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Angular velocity

Angular velocity is a fundamental concept in physics that describes the rate of change of an object's angular position with respect to time, quantifying how quickly it rotates around an . It is mathematically defined as the derivative of the θ with respect to time t, expressed as ω = dθ/dt, where θ is typically measured in ians. The standard unit of angular velocity is ians per second (/s), reflecting its role in rotational analogous to linear in translational motion. As a quantity, angular velocity ω has both magnitude and direction, with the latter determined by the : if the fingers of the right hand curl in the direction of rotation, the thumb points along the in the direction of the vector. For a rotating about a fixed , every point on the body shares the same angular velocity, simplifying the analysis of rotational dynamics. This vector nature enables the application of in describing complex motions, such as in planetary orbits or mechanical systems. Angular velocity relates directly to linear (tangential) velocity v through the equation v = r ω, where r is the radial distance from the axis of , linking rotational and translational descriptions of motion. In practical contexts, it is essential for understanding phenomena like the rotation of wheels in vehicles, the spin of celestial bodies, and the dynamics of gyroscopes in applications. Variations in angular velocity lead to , further extending its importance in Newton's laws for rotation.

Fundamentals

Definition and scalar form

Angular velocity in its scalar form describes the rate at which an object rotates about an , defined as the time of the \theta, expressed as \omega = \frac{d\theta}{dt}, where \theta is measured in radians. This scalar quantity captures the speed of rotation without regard to direction, serving as the rotational analog to linear velocity in . In the International System of Units (SI), angular velocity is measured in per second (rad/s); since the is dimensionless, the base unit is effectively the inverse second (s⁻¹), though rad/s is the conventional notation to highlight the angular nature. This definition presupposes familiarity with linear kinematics, where velocity is v = \frac{ds}{dt} and s represents along the path. For instance, in uniform , a constant scalar angular velocity \omega yields a constant tangential speed given by v = \omega r, with r as the radius of the path.

Vector form

In three-dimensional space, angular velocity is represented as a \vec{\omega}, which is a whose |\vec{\omega}| equals the scalar angular speed d\theta/dt, where \theta is the angle of . The direction of \vec{\omega} points along the instantaneous of , determined by the : the thumb aligns with \vec{\omega} while the fingers curl in the direction of the body's . This formulation extends the scalar concept by incorporating the orientation of the , essential for describing arbitrary rotations beyond planar motion. The angular velocity vector relates to infinitesimal rotations through the equation d\vec{\theta} = \vec{\omega} \, dt, where d\vec{\theta} is the rotation vector, representing an incremental change in over a small time dt. This linkage allows \vec{\omega} to parameterize the instantaneous rotational motion as a directed quantity. Unlike the vector, which accumulates over finite intervals and does not generally add commutatively due to the non-vector nature of large rotations, \vec{\omega} describes the local rate of change and supports vector addition for composing infinitesimal rotations. As a , \vec{\omega} transforms under parity inversion (spatial reflection through the origin) by changing sign, distinguishing it from polar vectors while behaving like a true under proper rotations. It is also termed an axial vector in certain contexts, reflecting its origin from cross products of polar vectors, such as in derivations of . For small angular displacements, the change in a position \vec{r} due to rotation by \vec{\omega} over time dt can be approximated using the infinitesimal form of Rodrigues' rotation formula, where the rotated vector is \vec{r}' \approx \vec{r} + d\vec{\theta} \times \vec{r}. However, composing finite rotations via vector addition of \vec{\omega} is only valid infinitesimally; larger composite rotations are non-commutative, meaning the order of successive rotations affects the final orientation, as rotations do not form a vector space.

Orbital angular velocity for point particles

In two dimensions

In two dimensions, the orbital angular velocity of a confined to the is defined as the scalar quantity measuring the rate of around the origin. This scalar can be expressed in terms of the particle's position coordinates ([x, y](/page/X&Y)) and velocity components (v_x, v_y) as \omega = \frac{x v_y - y v_x}{x^2 + y^2}, where the denominator is the square of the radial r^2 = x^2 + y^2. This expression arises from the kinematic definition of angular as the component perpendicular to the plane of motion. The sign of \omega indicates the of : positive for counterclockwise and negative for motion when viewed from the positive z-axis. For a point particle undergoing uniform circular motion in the plane, with constant tangential speed v and radius r, the angular velocity simplifies to the magnitude \omega = v / r. In this case, the direction is out of the plane along the positive z-axis for counterclockwise rotation, consistent with the right-hand rule. The orbital angular momentum scalar l for the particle of mass m relates kinematically to this as l = m r^2 \omega, providing a measure of the rotational tendency without delving into dynamic forces. This formulation of \omega applies more generally to any of the point particle in the , yielding the instantaneous angular velocity at any along the , regardless of whether the trajectory is circular. For example, consider a particle in an elliptical around a central point, such as a around a ; here, \omega = d\theta / dt varies along the path, reaching its maximum value at periapsis (closest approach) where the radial distance is minimized and tangential speed is maximized.

