Posts Tagged 'Applied Maths'

Samuel Haughton and the Twelve Faithless Hangmaids

In his study of humane methods of hanging, Samuel Haughton (1866) considered the earliest recorded account of execution by hanging (see Haughton’s Drop on this site). In the twenty-second book of the Odyssey, Homer described how the twelve faithless handmaids of Penelope “lay by night enfolded in the arms of the suitors” who were vying for Penelope’s hand in marriage. Her son Telemachus, with the help of his comrades, hanged all twelve handmaids on a single rope.

Hangmaids-05-COL

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Samuel Haughton and the Humane Drop

Samuel-Haughton

Samuel Haughton (1821-1897).

Samuel Haughton was born in Co. Carlow in 1821. He entered Trinity College Dublin aged just sixteen and graduated in 1843. He was elected a fellow in 1844 and was appointed professor of geology in 1851. He took up the study of medicine and graduated as a Doctor of Medicine in 1862, aged 40 [TM182 or search for “thatsmaths” at irishtimes.com].

In addition to his expertise in geology and medicine, Haughton was a highly talented applied mathematician. His mathematical investigations included the study of the motion of solid and fluid bodies, solar radiation, climatology, animal mechanics and ocean tides. One of his more bizarre applications of mathematics was to demonstrate a humane method of execution by hanging, by lengthening the drop to ensure instant death.

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Zhukovsky’s Airfoil

A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

\displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)}

and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section Joukowsky Airfoil.

Zhukovsky-Animation

Visualization of airflow around a Joukowsky airfoil. Image generated using code on this website.

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Joseph Fourier and the Greenhouse Effect

Jean-Baptiste Joseph Fourier, French mathematician and physicist, was born in Auxerre 251 years ago today. He is best known for the mathematical techniques that he developed in his analytical theory of heat transfer. Over the past two centuries, his methods have evolved into a major subject, harmonic analysis, with widespread applications in number theory, signal processing, quantum mechanics, weather prediction and a broad range of other fields [TM159 or search for “thatsmaths” at irishtimes.com].

GreenhouseEffect

Greenhouse Effect [Image Wikimedia Commons]

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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, is the region embracing all the points that can be reached by a sniper’s bullet, given a fixed muzzle velocity).

Sniper-Killzone-1 Family of trajectories with fixed initial speed and varying launch angles. Two particular trajectories are shown in black. Continue reading ‘The Kill-zone: How to Dodge a Sniper’s Bullet’

Euler and the Fountains of Sanssouci

When Frederick the Great was crowned King of Prussia in 1740 he immediately revived the Berlin Academy of Sciences and invited scholars from throughout Europe to Berlin. The most luminous of these was Leonhard Euler, who arrived at the academy in 1741. Euler was an outstanding genius, brilliant in both mathematics and physics. Yet, a myth persists that he failed spectacularly to solve a problem posed by Frederick. Euler is reputed to have bungled his mathematical analysis. In truth, there was much bungling, but the responsibility lay elsewhere. [TM122 or search for “thatsmaths” at irishtimes.com].

Sanssouci-Fountain

Sanssouci Palace, the summer home of Frederick the Great in Potsdam.
PHOTO: https://www.flickr.com/photos/b_hurwitz/4064337397/

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Inertial Oscillations and Phugoid Flight

The English aviation pioneer Frederick Lanchester (1868–1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term “phugoid” to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the basic modes of aircraft dynamics, and is clearly illustrated by the flight of gliders.

Glider-Loop-20

Glider in phugoid loop [photograph by Dave Jones on website of Dave Harrison]

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