MECH 260-102 Assignment 1 Solution
Problem 1 Mechanics of materials mainly concerns the “internal” effect, such as stresses and strains caused by external loads acting on a deformable body/structure. It is important in engineering because it is fundamental and it has wide application fields, such as in mechanical engineering, civil engineering.
Problem 2 Archimedes Biography: Archimedes of Syracuse (c. 287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors. Contribution: This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
Da Vinci Biography: Leonardo da Vinci (1452-1519) was born in Florence, Italy, and was a prestigious artist, inventor, engineer and scientist. Throughout his lifetime, he also lived in Milan, Bologna, Rome and Venice. Despite being Italian, he spent the last years of his life in a house that he was given in France.
Contribution: Leonardo da Vinci used the concept of gears and torque in his inventions. In doing so, he created the concept of moments, which are very important in modern day statics and in the mechanics of materials.
Galileo Biography: Galileo Galilei 15 February 1564– 8 January 1642) was an Italian physicist, mathematician, astronomer and philosopher who played a major role in the Scientific Revolution. Galileo was born in Pisa (then part of the Duchy of Florence), Italy, and the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia Ammannati. Four of their six children survived infancy, and the youngest Michelangelo (or Michelagnolo) also became a noted lutenist and composer. Galileo's full name was Galileo di Vincenzo Bonaiuti de' Galilei. At the age of 8, his family moved to Florence, but he was left with Jacopo Borghini for two years. He then was educated in the Camaldolese Monastery at Vallombrosa, 35 km southeast of Florence. Contribution: Galileo's theoretical and experimental work on the motions of bodies, along with the largely independent work of Kepler and RenéDescartes, was a precursor of the classical mechanics developed by Sir Isaac Newton.
Newton Biography: Sir Isaac Newton (4 January 1643 – 31 March 1727 [OS: 25 December 1642 – 20 March 1726])was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is considered by many scholars and members of the general public to be one of the most influential people in human history. Contribution: His 1687 publication of the Philosophiæ Naturalis Principia Mathematica (usually called the Principia) is considered to be among the most influential books in the history of science, laying the groundwork for most of classical mechanics. In this work, Newton described universal gravitation and the three laws of motion which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation, thus removing the last doubts about heliocentrism and advancing the Scientific Revolution. Newton built the first practical reflecting telescope and developed a theory of colour based on the observation that a prism decomposes white light into the many colours that form the visible spectrum. He also formulated an empirical law of cooling and studied the speed of sound.
In mathematics, Newton shares the credit with Gottfried Leibniz for the development of the differential and integral calculus. He also demonstrated the generalized binomial theorem, developed Newton's method for approximating the roots of a function, and contributed to the study of power series. Newton remains uniquely influential to scientists, as demonstrated by a 2005 survey of members of Britain's Royal Society asking who had the greater effect on the history of science and made the greater contribution to humankind, Newton or Albert Einstein. Royal Society scientists deemed Newton to have made the greater overall contribution on both.
Bernoulli Biography: Jacob Bernoulli (also known as James or Jacques) (27 December 1654 – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. Jacob Bernoulli was born in Basel, Switzerland. Following his father's wish, he studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences. This included the work of Robert Boyle and Robert Hooke. Contribution: Euler–Bernoulli beam theory (also known as engineer's beam theory, classical beam theory or just beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case for small deflections of a beam which is subjected to lateral loads only. It is thus a special case of Timoshenko beam theory which accounts for shear deformation and is applicable for thick beams. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution. Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.
Hooke
Biography: Robert Hooke FRS (18 July 1635 – 3 March 1703) was an English natural philosopher, architect and polymath who played an important role in the scientific revolution, through both experimental and theoretical work. His adult life comprised three distinct periods: as a brilliant scientific inquirer lacking money; achieving great wealth and standing through his reputation for hard work and scrupulous honesty following the great fire of 1666 (Section:Hooke the architect), but eventually becoming ill and party to jealous intellectual disputes. These issues may have contributed to his relative historical obscurity (section: Personality and disputes). Hooke studied at Wadham College during the Protectorate where he became one of a tightly-knit group of ardent Royalists centred on John Wilkins. Here he was employed as an assistant to Thomas Willis and to Robert Boyle, for whom he built the vacuum pumps used in Boyle's gas law experiments. He built some of the earliest Gregorian telescopes, observed the rotations of Mars and Jupiter, and, based on his observations of fossils, was an early proponent of biological evolution. He investigated the phenomenon of refraction, deducing the wave theory of light, and was the first to suggest that matter expands when heated and that air is made of small particles separated by relatively large distances. He performed pioneering work in the field of surveying and map-making and was involved in the work that led to the first modern plan-form map, though his plan for London on a grid system was rejected in favour of rebuilding along the existing routes. He also came near to deducing that gravity follows an inverse square law, and that such a relation governs the motions of the planets, an idea which was subsequently developed by Newton.[4] Much of Hooke's scientific work was conducted in his capacity as curator of experiments of the Royal Society, a post he held from 1662, or as part of the household of Robert Boyle. Contribution: Hooke is known for his law of elasticity (Hooke's law), his book, Micrographia, and for first applying the word "cell" to describe the basic unit of life. Even now there is much less written about him than might be expected from the sheer industry of his life: he was at one time simultaneously the curator of experiments of the Royal Society and a member of its council, Gresham Professor of Geometry and a Surveyor to the City of London after the Great Fire of London, in which capacity he appears to have performed more than half of all the surveys after the fire. He was also an important architect of his time, though few of his buildings now survive and some of those are generally misattributed, and was instrumental in devising a set of planning controls for London whose influence remains today. Allan Chapman has characterised him as "England's Leonardo".
