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Rotation of a rigid object P about a fixed object about a fixed axis O.

Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time (t). When dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the objects motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

Example

In the example illustrated to the right, a particle on object P at a fixed distance r from the origin, O, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (r, θ). In this particular example, the value of θ is changing, while the value of the radius remains the same. (In rectangular coordinates (x, y) both x and y vary with time). As the particle moves along the circle, it travels an arc length s, which becomes related to the angular position through the relationship:

s=r\theta \,

Measurements of angular displacement

Angular displacement may be measured in radians or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the centre.

\theta ={\frac {s}{r}}

For example if an object rotates 360 degrees around a circle of radius r, the angular displacement is given by the distance traveled around the circumference - which is 2πr divided by the radius: $\theta ={\frac {2\pi r}{r}}$ which easily simplifies to $\theta =2\pi$ . Therefore 1 revolution is $2\pi$ radians.

When object travels from point P to point Q, as it does in the illustration to the left, over $\delta t$ the radius of the circle goes around a change in angle. $\Delta \theta =\Delta \theta _{2}-\Delta \theta _{1}$ which equals the Angular Displacement.

Three dimensions

In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the Euler's rotation theorem; the magnitude specifies the rotation in radians about that axis (using the right-hand rule to determine direction).

Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition.^[1]

Matrix notation

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being $A_{0}$ and $A_{f}$ two matrices, the angular displacement matrix between them can be obtained as $dA=A_{f}.A_{0}^{-1}$

