Bound vector

Vector illustration

In technical mechanics, a bound vector or localized vector is a vector that describes a directed physical quantity that is assigned to a specific point in space. An example is a force given by the vector that acts on a body: depending on the point of attack, it can have different effects. If a bound vector is represented by an arrow, it cannot be moved to any point in space without changing the corresponding physical effect. The opposite term " free vector " applies to vector quantities that have the same meaning regardless of any reference point. An example is the speed of the mass points of a rigid body that only moves in a translatory manner, or the torque of a force couple. ${\displaystyle {\vec {F}}}$ ${\displaystyle {\vec {v}}}$

The term was coined by Heinrich Emil Timerding and related to a vector bound to a line of action , such as applies to the force on a rigid body in the case of static equilibrium. August Föppl later introduced the term volatile vector for this . In some older textbooks on the mechanics of rigid bodies, the term bound vector describes line- volatile vectors that can be moved along their line of action, but not across it.

Bound vectors are mainly used in statics to determine the overall effect of a system made up of several simultaneously acting forces with different directions and points of application. The term is not used in textbooks on theoretical mechanics as a branch of physics.

description

A vector bound to a point starts from a fixed point A and is determined by its magnitude and direction. The end point B is determined by the amount represented as the length and the direction. A bound vector is identified with either a pair of letters that describe the starting point and the end point and an arrow that spans both letters; or a single letter describes the name of the vector and is represented as this letter with an arrow:

${\displaystyle {\overrightarrow {AB}}}$or ${\displaystyle {\vec {A}}}$

A special case of the bound vector is the position vector , which is always bound to the coordinate origin and points from zero to (here) a point A.

${\displaystyle {\overrightarrow {0A}}}$or ${\displaystyle {\vec {r}}}$

In a gravitational field, a bound vector is assigned to each point, which is shown as an arrow. This must not be moved without changing its properties.

Mathematically, a bound vector can be represented by the symbol ( ), which indicates the vector component and the position vector of the reference point A: ${\displaystyle {\vec {r}}_{\!A},\,{\vec {A}}}$ ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {r}}_{\!A}}$

${\displaystyle {\begin{Bmatrix}{\overrightarrow {0A}}\,,{\overrightarrow {AB}}\end{Bmatrix}}}$or ${\displaystyle {\begin{Bmatrix}{\vec {r}}\,,{\vec {A}}\end{Bmatrix}}}$

Two bound vectors are equal if and only if they have the same vector component and the same reference point. When calculating with bound vectors, the usual rules of vector algebra apply to the vector component . Examples are the formulas for the center of mass and the total force in the principle of centroid in mechanics. However, it must be examined separately whether the result is again a bound vector and which reference point this has, if any. Which - bound or free - vector may be used to replace a system of several bound vectors must be determined according to the rules for the additional terms moment and equivalence . ${\displaystyle {\vec {A}}}$

The moment of a bound vector is the cross product of the position vector of its point of attack and its vector component.

${\displaystyle {\vec {M}}_{P}\,=\,{\vec {r}}\times {\vec {A}}}$

Moments of bound vectors can be added vectorially as long as they have the same reference point. Two systems of bound vectors are equivalent to each other (i.e. equivalent) if they lead to the same resulting moment regardless of the reference point. A bound vector system is equivalent in effect to a pair of bound vectors called a vector winder .

Web links

Wikibooks: Linear Algebra for Mechanics, Chapter Classification of Physical Vectors  - Learning and Teaching Materials

• Static scripts in maschinenbau-wissen.de

Individual evidence

1. ^ Heinz Ulbrich, Hans-Jürgen Weidemann, Friedrich Pfeiffer: Technical mechanics in formulas, tasks and solutions. Textbook for mechanical engineering. Springer, 2006, ISBN 3-8351-9058-X . (limited preview) (accessed April 8, 2013)
2. ↑ Lothar Papula: Papula, Mathematics 1. A textbook and workbook for basic studies. With numerous examples from science and technology. With 307 exercises with detailed solutions. Springer, 2007, ISBN 978-3-8348-9220-1 . ( online , accessed April 9, 2013)
3. ↑ Akshay Ranjan Paul, Pijush Roy, Sachayan Mukherjee: Mechanical sciences: engineering mechanics and strength of materials. PHI Learning, 2004, ISBN 81-203-2611-3 . ( limited preview , accessed May 8, 2013)
4. ^ VP Bhatnagar: A Complete Course in ISC Physics. Vol I, 1997, ISBN 81-209-0385-4 . ( limited preview , accessed May 8, 2013)
5. ↑ James H. Allen: Statics for mechanical engineers for dummies. John Wiley & Sons Publisher, 2012, ISBN 978-3-527-70761-4 . ( limited preview , accessed April 9, 2013)
6. ^ Waldemar Koestler, Moritz Tramer: Differential and integral calculus: infinitesimal calculus for engineers, especially for self-study. J. Springer, 1913, p. 70 (footnote)
7. ^ ^{A b} Kurt Meyberg, Peter Vachenauer: Higher Mathematics, 1st differential and integral calculus, vector and matrix calculation. 6th, corrected edition. Springer-Verlag, 2001, ISBN 3-642-56654-5 . ( limited preview )
8. ↑ P. Eberhard, M, Hanss: Technical Mechanics 1. Systems of Bound Vectors, Institute for Technical and Numerical Mechanics ( online ( Memento of the original from September 22, 2013 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this note. , PDF; 132 kB, accessed on April 10, 2013) 
9. ↑ Kurt Magnus, Hans Heinrich Müller-Slany: Fundamentals of technical mechanics . BG Teubner Verlag / GWV Fachverlage, Wiesbaden 2005, ISBN 3-8351-0007-6 ( limited preview in the Google book search).
10. ^ Klaus Fritzsche: Mathematics 1 for electrical engineering and IT. Chapter 2 Vector Calculation. (Lecture preparation for the winter semester 2001/2002) (online)