Angular velocity
Angular velocity | ||
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coord transformation | pseudovector | |
Derivations from other quantities | ω = dθ / dt | |
Dimension |
Part of a series on |
Classical mechanics |
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In physics, angular velocity (symbol ω or , the lowercase Greek letter
The magnitude of the pseudovector, , represents the
There are two types of angular velocity:
- Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. [citation needed]
- Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotationand is independent of the choice of origin, in contrast to orbital angular velocity.
Angular velocity has
The sense of angular velocity is conventionally specified by the
For example, a
Orbital angular velocity of a point particle
Particle in two dimensions
In the simplest case of circular motion at radius , with position given by the angular displacement from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: . If is measured in radians, the arc-length from the positive x-axis around the circle to the particle is , and the linear velocity is , so that .
In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector from the origin to a particle , with its
The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:
Here the cross-radial speed is the signed magnitude of , positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity gives magnitude (linear speed) and angle relative to the radius vector; in these terms, , so that
These formulas may be derived doing , being a function of the distance to the origin with respect to time, and a function of the angle between the vector and the x axis. Then:
Knowing , we conclude that the radial component of the velocity is given by , because is a radial unit vector; and the perpendicular component is given by because is a perpendicular unit vector.
In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes.
Particle in three dimensions
In three-dimensional space, we again have the position vector r of a moving particle. Here, orbital angular velocity is a pseudovector whose magnitude is the rate at which r sweeps out angle (in radians per unit of time), and whose direction is perpendicular to the instantaneous plane in which r sweeps out angle (i.e. the plane spanned by r and v). However, as there are two directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the right-hand rule is used.
Let the pseudovector be the unit vector perpendicular to the plane spanned by r and v, so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of ). Taking polar coordinates in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:
where θ is the angle between r and v. In terms of the cross product, this is:
From the above equation, one can recover the tangential velocity as:
Spin angular velocity of a rigid body or reference frame
Given a rotating frame of three unit coordinate vectors, all the three must have the same angular speed at each instant. In such a frame, each vector may be considered as a moving particle with constant scalar radius.
The rotating frame appears in the context of rigid bodies, and special tools have been developed for it: the spin angular velocity may be described as a vector or equivalently as a tensor.
Consistent with the general definition, the spin angular velocity of a frame is defined as the orbital angular velocity of any of the three vectors (same for all) with respect to its own center of rotation. The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear movements), and can be useful to decompose the rotation as in a gimbal. All components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative: .
By
If we choose a reference point fixed in the rigid body, the velocity of any point in the body is given by
Components from the basis vectors of a body-fixed frame
Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectors fixed to the body and with their common origin at O. The spin angular velocity vector of both frame and body about O is then
where is the time rate of change of the frame vector due to the rotation.
This formula is incompatible with the expression for orbital angular velocity
as that formula defines angular velocity for a single point about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body a single has to account for the motion of all particles in the body.
Components from Euler angles
The components of the spin angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and the use of an intermediate frame:
- One axis of the reference frame (the precession axis)
- The line of nodes of the moving frame with respect to the reference frame (nutation axis)
- One axis of the moving frame (the intrinsic rotation axis)
Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous
This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
where are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.[citation needed]
Tensor
The angular velocity tensor is a skew-symmetric matrix defined by:
The scalar elements above correspond to the
This is an infinitesimal rotation matrix. The linear mapping Ω acts as a cross product :
where is a
See also
- Angular acceleration
- Angular frequency
- Angular momentum
- Areal velocity
- Isometry
- Orthogonal group
- Rigid body dynamics
- Vorticity
References
- ISBN 978-81-265-0882-2.(UP1)
- ^ "Units with special names and symbols; units that incorporate special names and symbols".
- ISBN 978-0-13-607791-6.(EM1)
- ^ Singh, Sunil K. Angular Velocity. Rice University. Retrieved 21 May 2021 – via OpenStax.
- ^ K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics
- Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 978-0-201-07392-8.
- ISBN 978-0-7506-2896-9.
External links
- A college text-book of physics By Arthur Lalanne Kimball (Angular Velocity of a particle)
- Pickering, Steve (2009). "ω Speed of Rotation [Angular Velocity]". Sixty Symbols. Brady Haran for the University of Nottingham.