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Angular velocity

From Wikipedia, the free encyclopedia
Direction and rate of rotation

Angular velocity
Common symbols
ω
SI unitrad ⋅ s−1
InSI base unitss−1
Extensive?yes
Intensive?yes (forrigid body only)
Conserved?no
Behaviour under
coord transformation
pseudovector
Derivations from
other quantities
ω = dθ / dt
DimensionT−1{\displaystyle {\mathsf {T}}^{-1}}
Part of a series on
Classical mechanics
F=dpdt{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}

Inphysics,angular velocity (symbolω orω→{\displaystyle {\vec {\omega }}}, the lowercase Greek letteromega), also known as theangular frequency vector,[1] is apseudovector representation of how theangular position ororientation of an object changes with time, i.e. how quickly an objectrotates (spins or revolves) around an axis of rotation and how fast the axis itself changesdirection.[2]

The magnitude of the pseudovector,ω=‖ω‖{\displaystyle \omega =\|{\boldsymbol {\omega }}\|}, represents theangular speed (orangular frequency), the angular rate at which the object rotates (spins or revolves). The pseudovector directionω^=ω/ω{\displaystyle {\hat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}/\omega } isnormal to the instantaneousplane of rotation orangular displacement.

There are two types of angular velocity:

  • Orbital angular velocity refers to how fast a point objectrevolves about a fixed origin, i.e. the time rate of change of its angular position relative to theorigin.[citation needed]
  • Spin angular velocity refers to how fast a rigid body rotates around a fixed axis of rotation, and is independent of the choice of origin, in contrast to orbital angular velocity.

Angular velocity hasdimension of angle per unit time; this is analogous to linearvelocity, with angle replacingdistance, with time in common. TheSI unit of angular velocity isradians per second,[3] althoughdegrees per second (°/s) is also common. Theradian is adimensionless quantity, thus the SI units of angular velocity are dimensionally equivalent toreciprocal seconds, s−1, although rad/s is preferable to avoid confusion withrotation velocity in units ofhertz (also equivalent to s−1).[4]

The sense of angular velocity is conventionally specified by theright-hand rule, implyingclockwise rotations (as viewed on the plane of rotation);negation (multiplication by −1) leaves the magnitude unchanged but flips the axis in theopposite direction.[5]

For example, ageostationary satellite completes one orbit per day above theequator (360 degrees per 24 hours)a 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 (aunit vector) parallel toEarth's rotation axis (ω^=Z^{\displaystyle {\hat {\omega }}={\hat {Z}}}, in thegeocentric coordinate system). If angle is measured in radians, the linear velocity is the radius times the angular velocity,v=rω{\displaystyle v=r\omega }. With orbital radius 42,000 km from the Earth's center, the satellite'stangential speed through space is thusv = 42,000 km × 0.26/h ≈ 11,000 km/h. The angular velocity is positive since the satellite travelsprograde with the Earth's rotation (the same direction as the rotation of Earth).

^a Geosynchronous satellites actually orbit based on a sidereal day which is 23h 56m 04s, but 24h is assumed in this example for simplicity.

Orbital angular velocity of a point particle

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Particle in two dimensions

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The angular velocity of the particle atP with respect to the originO is determined by theperpendicular component of the velocity vectorv.

In the simplest case of circular motion at radiusr{\displaystyle r}, with position given by the angular displacementϕ(t){\displaystyle \phi (t)} from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time:ω=dϕdt{\textstyle \omega ={\frac {d\phi }{dt}}}. Ifϕ{\displaystyle \phi } is measured inradians, the arc-length from the positive x-axis around the circle to the particle isℓ=rϕ{\displaystyle \ell =r\phi }, and the linear velocity isv(t)=dℓdt=rω(t){\textstyle v(t)={\frac {d\ell }{dt}}=r\omega (t)}, so thatω=vr{\textstyle \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 vectorr{\displaystyle \mathbf {r} } from the originO{\displaystyle O} to a particleP{\displaystyle P}, with itspolar coordinates(r,ϕ){\displaystyle (r,\phi )}. (All variables are functions of timet{\displaystyle t}.) The particle has linear velocity splitting asv=v‖+v⊥{\displaystyle \mathbf {v} =\mathbf {v} _{\|}+\mathbf {v} _{\perp }}, with the radial componentv‖{\displaystyle \mathbf {v} _{\|}} parallel to the radius, and the cross-radial (or tangential) componentv⊥{\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 velocityω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:

