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Concept# Angular velocity

Summary

In physics, angular velocity (symbol ω, sometimes Ω), also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates (spins or revolves) around an axis of rotation and how fast the axis itself changes direction.
The magnitude of the pseudovector, , represents the angular speed (or angular frequency), the rate at which the object rotates (spins or revolves). The pseudovector direction is normal to the instantaneous plane of rotation or angular displacement.
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.
Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity.
Angular velocity has dimension of angle per unit time; this is analogous to linear velocity, with angle replacing distance, with time in common. The SI unit of angular velocity is radians per second, although degrees per second (°/s) is also common. The radian is a dimensionless quantity, thus the SI units of angular velocity are dimensionally equivalent to reciprocal seconds, s−1, although rad/s is preferable.
The sense of angular velocity is conventionally specified by the right-hand rule, implying clockwise rotations (as viewed on the plane of rotation); negation (multiplication by -1) leaves the magnitude unchanged but flips the axis in the opposite direction.
For example, a geostationary satellite completes one orbit per day above the equator (360 degrees per 24 hours) has angular velocity magnitude (angular speed) ω = 360°/24 h = 15°/h (or 2π rad/24 h ≈ 0.26 rad/h) and angular velocity direction (a unit vector) parallel to Earth's rotation axis (, in the geocentric coordinate system). If angle is measured in radians, the linear velocity is the radius times the angular velocity, .

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Right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding the orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of the cross product of two vectors. Rather than a mathematical fact, it is a convention, closely related to the convention that rotation around a vertical axis is positive if it is counterclockwise and negative if it is clockwise. Most left-hand and right-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations.

Angular displacement

The angular displacement (symbol θ, , or φ), also called angle of rotation or rotational displacement, of a physical body is the angle (in units of radians, degrees, turns, etc.) through which the body rotates (revolves or spins) around a centre or axis of rotation. Angular displacement may be signed, indicating the sense of rotation (e.g., clockwise); it may also be greater (in absolute value) than a full turn. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time.

Cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.

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