The rate of change of angular position of a rotating object.Angular velocity is a measure of how fast an object rotates or revolves around a fixed point or axis. Unlike linear velocity, which describes how quickly a point moves in a straight line, angular velocity describes how quickly the angle of a rotating object changes. It is a vector quantity, meaning it has both a magnitude (how fast the rotation is) and a direction (along the axis of rotation, following the right-hand rule).
The concept of angular velocity has roots in ancient astronomy, but it was formalized in the 17th century by scientists like Galileo and Huygens, who studied circular motion. Leonhard Euler later developed the mathematical framework for rigid body rotation in the 18th century.
For an object rotating around an axis, angular velocity is calculated as the change in the angular position (measured in radians) divided by the change in time. It is often denoted by the Greek letter omega (ω). The relationship between angular velocity and linear velocity for a point at a distance r from the axis is v = ωr, which shows that points farther from the axis move faster linearly even if the angular velocity is the same.
Angular velocity is essential in engineering for designing rotating machinery, in navigation for gyroscopes, and in astrophysics for determining the rotation rates of stars, planets, and galaxies.
The angular velocity of the Earth's rotation is about 7.27 × 10⁻⁵ radians per second, which means a point on the equator moves at roughly 1670 kilometers per hour.
| ω | variable | frequency of the wave — angular frequency radian per second |
| m | variable | azimuthal quantum number — azimuthal quantum number |
| Ω_H | variable | angular velocity of the black hole horizon — angular velocity radian per second |
| Re | operator | extracts the real component of a complex quantity — real part |
| \rho_{\rm GJ} | variable | Goldreich-Julian charge density — charge density coulomb per cubic metre |
| \mathbf{\Omega} | variable | Angular velocity vector of the neutron star — angular velocity radian per second |
| \mathbf{B} | variable | Magnetic field vector — magnetic field tesla |
| c | constant | Universal constant, speed of light in vacuum — speed of light = 299792458 metre per second |
| 2\pi | constant | Mathematical constant, 2π — two pi = 6.283 |
| \cdot | operator | Scalar product of two vectors — dot product |
| δM | variable | Infinitesimal change in black hole mass — mass kilogram |
| κ | variable | Surface gravity of the black hole — surface gravity metre per second squared |
| δA | variable | Infinitesimal change in horizon area — area square metre |
| Ω_H | variable | Angular velocity of the horizon — angular velocity radian per second |
| δJ | variable | Infinitesimal change in angular momentum — angular momentum kilogram square metre per second |
| δ | operator | Infinitesimal variation of a quantity — variation |
| \dot{\omega} | variable | rate of periastron advance — angular velocity radian per second |
| M | variable | total mass of the binary system — mass kilogram |
| a | variable | semi-major axis of the orbit — length metre |
| e | variable | orbital eccentricity — eccentricity |
| G | constant | Constant of universal gravitation. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| c | constant | Speed of light in vacuum. — speed of light = 299792458 metre per second |
| I | variable | moment of inertia — moment of inertia kilogram metre squared |
| Ω | variable | angular velocity — angular velocity radian per second |
| Ω̇ | variable | time derivative of angular velocity — angular acceleration radian per second squared |
| t_E | variable | Einstein crossing time — time second |
| \theta_E | variable | Angular radius of the Einstein ring — einstein radius angle radian |
| \mu | variable | Relative proper motion between lens and source — angular velocity radian per second |
