and center. The pendulum, subtle in its power and long in its reach, has faithfully accompanied humanity on its quest to understand the universe. And, while the narrative of the pendulum begins in our ancient past, its story continues to have great contemporary relevance, and there is every reason to suppose that both its metaphorical and experimental importance will continue into the distant future.

Table of Contents

Chapter 1:The mathematical pendulum

    Dimensional analysis

    A first approximate equation

    The Seconds Pendulum

    Gravimetry and Kater’s pendulum

    Measuring height by time

    The Devil is in the details

    More details and complications

    Conservation of momentum and Newton’s cradle

    Conservation of energy and the ballistic pendulum

    The tautochrone

    Both isochronal and fastest

    A pendulum that can never be: I

    Circular orbits

    The Newtonian domain

    Our strange 3-D world

    The conical pendulum

    The governor

    The paraconical pendulum

    The eclipse effect: fact or fiction?

    Galileo - again

Chapter 2: Making time

    Absolute time and relativity

    A pendulum that can never be: II

    The movement of time

    Ancient time

    Celestial time

    Mechanical time

    The chaos of Pluto

    The heartbeat of time

    The clock of Boys

    The pendulum transformed

    International Atomic Time and the leap second

    Time bounds the Earth

    Hooke’s animadversions

    The pendulum is dead. Long live the pendulum

Chapter 3: The Shape of the Earth

    Little Pendulum Island

    A classic result

    Shadow lands

    There and back again

    A map without wrinkles

    Richter’s adventures

    Newton’s equatorial bulge

    Discontent goes to Peru

    Maupertius goes to Lapland

    Running hot and cold

    Give a little, take a little

    The Great Arc

    The Challenger Deep

    The Figure of the Earth

    Earthquakes and Polar Drift

    Sympathetic vibrations

Chapter 4: Weighing the Earth

    In the balance

    Weighing gravity

    Packing it in

    Defining up

    The attraction of mountains

    Maskelyne heads north

    Airy down a coalmine

    The Cavendish experiment

    The measure of big-G

    A pendulum that can never be: III

    The roots of mountains

    Gravitational aether

    The weak equivalence principle

    How constant is big-G

    An aside on the TOE

    The nanomechanical whisker

Chapter 5: Moving Heaven and Earth

    Hipparchus finds precession

    The ever-inventive Robert Hooke

    Newton’s alternative way

    Bradley’s aberration

    Ole Rømer sees the light

    Bradley gets the nod

    Halley finds a proper motion

    Trigonometric parallax measured

    Parallax found

    Passing ships

Chapter 6: All in a spin

    What goes up …..

    A technical aside on spin dynamics

    A letter from Newton

    At Garraways

    Flamsteed checks for consistency

    The pendulum at the Parthenon

    A technical aside: the sin(φ) term

    A centrifugal correction and spin-apart

    The Eötvös effect: a final correction

    The gyroscope and the spin that isn’t

    The Croll-Melankovich cycle

    Inconstant spin, and spin down

    Moon drift and the end of all things

    The inverted pendulum

Chapter 7: From beauty, to resonance, to chaos

    The harmonograph

    Lissajous figures

    Dean’s two-point pendulum

    Huygens’s odd sympathy and the spark-gap oscillator

    Barton’s pendulum, Kirkwood gaps and the Lorentz atom

    The pendulum goes discrete

    The quantum oscillator – one step too far

    Duffing’s non-linear oscillator

    Constancy meets chaos

    The Lorenz attractor

    The double compound pendulum


Further notes and references