References

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-{{Transformation rules}}
+{{Classical mechanics|cTopic=Fundamental concepts}}
+[[Image:angulardisplacement1.jpg|300px|right|thumb|Rotation of a rigid object ''P'' about a fixed object about a fixed axis ''O''.]]
-In [[propositional calculus|propositional logic]] and [[boolean algebra]], '''De Morgan's laws'''<ref>Copi and Cohen</ref><ref>Hurley</ref><ref>Moore and Parker</ref> are a pair of transformation rules that are both [[validity|valid]] [[rule of inference|rules of inference]]. The rules allow the expression of [[Logical conjunction|conjunctions]] and [[Logical disjunction|disjunctions]] purely in terms of each other via [[logical negation|negation]].
+'''Angular displacement''' of a body is the [[angle]] in [[radian]]s ([[degree (angle)|degree]]s, [[turn (geometry)|revolutions]]) through which a point or line has been rotated in a specified sense about a specified [[rotation|axis]]. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time (''t'').  When dealing with the rotation of an object, it becomes simpler to consider the body itself rigid.  A body is generally considered rigid when the separations between all the particles remains constant throughout the objects motion, so for example parts of its mass are not flying off.  In a realistic sense, all things can be deformable, however this impact is minimal and negligible.  Thus the rotation of a rigid body over a fixed axis is referred to as [[rotational motion]].
-The rules can be expressed in English as:
+==Example==
-<blockquote>The negation of a conjunction is the disjunction of the negations.<br>
+In the example illustrated to the right, a particle on object P at a fixed distance ''r'' from the origin, ''O'', rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates (''r'', ''θ'').  In this particular example, the value of ''θ'' is changing, while the value of the radius remains the same.  (In rectangular coordinates (''x'', ''y'') both ''x'' and ''y'' vary with time). As the particle moves along the circle, it travels an [[Arc (geometry)|arc length]] ''s'', which becomes related to the angular position through the relationship:
-The negation of a disjunction is the conjunction of the negations.</blockquote>
-or informally as:
-<blockquote>"'''''not (A and B)'''''" is the same as "'''''(not A) or (not B)'''''"<br>
-<br>
-and also,<br>
-<br>
-"'''''not (A or B)'''''" is the same as "'''''(not A) and (not B)'''''"</blockquote>
-The rules can be expressed in [[formal language]] with two propositions ''P'' and ''Q'' as:
+:<math>
+s=r\theta \,</math>
-:<math>\neg(P\land Q)\iff(\neg P)\lor(\neg Q)</math>
+==Measurements of angular displacement==
-:<math>\neg(P\lor Q)\iff(\neg P)\land(\neg Q)</math>
+Angular displacement may be measured in [[radian]]s or degrees. If using radians, it provides a very simple relationship between distance traveled around the circle and the distance ''r'' from the centre.
-where:
+:<math>\theta=\frac sr</math>
-*¬ is the negation operator (NOT)
-*<math>\land</math> is the conjunction operator (AND)
-*<math>\lor</math> is the disjunction operator (OR)
-*⇔  is a [[metalogic]]al symbol meaning "can be replaced in a [[formal proof|logical proof]] with"
-Applications of the rules include simplification of logical [[Expression (computer science)|expressions]] in [[computer program]]s and digital circuit designs. De Morgan's laws are an example of a more general concept of [[duality (mathematics)|mathematical duality]].
+For example if an object rotates 360 degrees around a circle of radius ''r'', the angular displacement is given by the distance traveled around the circumference - which is 2π''r''
+divided by the radius: <math>\theta= \frac{2\pi r}r</math> which easily simplifies to <math>\theta=2\pi</math>. Therefore 1 revolution is <math>2\pi</math> radians.
-== Formal notation ==
+<!-- Image with unknown copyright status removed: [[Image:angulardisplacement2.jpg|250px|left|thumb|A particle that is rotating from point P to point Q along the acr of the circle.  In the time that elapses, the change in time is equal to the final time minus the original time, and the radius travels an angle theta, or the original angle subtracted from the final angle.]] -->
-The ''negation of conjunction'' rule may be written in [[sequent]] notation:
+When object travels from point P to point Q, as it does in the illustration to the left, over <math>\delta t</math> the radius of the circle goes around a change in angle. <math>\Delta \theta = \Delta \theta_2 - \Delta \theta_1 </math>  which equals the '''Angular Displacement'''.
-:<math>\neg(P \and Q) \vdash (\neg P \or \neg Q)</math>
-The ''negation of disjunction'' rule may be written as:
+== Three dimensions ==
-:<math>\neg(P \or Q) \vdash (\neg P \and \neg Q)</math>
+In three dimensions, angular displacement is an entity with a direction and a magnitude.  The direction specifies the axis of rotation, which always exists by virtue of the [[Euler's rotation theorem]]; the magnitude specifies the rotation in [[radian]]s about that axis (using the [[right-hand rule]] to determine direction).
-In [[Rule of inference|rule form]]:
+Despite having direction and magnitude, angular displacement is not a [[vector (geometry)|vector]] because it does not obey the [[commutative law]] for addition.<ref>{{cite book|last1=Kleppner|first1=Daniel|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|publisher=McGraw-Hill|year=1973|pages=288–89}}</ref>
-''negation of conjunction''
-:<math>\frac{\neg (P \and Q)}{\therefore \neg P \or \neg Q}</math>
-and
+=== Matrix notation ===
