HOME

TheInfoList



OR:

The angular displacement (symbol θ, , or φ) – also called angle of rotation, rotational displacement, or rotary displacement – of a
physical body In natural language and physical science, a physical object or material object (or simply an object or body) is a wiktionary:contiguous, contiguous collection of matter, within a defined boundary (or surface), that exists in space and time. Usual ...
is the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
(in
units Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
of
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s, degrees, turns, etc.) through which the body rotates (revolves or spins) around a centre or
axis of rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
. Angular displacement may be signed, indicating the sense of rotation (e.g.,
clockwise Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...
); it may also be greater (in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
) than a full turn.


Context

When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in
circular motion In physics, circular motion is movement of an object along the circumference of a circle or rotation along a circular arc. It can be uniform, with a constant rate of rotation and constant tangential speed, or non-uniform with a changing rate ...
it undergoes a changing velocity and acceleration at any time. When dealing with the rotation of a body, 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 body's 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.


Example

In the example illustrated to the right (or above in some mobile versions), a particle or body P is 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 Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
''s'', which becomes related to the angular position through the relationship: :s = r\theta .


Definition and units

Angular displacement may be expressed in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s or degrees. Using radians provides a very simple relationship between distance traveled around the circle (''
circular arc A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than radians (180 ...
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
'') and the distance ''r'' from the centre (''
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''): :\theta = \frac \mathrm For example, if a body rotates 360° 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: \theta= \fracr which easily simplifies to: \theta=2\pi. Therefore, 1
revolution In political science, a revolution (, 'a turn around') is a rapid, fundamental transformation of a society's class, state, ethnic or religious structures. According to sociologist Jack Goldstone, all revolutions contain "a common set of elements ...
is 2\pi radians. The above definition is part of the
International System of Quantities The International System of Quantities (ISQ) is a standard system of Quantity, quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This ...
(ISQ), formalized in the international standard
ISO 80000-3 ISO/IEC 80000, ''Quantities and units'', is an international standard describing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the Intern ...
(Space and time),
(11 pages)
and adopted in the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI).
/ref> Angular displacement may be signed, indicating the sense of rotation (e.g.,
clockwise Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...
); it may also be greater (in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
) than a full turn. In the ISQ/SI, angular displacement is used to define the '' number of revolutions'', ''N''θ/(2π rad), a ratio-type quantity of dimension one.


In 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 In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed po ...
; the magnitude specifies the rotation in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s about that axis (using the
right-hand rule In mathematics and physics, the right-hand rule is a Convention (norm), convention and a mnemonic, utilized to define the orientation (vector space), orientation of Cartesian coordinate system, axes in three-dimensional space and to determine the ...
to determine direction). This entity is called an axis-angle. Despite having direction and magnitude, angular displacement is not a
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
because it does not obey the
commutative law In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.


Rotation matrices

Several ways to describe rotations exist, like rotation matrices or
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189� ...
. See
charts on SO(3) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot ...
for others. 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 \Delta A = A_f A_0^. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity. In the limit, we will have an infinitesimal rotation matrix.


Infinitesimal rotation matrices


See also

*
Angular distance Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtende ...
*
Angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
* Angular position *
Angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
*
Azimuth An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system. Mathematically, the relative position vector from an observer ( origin) to a point ...
*
Infinitesimal rotation An infinitesimal rotation matrix or differential rotation matrix is a matrix (mathematics), matrix representing an infinitesimal, infinitely small rotation. While a rotation matrix is an orthogonal matrix R^\mathsf = R^ representing an element of S ...
*
Linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
* Second moment of area * Unwrapped phase


References


Sources

* * {{Classical mechanics derived SI units Angle Rotation Sign (mathematics)