Tag: Mechanics

Resonant Vibrations from Atoms to the Far Horizons of the Cosmos

Panta Rhei — everything flows — said Heraclites, describing the impermanence of the world. He might well have said “everything vibrates”. From sub-atomic particles to the farthest reaches of the cosmos we find oscillations. Vibration is key for aircraft wing, motor engine and optical system design. Ocean tides forced by the Moon and seasonal variations … Continue reading Resonant Vibrations from Atoms to the Far Horizons of the Cosmos →

Mamikon’s Visual Calculus and Hamilton’s Hodograph

[This is a condensed version of an article [5] in Mathematics Today] A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon's Theorem states that `The area of a tangent sweep of a curve is equal to … Continue reading Mamikon’s Visual Calculus and Hamilton’s Hodograph →

Émilie Du Châtelet and the Conservation of Energy

A remarkable French natural philosopher and mathematician who lived in the early eighteenth century, Émilie Du Châtalet, is generally remembered for her translation of Isaac Newton's Principia Mathematica, but her work was much more than a simple translation: she added an extensive commentary in which she included new developments in mechanics, the most important being … Continue reading Émilie Du Châtelet and the Conservation of Energy →

Embedding: Reconstructing Solutions from a Delay Map

M In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured … Continue reading Embedding: Reconstructing Solutions from a Delay Map →

Cornelius Lanczos – Inspired by Hamilton’s Quaternions

In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera's keen interest in … Continue reading Cornelius Lanczos – Inspired by Hamilton’s Quaternions →

A Ring of Water Shows the Earth’s Spin

Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus. Compton (1892--1962) won the Nobel Prize in Physics in 1927 for his work on scattering of EM radiation. This phenomenon, now called the Compton effect, confirmed the particle nature of … Continue reading A Ring of Water Shows the Earth’s Spin →

The “extraordinary talent and superior genius” of Sophie Germain

When a guitar string is plucked, we don't see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as … Continue reading The “extraordinary talent and superior genius” of Sophie Germain →

The curious behaviour of the Wilberforce Spring.

The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to stretch up and down and to twist about its axis. In equilibrium, the spring hangs down with the pull of gravity … Continue reading The curious behaviour of the Wilberforce Spring. →

Boxes and Loops

We will describe some generic behaviour patterns of dynamical systems. In many systems, the orbits exhibit characteristic patterns called boxes and loops. We first describe orbits for a simple pendulum, and then look at some systems in higher dimensions. Libration and Rotation of a Pendulum The simple pendulum, with one degree of freedom, provides a … Continue reading Boxes and Loops →

Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph

The Greeks regarded the heavens as the epitome of perfection. All flaws and blemishes were confined to the terrestrial domain. Since the circle is perfect in its infinite symmetry, it was concluded by Aristotle that the Sun and planets move in circles around the Earth. Later, the astronomer Ptolemy accounted for deviations by means of … Continue reading Kepler’s Vanishing Circles Hidden in Hamilton’s Hodograph →

The Kill-zone: How to Dodge a Sniper’s Bullet

Under mild simplifying assumptions, a projectile follows a parabolic trajectory. This results from Newton's law of motion. Thus, for a fixed energy, there is an accessible region around the firing point comprising all the points that can be reached. We will derive a mathematical description for this kill-zone (the term kill-zone, used for dramatic effect, … Continue reading The Kill-zone: How to Dodge a Sniper’s Bullet →

Trappist-1 & the Age of Aquarius

The Pythagoreans believed that the planets generate sounds as they move through the cosmos. The idea of the harmony of the spheres was brought to a high level by Johannes Kepler in his book Harmonices Mundi, where he identified many simple relationships between the orbital periods of the planets [TM154 or search for “thatsmaths” at irishtimes.com]. Kepler's … Continue reading Trappist-1 & the Age of Aquarius →

Falling Bodies [1]: Sky-diving

Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle's theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a … Continue reading Falling Bodies [1]: Sky-diving →

New Estimate of the Speed of Light

A team of German scientists have recently discovered a new method of measuring the speed of light using Einstein's famous equation E = m c2 Scientists from SFZ, the Spätenheim Forschungszentrum in Bavaria, assembled a group of twenty volunteer climbers at a local mountain, Schmerzenberg. Using high-precision Mettler balance equipment, each climber was weighed at … Continue reading New Estimate of the Speed of Light →