In three dimensions

In three dimensions, the orbital angular velocity of a generalizes the two-dimensional case to allow for non-planar trajectories, using a formulation that captures both the rate and of instantaneous about a chosen . The angular velocity \boldsymbol{\omega} is defined as \boldsymbol{\omega} = \frac{\boldsymbol{r} \times \boldsymbol{v}}{|\boldsymbol{r}|^2}, where \boldsymbol{r} is the of the particle relative to the and \boldsymbol{v} is its linear . This expression arises from decomposing the velocity into radial and tangential components, where the tangential component corresponds to about the . The magnitude of \boldsymbol{\omega} is |\boldsymbol{\omega}| = \frac{v \sin \phi}{r}, where r = |\boldsymbol{r}|, v = |\boldsymbol{v}|, and \phi is between \boldsymbol{r} and \boldsymbol{v}. This magnitude represents the instantaneous rotational speed, equivalent to the speed of the velocity component to \boldsymbol{r} divided by the from the . The direction of \boldsymbol{\omega} is to the plane instantaneously spanned by \boldsymbol{r} and \boldsymbol{v}, determined by the applied to the . For arbitrary motion, \boldsymbol{\omega} specifies the instantaneous axis of rotation passing through the , along which the particle appears stationary at that instant, with the motion decomposing into about this plus possible radial motion. However, \boldsymbol{\omega} is generally not constant, varying with the particle's unless the motion is uniform perpendicular to a fixed . A representative example is a particle following a uniform helical path, such as a in a uniform magnetic field with velocity components parallel and perpendicular to the field; here, \boldsymbol{\omega} has a constant component along the helix axis (the field direction), with magnitude determined by the cyclotron frequency \omega = qB/m, while the parallel velocity contributes no angular component. This formulation is origin-dependent: changing the reference point alters \boldsymbol{r} and thus \boldsymbol{\omega}, requiring adjustment of the position and velocity vectors relative to the new origin for consistency.

Spin angular velocity for rigid bodies

Definition and properties

In rigid body dynamics, the spin angular velocity \vec{\omega} is defined as the vector that describes the instantaneous rotational motion of the entire rigid body. The linear velocity \vec{v} of any point P in the body relative to a reference point O (such as the center of mass) is then given by \vec{v} = \vec{v}_O + \vec{\omega} \times \vec{r}, where \vec{r} is the position vector from O to P, and \vec{v}_O is the velocity of O. This relation holds because the rigid body constraint requires that distances between any two points remain constant during motion, ensuring no deformation. The general motion can be decomposed into translation of O plus rotation about an axis through O parallel to \vec{\omega}. A key property of \vec{\omega} is that it is identical for every point in the rigid body at any given instant, independent of the choice of origin, in contrast to the orbital angular velocity of a point particle. The instantaneous axis of rotation is the line parallel to \vec{\omega} along which points instantaneously have zero velocity. Its location relative to the reference point O is displaced by \vec{d} = (\vec{v}_O \times \vec{\omega}) / |\vec{\omega}|^2 from O. For example, in a spinning top, \vec{\omega} aligns with the body's symmetry , enabling stable rotation about that . Similarly, for a undergoing pure rolling without slipping, \vec{\omega} is directed to the of the and its relates to the center's by v_O = \omega r, but the instantaneous passes through the contact point. In general rigid body motion, which may combine rotation and translation, the motion corresponds to a screw motion about an instantaneous screw axis, with zero pitch for pure rotation (no translation component parallel to \vec{\omega}).

