Angular velocity

In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a rigid body's centre of rotation revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise. In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a rigid body's centre of rotation revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise. For example, a geostationary satellite completes one orbit per day above the equator, or 360 degrees per 24 hours, and has angular velocity ω = 360 / 24 = 15 degrees per hour, or 2π / 24 ≈ 0.26 radians per hour. If angle is measured in radians, the linear velocity is the radius times the angular velocity, v = r ω {displaystyle v=romega } . With orbital radius 42,000 km from the earth's center, the satellite's speed through space is thus v = 42,000 × 0.26 ≈ 11,000 km/hr. The angular velocity is positive since the satellite travels eastward with the Earth's rotation (counter-clockwise from above the north pole.) In three dimensions, angular velocity is a pseudovector, with its magnitude measuring the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule. In the simplest case of circular motion at radius r {displaystyle r} , with position given by the angular displacement ϕ ( t ) {displaystyle phi (t)} from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: ω = d ϕ d t {displaystyle omega ={ frac {dphi }{dt}}} . If ϕ {displaystyle phi } is measured in radians, the distance from the x-axis around the circle to the particle is ℓ = r ϕ {displaystyle ell =rphi } , and the linear velocity is v ( t ) = d ℓ d t = r ω ( t ) {displaystyle v(t)={ frac {dell }{dt}}=romega (t)} , so that ω = v r {displaystyle omega ={ frac {v}{r}}} . 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 r {displaystyle mathbf {r} } from the origin O {displaystyle O} to a particle P {displaystyle P} , with its polar coordinates ( r , ϕ ) {displaystyle (r,phi )} . (All variables are functions of time t {displaystyle t} .) The particle has linear velocity splitting as v = v ‖ + v ⊥ {displaystyle mathbf {v} =mathbf {v} _{|}+mathbf {v} _{perp }} , with the radial component v ‖ {displaystyle mathbf {v} _{|}} parallel to the radius, and the cross-radial (or tangential) component v ⊥ {displaystyle mathbf {v} _{perp }} perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity. 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 v ⊥ {displaystyle v_{perp }} is the signed magnitude of v ⊥ {displaystyle mathbf {v} _{perp }} , positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity v {displaystyle mathbf {v} } gives magnitude v {displaystyle v} (linear speed) and angle θ {displaystyle heta } relative to the radius vector; in these terms, v ⊥ = v sin ⁡ ( θ ) {displaystyle v_{perp }=vsin( heta )} , so that These formulas may be derived from r = ( x ( t ) , y ( t ) ) {displaystyle mathbf {r} =(x(t),y(t))} , v = ( x ′ ( t ) , y ′ ( t ) ) {displaystyle mathbf {v} =(x'(t),y'(t))} and ϕ = arctan ⁡ ( y ( t ) / x ( t ) ) {displaystyle phi =arctan(y(t)/x(t))} , together with the projection formula v ⊥ = r ⊥ r ⋅ v {displaystyle v_{perp }={ frac {mathbf {r} ^{perp }!!}{r}}cdot mathbf {v} } , where r ⊥ = ( − y , x ) {displaystyle mathbf {r} ^{perp }=(-y,x)} . 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.