All Topics  
Angular velocity

 

 
   Email Print
   Bookmark   Link
 

 

Angular velocity
 
 
  Topics Angular velocity Topic Home

Do not confuse with angular frequency
Angular frequency

In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....


In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the angular velocity is a vector quantity (more precisely, a pseudovector
Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper Rotation , i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation...
) which specifies the angular speed, and axis about which an object is rotating. The SI
Si

Si, si, or SI may refer to :...
 unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g.






Discussion
Ask a question about 'Angular velocity'
Start a new discussion about 'Angular velocity'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Do not confuse with angular frequency
Angular frequency

In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....


In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, the angular velocity is a vector quantity (more precisely, a pseudovector
Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper Rotation , i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation...
) which specifies the angular speed, and axis about which an object is rotating. The SI
Si

Si, si, or SI may refer to :...
 unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g. revolutions per minute
Revolutions per minute

Revolutions per minute is a units of measurement of frequency: the number of Turn completed in one minute around a rotation around a fixed axis....
), it is often called the rotational velocity and its magnitude the rotational speed
Rotational speed

Rotational speed indicates, for example, how fast a motor is running. Rotational speed is equivalent to angular speed, but with different units....
. Angular velocity is usually represented by the symbol omega
Omega

Omega is the 24th and last letter of the Greek alphabet. In the Greek numerals it has a value of 800. The word literally means "great O" , as opposed to Omicron, which means "little O" ....
  (O or ?). The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the Right hand grip rule
Right hand grip rule

The right hand grip rule is a physics principle applied to electric current passing through a solenoid, resulting in a magnetic field. When you wrap your right hand around the solenoid with your fingers in the direction of the conventional current, your thumb points in the direction of the magnetic north pole....
.

The angular velocity of a particle


Two dimensions


The angular velocity of a particle in a 2-dimensional plane is the easiest to understand. As shown in the figure on the right (typically expressing the angular measures f and ? in radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s), if we draw a line from the origin (O) to the particle (P), then the velocity vector (v) of the particle will have a component along the radius (radial component, v?) and a component perpendicular to the radius (tangential component, v).

A radial motion produces no rotation of the particle (relative to the origin), so for purposes of finding the angular velocity the parallel (radial) component can be ignored. Therefore, the rotation is completely produced by the tangential motion (like that of a particle moving along a circumference), and the angular velocity is completely determined by the perpendicular (tangential) component.

It can be seen that the rate of change of the angular position of the particle is related to the tangential velocity by: Utilizing ?, the angle between vectors v? and v, or equivalently as the angle between vectors r and v, gives:

Combining the above two equations and defining the angular velocity as ?=df/dt yields:

In two dimensions the angular velocity is a single number which has no direction. A single number which has no direction is either a scalar
Scalar

A scalar is a variable that only has magnitude , e.g. a speed of 40 km/h. Compare it with vector, a quantity comprising both magnitude and Direction , e.g....
 or a pseudoscalar
Pseudoscalar

In physics, a pseudoscalar is a quantity that behaves like a scalar , except that it changes sign under a Parity such as improper rotations while a true scalar does not....
, the difference being that a scalar does not change its sign when the x and y axes are exchanged (or inverted), while a pseudoscalar does. The angle as well as the angular velocity is a pseudoscalar. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis. If the axes are inverted, but the sense of a rotation does not, then the sign of the angle of rotation, and therefore the angular velocity as well, will change.

It is important to note that the pseudoscalar angular velocity of a particle depends upon the choice of the origin and upon the orientation of the coordinate axes.

The equation allows the angular velocity to be found.

Three dimensions


In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovector
Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper Rotation , i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation...
. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right-hand rule
Right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vector in 3 dimensions. It was invented for use in electromagnetism by British physicist Zachariah William Cole in the late 1800s....
 indicates the positive direction of the angular velocity pseudovector, namely:

If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.


Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line normal to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Define a unit vector which points in the direction of the angular velocity pseudovector. The angular velocity may be written in a manner similar to that for two dimensions:

which, by the definition of the cross product
Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
, can be written:

Higher dimensions


In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor. This tensor will have n(n-1)/2 independent components and this number is the dimension of the Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 of the Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 of rotations of an n-dimensional inner product space. It turns out that in three dimensional space angular velocity can be represented by vector because number of independent components is equal to number of dimensions of space.

Angular velocity of a rigid body


In order to deal with the motion of a rigid body
Rigid body

In physics, a rigid body is an idealization of a solid Physical body of finite size in which deformation is neglected. In other words, the distance between any two given Point s of a rigid body remains constant in time regardless of external forces exerted on it....
, it is best to consider a coordinate system that is fixed with respect to the rigid body, and to study the coordinate transformations between this coordinate and the fixed "laboratory" system. As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is Ri in the lab frame, and at position ri in the body frame. It is seen that the position of the particle can be written:

The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector is unchanging. By Euler's rotation theorem
Euler's rotation theorem

In kinematics, 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 rotation about a fixed axis through that point....
, we may replace the vector with where is a rotation matrix
Rotation matrix

In matrix theory, a rotation matrix is a real number square matrix whose transpose is its invertible matrix and whose determinant is 1 The matrix is so-called because it geometrically corresponds to a linear map that sends vectors to a corresponding vector rotated about the origin by a fixed angle....
 and is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix which is changing in time and not the reference vector , as the rigid body rotates about point O'. The position of the particle is now written as:

Taking the time derivative yields the velocity of the particle:

where Vi is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). Since is a rotation matrix its inverse is its transpose. So we substitute :

Continue by taking the time derivative of :

Applying the formula (AB)T = BTAT:

is the negative of its transpose. Therefore it is a skew symmetric 3x3 matrix
Skew-symmetric matrix

In linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:...
. We can therefore take its dual to get a 3 dimensional vector. is called the angular velocity tensor
Angular velocity tensor

In physics, the angular velocity tensor is defined as a matrix T such that:It allows us to express the cross productas a matrix multiplication....
. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ?.

Substituting ? into the above velocity expression:

It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.

It is an important point that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the angular momentum
Angular momentum

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
 of the rigid body.

If the reference point is the "Instantaneous axis of rotation" the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.

See also

  • Angular frequency
    Angular frequency

    In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....
  • Angular acceleration
    Angular acceleration

    Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha ....
  • Angular momentum
    Angular momentum

    In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector....
  • Areal velocity
    Areal velocity

    Areal velocity is the rate at which area is swept out by a particle as it moves along a curve. In many applications, the curve lies in a plane, but in others, it is a space curve....
  • Isometry
    Isometry

    In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
  • Lie algebra
    Lie algebra

    In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
  • Orthogonal group
    Orthogonal group

    In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
  • Rigid body dynamics
    Rigid body dynamics

    In physics, rigid body dynamics is the study of the dynamics of rigid bodies. Unlike Point particle, which move only in three Degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moment of inertia, etc., that characterize motion in six Degrees of freedom ....
  • Rotation group
    Rotation group

    In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....