-''negation of disjunction''
+Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being <math>A_0</math> and <math>A_f</math> two matrices, the angular displacement matrix between them can be obtained as <math>dA =  A_f . A_0^{-1}</math>
-:<math>\frac{\neg (P \or Q)}{\therefore \neg P \and \neg Q}</math>
-and expressed as a truth-functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic:
+== References ==
+<references />
-:<math>\neg (P \and Q) \to (\neg P \or \neg Q)</math>
-:<math>\neg (P \or Q) \to (\neg P \and \neg Q)</math>
-where <math>P</math>, and <math>Q</math> are propositions expressed in some formal system.
-===Substitution form===
-De Morgan's laws are normally shown in the compact form above, with negation of the output on the left and negation of the inputs on the right.  A clearer form for substitution can be stated as:
-:<math>(P \and Q) \equiv \neg (\neg P \or \neg Q)</math>
-:<math>(P \or Q) \equiv \neg (\neg P \and \neg Q)</math>
-This emphasizes the need to invert both the inputs and the output, as well as change the operator, when doing a substitution.
-=== Set theory and Boolean algebra ===
-In set theory and [[Boolean algebra (logic)|Boolean algebra]], it is often stated as "Union and intersection interchange under complementation",<ref>''Boolean Algebra'' By R. L. Goodstein. ISBN 0-486-45894-6</ref> which can be formally expressed as:
-*<math>\overline{A \cup B}\equiv\overline{A} \cap \overline{B}</math>
-*<math>\overline{A \cap B}\equiv\overline{A} \cup \overline{B}</math>
-where:
-*{{overline|''A''}} is the negation of A, the [[overline]] being written above the terms to be negated
-*∩ is the [[Intersection (set theory)|intersection]] operator (AND)
-*∪ is the [[Union (set theory)|union]] operator (OR)
-The generalized form is:
-: <math>\overline{\bigcap_{i \in I} A_{i}}\equiv\bigcup_{i \in I} \overline{A_{i}}</math>
-: <math>\overline{\bigcup_{i \in I} A_{i}}\equiv\bigcap_{i \in I} \overline{A_{i}}</math>
-where ''I'' is some, possibly uncountable, indexing set.
-In set notation, De Morgan's law can be remembered using the [[mnemonic]] "break the line, change the sign".<ref>[http://books.google.com/books?id=NdAjEDP5mDsC&pg=PA81&lpg=PA81&dq=break+the+line+change+the+sign&source=web&ots=BtUl4oQOja&sig=H1Wz9e6Uv_bNeSbTvN6lr3s47PQ#PPA81,M1 2000 Solved Problems in Digital Electronics] By S. P. Bali</ref>
-=== Engineering ===
-In [[electrical and computer engineering]], De Morgan's law is commonly written as:
-: <math>\overline{A \cdot B} \equiv \overline {A} + \overline {B}</math>
-: <math>\overline{A + B} \equiv \overline {A} \cdot \overline {B}</math>
-where:
-* <math> \cdot </math> is a logical AND
-* <math>+</math> is a logical OR
-* the {{overline|overbar}} is the logical NOT of what is underneath the overbar.
-==History==
-The law is named after [[Augustus De Morgan]] (1806–1871)<ref>''[http://www.mtsu.edu/~phys2020/Lectures/L19-L25/L3/DeMorgan/body_demorgan.html DeMorgan’s Theorems]'' at mtsu.edu</ref> who introduced a formal version of the laws to classical [[propositional logic]]. De Morgan's formulation was influenced by algebraization of logic undertaken by [[George Boole]], which later cemented De Morgan's claim to the find. Although a similar observation was made by [[Aristotle]] and was known to Greek and Medieval logicians<ref>Bocheński's ''History of Formal Logic''</ref> (in the 14th century, [[William of Ockham]] wrote down the words that would result by reading the laws out),<ref>William of Ockham, Summa Logicae, part II, sections 32 & 33.</ref> De Morgan is given credit for stating the laws formally and incorporating them into the language of logic. De Morgan's Laws can be proved easily, and may even seem trivial.<ref>[http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/ademo.htm Augustus De Morgan (1806 -1871)] by Robert H. Orr</ref> Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments.
-==Informal proof==
-De Morgan's theorem may be applied to the negation of a [[disjunction]] or the negation of a [[Logical conjunction|conjunction]] in all or part of a formula.
-===Negation of a disjunction===
-In the case of its application to a disjunction, consider the following claim: "it is false that either of A or B is true", which is written as:
-:<math>\neg(A\lor B)</math>
-In that it has been established that ''neither'' A nor B is true, then it must follow that both A is not true [[logical AND|and]] B is not true, which may be written directly as:
-:<math>(\neg A)\wedge(\neg B)</math>
-If either A or B ''were'' true, then the disjunction of A and B would be true, making its negation false. Presented in English, this follows the logic that "Since two things are both false, it is also false that either of them is true."
-Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first claim.
-===Negation of a conjunction===
-The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale.  Consider the following claim: "it is false that A and B are both true", which is written as:
-:<math>\neg(A\land B)</math>
-In order for this claim to be true, either or both of A or B must be false, for if they both were true, then the conjunction of A and B would be true, making its negation false. Thus, [[inclusive or|one (at least) or more]] of A and B must be false (or equivalently, one or more of "not A" and "not B" must be true). This may be written directly as:
-:<math>(\neg A)\lor(\neg B)</math>
-Presented in English, this follows the logic that "Since it is false that two things are both true, at least one of them must be false."
-Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction thus results in a true expression, and this expression is identical to the first claim.