Determination of components

In the body-fixed frame attached to a rigid body, the spin angular velocity \vec{\omega} is expressed in terms of its components along the body basis vectors \hat{e}_1, \hat{e}_2, \hat{e}_3 as \vec{\omega} = \omega_1 \hat{e}_1 + \omega_2 \hat{e}_2 + \omega_3 \hat{e}_3. These components characterize the instantaneous rotation of the body relative to an inertial frame, with the basis vectors themselves evolving according to the relation \frac{d \hat{e}_i}{dt} = \vec{\omega} \times \hat{e}_i (evaluated in the body frame), a fundamental kinematic relation for rigid rotation that ensures the frame rotates with the body. This decomposition allows for the analysis of rotational dynamics using the body's principal axes, where the components \omega_i directly enter Euler's equations of motion. One common method to compute these components analytically is through , which parameterize the orientation of the body frame relative to the inertial frame via successive rotations. For the 3-1-3 Euler angle convention—often used for symmetric bodies like gyroscopes—the components in the body frame are given by: \begin{align*} \omega_x &= \dot{\phi} \sin \theta \sin \psi + \dot{\theta} \cos \psi, \\ \omega_y &= \dot{\phi} \sin \theta \cos \psi - \dot{\theta} \sin \psi, \\ \omega_z &= \dot{\phi} \cos \theta + \dot{\psi}, \end{align*} where \phi is the precession angle, \theta is the nutation angle, and \psi is the spin angle, with dots denoting time derivatives. These expressions arise from composing the angular velocity contributions of each Euler rotation, projecting them onto the body axes; for instance, \dot{\psi} primarily contributes to the spin along the body z-axis, while \dot{\phi} and \dot{\theta} introduce transverse components modulated by the orientation angles. In experimental or numerical contexts, the components can be determined practically by observing the linear velocities \vec{v}_i of at least three non-collinear points on the body relative to a reference point, solving the \vec{v}_i = \vec{v}_O + \vec{\omega} \times \vec{r}_i (where \vec{r}_i are position vectors from the reference) via least-squares minimization to isolate \vec{\omega}. This approach is particularly useful in applications like or , where sensor data provides the \vec{v}_i. A illustrative example is the steady precession of a gyroscope, where the body-frame components are time-varying due to the combined effects of high spin \dot{\psi} along the symmetry axis (yielding a dominant \omega_z \approx \dot{\psi}) and slower precession \dot{\phi}, which induces oscillating transverse components \omega_x and \omega_y at the precession frequency. In this case, the nutation \dot{\theta} \approx 0 for steady motion simplifies the expressions, but the precession causes the angular velocity vector to trace a path in the body frame, highlighting the dynamic nature of the components during torque-induced rotation.

Mathematical representations

In coordinate frames

Angular velocity is typically expressed in specific coordinate frames to describe the rotation of a rigid body relative to a reference. The inertial frame, fixed and non-rotating, provides components \vec{\omega}^i that remain constant in direction for steady rotations, while the body-fixed frame, attached to the rotating body, yields components \vec{\omega}^b that account for the body's orientation. The transformation between these frames uses the direction cosine matrix (DCM) R, which rotates vectors from the body frame to the inertial frame, such that \vec{\omega}^i = R \vec{\omega}^b. This relation ensures that the magnitude of \vec{\omega} is frame-invariant, but the components differ due to the basis change. The kinematics linking angular velocity to the attitude representation involve the time derivative of the DCM. Specifically, \dot{R} = [\vec{\omega}^i \times] R, where [\vec{\omega}^i \times] denotes the skew-symmetric cross-product matrix formed from \vec{\omega}^i: [\vec{\omega}^i \times] = \begin{pmatrix} 0 & -\omega^i_z & \omega^i_y \\ \omega^i_z & 0 & -\omega^i_x \\ -\omega^i_y & \omega^i_x & 0 \end{pmatrix}. Equivalently, in body-frame components, \dot{R} = R [\vec{\omega}^b \times], allowing computation of orientation evolution from measured body rates. These equations derive from the requirement that vectors fixed in the body frame rotate with the body. Under a coordinate transformation corresponding to a proper rotation, the components of angular velocity transform as those of a : if the frame rotates by R, then \vec{\omega}' = R \vec{\omega}. However, as an axial (or ), \vec{\omega} reverses sign under parity inversion but behaves like a polar under rotations, preserving its utility in rotational dynamics. This transformation property is essential for consistency across frames in multi-body systems. In non-inertial reference frames rotating with respect to an inertial frame at angular velocity \vec{\Omega}, the effective angular velocity for describing motion includes this frame rotation. For instance, the absolute angular velocity of a body in the rotating frame becomes \vec{\omega} + \vec{\Omega}, influencing fictitious forces like the centrifugal term m \vec{\Omega} \times (\vec{\Omega} \times \vec{r}) and Coriolis term -2m \vec{\Omega} \times \vec{v}. This adjustment is crucial for analyzing dynamics in Earth-fixed coordinates, where \vec{\Omega} is the planet's rotation rate. A practical application occurs in control, where angular velocity components are derived from sensor data using quaternions or DCMs to represent the spacecraft's relative to an inertial . Quaternions propagate the via \dot{\mathbf{q}} = \frac{1}{2} \mathbf{q} \otimes [\vec{\omega}^b \times] \mathbf{e}_4, enabling laws that maintain pointing accuracy despite disturbances. To compute orientation from angular velocity measurements, numerical integration methods integrate \vec{\omega} over time, but Euler angles suffer from gimbal lock singularities at certain orientations (e.g., pitch near \pm 90^\circ). Quaternions or rotation vectors circumvent these issues by providing singularity-free representations, with integration schemes like Runge-Kutta applied to the quaternion kinematic equations for high-fidelity simulations in aerospace applications.

