Transcript A × B
Which came first: Vector Product or Torque? by Antonia Katsinos VECTOR ALGEBRA vs. TORQUE CROSS PRODUCT 1844 – Hermann Grassman 1843 – William Rowan Hamilton 1880-1884 Josiah Willard Gibbs 1880 - Oliver Heaviside TORQUE Rotational/angular force 300 BC – Archimedes work of on levers 1687 - Newton’s Second Law of Rotation 1922 - Jack Johnson patented the wrench Torque = Force applied x lever arm TWO TRADITIONS THE STUDY OF NUMBERS Natural numbers Negatives numbers, zero, fractions & irrational #s solving the equation x^2 +1 = 0 led to complex numbers Their geometrical representation in space led to vector analysis PHYSICAL PHENOMENA need to describe with magnitude and direction such as velocity plus the need of geometry to approach physical problems together brought forward to concept of a vector HERMAN GRASSMANN (April 15, 1809 – Sept. 26, 1877) 1832 was a high school math teacher at the Gymnasium in Germany while teaching continued his father’s research on the concept of product in geometry Recognized a relationship between sums and products; whether you multiply the sum of two displacements by a third displacement lying in the same plane, or the individual terms by the same displacement and add the products with due regard for their positive and negative values, the same result is obtained 1840 during the writing of further examinations he realized that he could apply the vector methods he had been researching to the essay topic of the theory of the tides his system is closer to our present day vector algebra, but unrecognized because of his obscurity and his books’ unreadability WILLIAM ROWAN HAMILTON (August 4, 1805 – September 2, 1865) one interest was the relationship between complex numbers and geometry sought an algebra of complex numbers that bears the same relationship with 3D geometry unexpectedly his search ended on 16 October 1843 when he realized that the appropriate algebra was not a triplet algebra but a 4D algebra of what he called "quaternions“ many mathematical terms in common use today - including scalar and vector - were introduced by Hamilton, as he developed the theory of quaternions QUATERNIONS A quaternion is a 4D complex number that is of the form q = w + xi + yj + zk, where i, j and k are all different square roots of 1. The quaternion can be regarded as an object composed of a scalar part, w, which is a real number, and a vector part, xi + yj + zk. the vector part may be represented, in magnitude and direction, by a line joining two points in 3D space. JOSIAH WILLARD GIBBS (February 11, 1839 – April 18, 1903) from 1880 to 1884, Gibbs combined the ideas of two mathematicians, the quaternions of William Rowan Hamilton and the exterior algebra of Herman Grassmann to obtain vector analysis Gibbs designed vector analysis to clarify and advance mathematical physics From 1882 to 1889, Gibbs refined his vector analysis, wrote on optics, and developed a new electrical theory of light OLIVER HEAVISIDE (May 18, 1850 – February 3, 1925) both Gibbs and Heaviside arrived at identical systems by modifying Hamilton’s quaternions algebra, working independently of each other while both Gibbs and Heaviside started with Hamilton’s methods, the system they both arrived at was closer to Grassmann’s in structure improved vector terminology promoted the use of vectors in his 1893 book “Electromagnetic Theory” Vector Analysis was created in the late 19th century when Hamilton’s quaternion system was adapted to the needs of physics by Clifford, Tait, Maxwell, Heaviside, Gibbs and others many earlier results obtained by Lagrange, Gauss, Green and others on hydrodynamics, sound and electricity, were then re-expressed in terms of vector analysis. many of the vector analysis topics are now taught in courses on the “calculus on manifolds.” BASIC VECTOR OPERATIONS Both a magnitude and a direction must be specified for a vector quantity Any number of vector quantities of the same type (i.e., same units) can be combined by basic vector operations VECTOR PRODUCT first mention of the CROSS PRODUCT is found on p. 61 of Vector Analysis, founded upon the lectures of J. Willard Gibbs, second edition, by Edwin Bidwell Wilson (1879-1964), published by Charles Scribner's Sons in 1909 the skew product is denoted by a cross as the direct product was by a dot it is written: C = A X B and read A cross b for this reason it is often called the cross product THE CROSS PRODUCT a type of “multiplication” law that turns our vector space into a vector algebra A and B must be 3D vectors The result is a 3D vector with Length: |A × B|=|A||B|sinθ, where θ, is the angle between A and B and Orientation: A × B is perpendicular to both A and B The choice of orientations is made by the right hand rule. Right hand Rule for the Direction of Cross Product Draw an arc starting from the vector A and finishing on vector B Curl your fingers the same way as the arc Your right thumb points in the direction of the cross product CCW rotation is in the +z direction CW rotation is in the – z direction HISTORY OF THE TORQUE The principle of moments is derived from Archimedes' discovery of the operating principle of the lever He used to say, "Give me a place to stand and with a lever I will move the whole world." In the lever one applies a force, in his day most often human muscle, to an arm, a beam of some sort Archimedes noted that the amount of force applied to the object, the moment of force, is defined as M = rF, where F is the applied force, and r is the distance from the applied force to object The Law of the Lever According to ARCHIMEDES….. “Magnitudes are in equilibrium at distances reciprocally proportional to their weights”. This is the statement of the Law of the Lever that Archimedes gives in Propositions 6 and 7 of Book I of his work entitled On the Equilibrium of Planes. Archimedes demonstrated mathematically that the ratio of the effort applied to the load raised is equal to the inverse ratio of the distances of the effort and load from the pivot or fulcrum of the lever The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. Newton’s Second Law of Rotation the Second Law explains how a force acts on an object in linear motion an object accelerates in the direction the force is moving it (F = ma) Second Law of Rotation explains the relationship between the net external torque and the angular acceleration of an object Rotational and Linear Example What is a TORQUE Today? TORQUE: FORCE APPLIED X WRENCH? The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force. RIGHT HAND RULE for TORQUE WHY THE WRENCH? TEACHING STRATEGIES Matrices Physics Geometrical Representation Component Form Can we co-exist?