-==Formal proof==
-The proof that <math>(A\cap B)^c = A^c \cup B^c</math> is done by first proving that <math>(A\cap B)^c \subseteq A^c \cup B^c</math>, and then by proving that <math>A^c \cup B^c \subseteq (A\cap B)^c</math>
-Let <math>x \in (A \cap B)^c</math>.  Then <math>x \not\in A \cap B</math>.  Because <math>A \cap B = \{y | y \in A \text{ and } y \in B\}</math>, then either <math>x \not\in A</math> or <math>x \not\in B</math>.  If <math>x \not\in A</math>, then <math>x \in A^c</math>, so then <math>x \in A^c \cup B^c</math>.  Otherwise, if <math>x \not\in B</math>, then <math>x \in B^c</math>, so <math>x \in A^c\cup B^c</math>.  Because this is true for any arbitrary <math>x \in (A\cap B)^c</math>, then <math>\forall x \in (A\cap B)^c, x \in A^c \cup B^c</math>, and so <math>(A\cap B)^c \subseteq A^c \cup B^c</math>.
-To prove the reverse direction, assume that <math>\exists x \in A^c \cup B^c</math> such that <math>x \not\in (A\cap B)^c</math>.  Then <math>x \in A\cap B</math>.  It follows that <math>x \in A</math> and <math>x \in B</math>.  Then <math>x \not\in A^c</math> and <math>x \not\in B^c</math>.  But then <math>x \not\in A^c \cup B^c</math>, in contradiction to the hypothesis that <math>x \in A^c \cup B^c</math>.  Therefore, <math>\forall x \in A^c \cup B^c, x \in (A\cap B)^c</math>, and <math>A^c \cup B^c \subseteq (A\cap B)^c</math>.
-Because <math>A^c \cup B^c \subseteq (A\cap B)^c</math> and <math>(A \cap B)^c \subseteq A^c \cup B^c</math>, then <math>(A\cap B)^c = A^c \cup B^c</math>, concluding the proof of De Morgan's Law.
-The other De Morgan's Law, that <math>(A\cup B)^c = A^c \cap B^c</math>, is proven similarly.
-==Extensions==
-In extensions of classical propositional logic, the duality still holds (that is, to any logical operator we can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another.  This leads to an important property of logics based on classical logic, namely the existence of [[negation normal form]]s: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula.  The existence of negation normal forms drives many applications, for example in [[digital circuit]] design, where it is used to manipulate the types of [[logic gate]]s, and in formal logic, where it is a prerequisite for finding the [[conjunctive normal form]] and [[disjunctive normal form]] of a formula.  Computer programmers use them to simplify or properly negate complicated [[Conditional (programming)|logical conditions]]. They are also often useful in computations in elementary [[probability theory]].
-Let us define the dual of any propositional operator P(''p'', ''q'', ...) depending on elementary propositions ''p'', ''q'', ... to be the operator <math>\mbox{P}^d</math> defined by
-:<math>\mbox{P}^d(p, q, ...) = \neg P(\neg p, \neg q, \dots).</math>
-This idea can be generalised to quantifiers, so for example the [[universal quantifier]] and [[existential quantifier]] are duals:
-:<math> \forall x \, P(x) \equiv \neg \exists x \, \neg P(x), </math>
-:<math> \exists x \, P(x) \equiv \neg \forall x \, \neg P(x). </math>
-To relate these quantifier dualities to the De Morgan laws, set up a [[model theory|model]] with some small number of elements in its domain ''D'', such as
-:''D'' = {''a'', ''b'', ''c''}.
-Then
-:<math> \forall x \, P(x) \equiv P(a) \land P(b) \land P(c) </math>
-and
-:<math> \exists x \, P(x) \equiv P(a) \lor P(b) \lor P(c).\, </math>
-But, using De Morgan's laws,
-:<math> P(a) \land P(b) \land P(c) \equiv \neg (\neg P(a) \lor \neg P(b) \lor \neg P(c)) </math>
-and
-:<math> P(a) \lor P(b) \lor P(c) \equiv \neg (\neg P(a) \land \neg P(b) \land \neg P(c)), </math>
-verifying the quantifier dualities in the model.
-Then, the quantifier dualities can be extended further to [[modal logic]], relating the box ("necessarily") and diamond ("possibly") operators:
-:<math> \Box p \equiv \neg \Diamond \neg p, </math>
-:<math> \Diamond p \equiv \neg \Box \neg p.\, </math>
-In its application to the [[Subjunctive possibility|alethic modalities]] of possibility and necessity, [[Aristotle]] observed this case, and in the case of [[normal modal logic]], the relationship of these modal operators to the quantification can be understood by setting up models using [[Kripke semantics]].
 ==See also==
-* [[Isomorphism]] (NOT operator as isomorphism between [[wikt:positive logic|positive logic]] and [[wikt:negative logic|negative logic]])
+*[[Second moment of area]]
-* [[List of Boolean algebra topics]]
+*[[Linear elasticity]]
+*[[Rotation_matrix#Infinitesimal_rotations|Infinitesimal rotation]]
-==References==
+*[[Angular distance]]
-{{reflist}}
+*[[Angular velocity]]
-==External links==
+{{Classical mechanics derived SI units}}
-* {{springer|title=Duality principle|id=p/d034130}}
-* {{MathWorld | urlname=deMorgansLaws | title=de Morgan's Laws}}
-* {{PlanetMath | urlname=DeMorgansLaws | title=de Morgan's laws | id=2308}}
-{{Set theory}}
+[[Category:Angle]]
-[[Category:Boolean algebra]]
-[[Category:Duality theories]]
-[[Category:Rules of inference]]
-[[Category:Articles containing proofs]]
-[[Category:Theorems in propositional logic]]

Main Page: Difference between revisions

Revision as of 21:03, 9 August 2014

Contents

Example

Measurements of angular displacement

Three dimensions

Matrix notation

References

See also

Navigation menu

Main Page: Difference between revisions

Revision as of 21:03, 9 August 2014

Example

Measurements of angular displacement

Three dimensions

Matrix notation

References

See also

Navigation menu

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