Tensor representation

The angular velocity \boldsymbol{\omega} in three-dimensional can be represented by a second-rank \boldsymbol{\Omega}, which facilitates formulations of rotational using linear . The components of this tensor are defined as \Omega_{ij} = -\epsilon_{ijk} \omega_k, where \epsilon_{ijk} is the and over repeated indices is implied. This ensures that the tensor applied to a \mathbf{r} reproduces the : (\boldsymbol{\Omega} \mathbf{r})_i = \epsilon_{ijk} \omega_j r_k = (\boldsymbol{\omega} \times \mathbf{r})_i. As a skew-symmetric tensor (\boldsymbol{\Omega}^T = -\boldsymbol{\Omega}), \boldsymbol{\Omega} has a vanishing trace, \operatorname{tr}(\boldsymbol{\Omega}) = 0. Its eigenvalues are $0, +i|\boldsymbol{\omega}|, and -i|\boldsymbol{\omega}|, arising from the characteristic equation \lambda^3 + |\boldsymbol{\omega}|^2 \lambda = 0. These properties underscore \boldsymbol{\Omega}'s role as an element of the Lie algebra \mathfrak{so}(3). The tensor \boldsymbol{\Omega} serves as the infinitesimal generator of rotations, where the exponential map \exp(t \boldsymbol{\Omega}) yields a rotation matrix in the special orthogonal group \mathrm{SO}(3). For small t, this approximates as \exp(t \boldsymbol{\Omega}) \approx \mathbf{I} + t \boldsymbol{\Omega}, linking local rotational increments to the global group structure. In continuum mechanics, the velocity gradient tensor \mathbf{L} = \nabla \mathbf{v} decomposes additively into a symmetric rate-of-strain tensor \mathbf{D} = (\mathbf{L} + \mathbf{L}^T)/2 and an antisymmetric spin tensor \mathbf{W} = (\mathbf{L} - \mathbf{L}^T)/2. The spin tensor \mathbf{W} captures the rotational component of the deformation, coinciding with \boldsymbol{\Omega} for rigid-body motions but representing the average rotation rate in deformable continua. A key application appears in , where the vector \boldsymbol{\zeta} = \nabla \times \mathbf{v} relates directly to the rotation tensor: the components satisfy \zeta_k = -\epsilon_{kij} W_{ij}, such that the magnitude corresponds to twice the angular speed encoded in \mathbf{W}. This connection quantifies local fluid element rotation as half the vector.