Euler Biography: Leonhard Euler (15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. Leonhard Euler was a Swiss mathematician and physicist. Euler studied infinitesimal calculus and graph theory and made a bunch of
important discoveries. Additionally, he has done some work in mechanics, fluid dynamics, optics and astronomy. It is arguable that Euler is one of the best mathematicians of all time.
Contribution: Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Together with Daniel Bernoulli, Euler developed the Euler-Bernoulli equation, which simplifies the law of elasticity. This discovery was of utter importance in advancing engineering and mechanics. His papers on optics also helped develop the wave theory of light, which was proposed by Huygens. D’Alembert
Biography: Jean-Baptiste le Rond d'Alembert (16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist and philosopher. He was also coeditor with Denis Diderot of the Encyclopédie. Contribution: The wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.
Lagrange Biography: Joseph-Louis Lagrange (25 January 1736, Turin, Piedmont – 10 April 1813, Paris), born Giuseppe Lodovico (Luigi) Lagrangia, was an Italian-born mathematician and astronomer, who lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. On the recommendation of Euler and D'Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing a large body of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique Analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888-89), written in Berlin and first published in 1788, offered the most comprehensive
treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. Contribution: Lagrange made the papers: “Mécanique Analytique”, in which he introduced the law of virtual work. Using this law and calculus of variations, he managed to deduce the whole of mechanics, including both solid and fluid mechanics. Coulomb Biography: Coulomb was born in Angoulême, France, to a well-to-do family. His father, Henri Coulomb, was inspector of the Royal Fields in Montpellier. His mother, Catherine Bajet, came from a wealthy family in the wool trade. When Coulomb was a boy, the family moved to Paris and there Coulomb studied at the prestigious Collège des QuatreNations. The courses he studied in mathematics there, under Pierre Charles Monnier, left him determined to pursue mathematics and similar subjects as a career. From 1757 to 1759 he joined his father's family in Montpellier and took part in the work of the academy of the city, directed by the mathematician Augustin Danyzy. With his father's approval, Coulomb returned to Paris in 1759 where he was successful in the entrance examination for the military school at Mézières. Contribution: He is best known for developing Coulomb's law, the definition of the electrostatic force of attraction and repulsion. The SI unit of charge, the coulomb, was named after him.
Laplace Biography: Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volumes Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace. Contribution: He formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in applied mathematics, is also named after him.
Poisson Biography: Siméon Denis Poisson (21 June 1781 – 25 April 1840), was a French mathematician, geometer, and physicist. Contribution: Poisson's is well-known for his correction of Laplace's second order partial differential equation for potential.Poisson's ratio named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio ν is a measure of the Poisson effect. The Poisson ratio is the ratio of the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes. Saint-Venant Biography & Contribution:: Adhémar Jean Claude Barréde Saint-Venant (August 23, 1797 – January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern hydraulic engineering. Although his surname was Barréde Saint-Venant in non-French mathematical literature he is known simply as Saint-Venant. His name is also associated with Saint-Venant's principle of statically equivalent systems of load, Saint-Venant's theorem and for Saint-Venant's compatibility condition, the integrability conditions for a symmetric tensor field to be a strain. Castigliano Biography: Carlo Alberto Castigliano (9 November 1847 – 25 October 1884) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial derivatives of strain energy. Contribution: Castigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy.
Galerkin
Biography: Boris Grigoryevich Galerkin (surname more accurately romanized as Galyorkin; March 4, 1871 – July 12, 1945), born in Polozk, Belarus, Russian Empire was a Russian/Soviet mathematician and an engineer. Contribution: In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem. In principle, it is the equivalent of applying the method of variation to a function space, by converting the equation to a weak formulation. Typically one then applies some constraints on the function space to characterize the space with a finite set of basis functions. Often when using a Galerkin method one also gives the name along with typical approximation methods used, such as Petrov-Galerkin method or Ritz-Galerkin method. Timoshenko Biography: Stephen P. Timoshenko (Ukrainian: Степан Прокопович Тимошенко, Russian: Степан Прокофьевич Тимошенко, also written as (transliterated: Stepan Prokopovych Tymoshenko), December 22, 1878 – May 29, 1972), is reputed to be the father of modern engineering mechanics. He wrote many of the seminal works in the areas of engineering mechanics, elasticity and strength of materials, many of which are still widely used today. Contribution: He wrote many of the seminal works in the areas of engineering mechanics, elasticity and strength of materials, many of which are still widely used today.