Related concepts

Angular acceleration

Angular acceleration is defined as the time of the vector, denoted as \boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt}. This describes the at which the changes in both magnitude and direction for a rotating . If the magnitude of \boldsymbol{\omega} varies while its direction remains fixed (as in rotation about a principal ), \boldsymbol{\alpha} aligns or antiparallel to \boldsymbol{\omega}. Conversely, during where the direction of \boldsymbol{\omega} shifts, \boldsymbol{\alpha} includes a component to \boldsymbol{\omega}, reflecting the instantaneous change in orientation. In an inertial reference frame, the components of for a are simply \alpha_i = \frac{d\omega_i}{dt} for i = x, y, z. In the body-fixed frame, which rotates with the body, the kinematic expression for the time of \boldsymbol{\omega} accounts for the frame's motion itself, leading to \left( \frac{d\boldsymbol{\omega}}{dt} \right)_{\text{inertial}} = \left( \frac{d\boldsymbol{\omega}}{dt} \right)_{\text{body}} + \boldsymbol{\omega} \times \boldsymbol{\omega}, though the second term vanishes, simplifying to the body-frame rate for the vector. This formulation emphasizes the kinematic perspective, distinct from dynamic interpretations involving torques. The finite change \Delta \boldsymbol{\omega} over a time interval is exactly \int \boldsymbol{\alpha} \, dt. However, due to the non-commutative nature of vector rotations in three dimensions, the corresponding finite rotation cannot be simply obtained by integrating \boldsymbol{\omega} in a commutative manner; for precise finite rotations, parametrizations like the (or vector) are employed to capture the compounded effects of sequential angular changes. Consider an example of a accelerating under a constant axial : the resulting is \boldsymbol{\alpha} = \frac{T}{I} \hat{n}, directed along the axis \hat{n}, where T is the torque magnitude and I is the about that axis. This follows analogously from the rotational form of Newton's second law, providing a straightforward kinematic description of the speeding-up motion. The contributes to the linear of points on the through the kinematic relation \mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r} + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}), where \mathbf{r} is the position vector from the rotation reference point to the point of interest, and \mathbf{a} is the . The \boldsymbol{\alpha} \times \mathbf{r} represents the tangential component arising directly from \boldsymbol{\alpha}, while the second accounts for centripetal effects; for points with \mathbf{v}_{\text{rel}} in the body (though zero in pure rigid ), an additional Coriolis $2 \boldsymbol{\omega} \times \mathbf{v}_{\text{rel}} appears.

Relation to angular momentum

In rigid body dynamics, the angular momentum \vec{L} about the center of mass is linearly related to the angular velocity \vec{\omega} through the inertia tensor \mathbf{I}, expressed as \vec{L} = \mathbf{I} \vec{\omega}. The inertia tensor \mathbf{I} is a symmetric 3×3 matrix that encapsulates the distribution of the body relative to the chosen coordinate frame. In the principal axis frame, where \mathbf{I} is diagonal with elements I_1, I_2, I_3, this relation simplifies component-wise to L_i = I_i \omega_i for i = 1, 2, 3. For a single point particle of mass m at position \vec{r} from the center of rotation in a rigid body, the linear momentum is \vec{p} = m \vec{v} = m (\vec{\omega} \times \vec{r}), so the angular momentum is \vec{L} = \vec{r} \times \vec{p} = m \vec{r} \times (\vec{\omega} \times \vec{r}). Using the vector triple product identity, this expands to \vec{L} = m [r^2 \vec{\omega} - (\vec{r} \cdot \vec{\omega}) \vec{r}], where r^2 = |\vec{r}|^2. The magnitude simplifies to L = m r^2 \omega_\perp when considering the component of \vec{\omega} perpendicular to \vec{r}, highlighting the perpendicular contribution to rotation. Angular velocity \vec{\omega} is fundamentally kinematic, characterizing the instantaneous rotation without regard to mass, whereas angular momentum \vec{L} is dynamic, incorporating the body's mass distribution via \mathbf{I}. Conservation of \vec{L} in the absence of external torques implies that variations in \mathbf{I} (e.g., due to reconfiguration) alter \vec{\omega} to maintain |\vec{L}| = I |\vec{\omega}| constant, as demonstrated by an ice skater who increases spin rate by pulling in their arms, reducing I and thus boosting \omega. The dynamical evolution of \vec{\omega} is governed by Euler's rigid body equations in the principal frame: \begin{align*} I_1 \dot{\omega}_1 + (I_3 - I_2) \omega_2 \omega_3 &= \tau_1, \\ I_2 \dot{\omega}_2 + (I_1 - I_3) \omega_3 \omega_1 &= \tau_2, \\ I_3 \dot{\omega}_3 + (I_2 - I_1) \omega_1 \omega_2 &= \tau_3, \end{align*} or in vector notation, \mathbf{I} \dot{\vec{\omega}} + \vec{\omega} \times (\mathbf{I} \vec{\omega}) = \vec{\tau}, where \vec{\tau} is the applied and \dot{\vec{\omega}} is the . For torque-free motion (\vec{\tau} = 0) of an asymmetric with distinct principal moments, \vec{\omega} undergoes free , tracing a closed path around the fixed \vec{L} direction in the body frame, with the precession rate depending on the differences in I_i.