ω=dϕdt=v⊥r.{\displaystyle \omega ={\frac {d\phi }{dt}}={\frac {v_{\perp }}{r}}.}

Here the cross-radial speedv⊥{\displaystyle v_{\perp }} is the signed magnitude ofv⊥{\displaystyle \mathbf {v} _{\perp }}, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocityv{\displaystyle \mathbf {v} } gives magnitudev{\displaystyle v} (linear speed) and angleθ{\displaystyle \theta } relative to the radius vector; in these terms,v⊥=vsin⁡(θ){\displaystyle v_{\perp }=v\sin(\theta )}, so that

ω=vsin⁡(θ)r.{\displaystyle \omega ={\frac {v\sin(\theta )}{r}}.}

These formulas may be derived doingr=(rcos⁡(φ),rsin⁡(φ)){\displaystyle \mathbf {r} =(r\cos(\varphi ),r\sin(\varphi ))}, beingr{\displaystyle r} a function of the distance to the origin with respect to time, andφ{\displaystyle \varphi } a function of the angle between the vector and the x axis. Then:drdt=(r˙cos⁡(φ)−rφ˙sin⁡(φ),r˙sin⁡(φ)+rφ˙cos⁡(φ)),{\displaystyle {\frac {d\mathbf {r} }{dt}}=({\dot {r}}\cos(\varphi )-r{\dot {\varphi }}\sin(\varphi ),{\dot {r}}\sin(\varphi )+r{\dot {\varphi }}\cos(\varphi )),}which is equal to:r˙(cos⁡(φ),sin⁡(φ))+rφ˙(−sin⁡(φ),cos⁡(φ))=r˙r^+rφ˙φ^{\displaystyle {\dot {r}}(\cos(\varphi ),\sin(\varphi ))+r{\dot {\varphi }}(-\sin(\varphi ),\cos(\varphi ))={\dot {r}}{\hat {r}}+r{\dot {\varphi }}{\hat {\varphi }}}(seeUnit vector in cylindrical coordinates).

Knowingdrdt=v{\textstyle {\frac {d\mathbf {r} }{dt}}=\mathbf {v} }, we conclude that the radial component of the velocity is given byr˙{\displaystyle {\dot {r}}}, becauser^{\displaystyle {\hat {r}}} is a radial unit vector; and the perpendicular component is given byrφ˙{\displaystyle r{\dot {\varphi }}} becauseφ^{\displaystyle {\hat {\varphi }}} 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 apseudoscalar, a numerical quantity which changes sign under aparity inversion, such as inverting one axis or switching the two axes.

Particle in three dimensions

[edit]
The orbital angular velocity vector encodes the time rate of change of angular position, as well as the instantaneous plane of angular displacement. In this case (counter-clockwise circular motion) the vector points up.

Inthree-dimensional space, we again have the position vectorr of a moving particle. Here, orbital angular velocity is apseudovector whose magnitude is the rate at whichr sweeps out angle (in radians per unit of time), and whose direction is perpendicular to the instantaneous plane in whichr sweeps out angle (i.e. the plane spanned byr andv). However, as there aretwo directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, theright-hand rule is used.

Let the pseudovectoru{\displaystyle \mathbf {u} } be the unit vector perpendicular to the plane spanned byr andv, so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top ofu{\displaystyle \mathbf {u} }). Taking polar coordinates(r,ϕ){\displaystyle (r,\phi )} in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:

ω=ωu=dϕdtu=vsin⁡(θ)ru,{\displaystyle {\boldsymbol {\omega }}=\omega \mathbf {u} ={\frac {d\phi }{dt}}\mathbf {u} ={\frac {v\sin(\theta )}{r}}\mathbf {u} ,}

whereθ is the angle betweenr andv. In terms of the cross product, this is:

ω=r×vr2.{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}.}[6]

From the above equation, one can recover the tangential velocity as:

v⊥=ω×r{\displaystyle \mathbf {v} _{\perp }={\boldsymbol {\omega }}\times \mathbf {r} }

Spin angular velocity of a rigid body or reference frame

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Given a rotating frame of three linearly independent unit coordinate vectors, at each instant in time, there always exists a common axis (called the axis of rotation) around which all three vectors rotate with the same angular speed and in the same angular direction (clockwise or counterclockwise). In such a frame, each vector may be considered as a moving particle with constant scalar radius. A collection of such particles is called a rigid body.

Euler's rotation theorem says that in a rotating frame, the axis of rotation one obtains from one choice of three linearly independent unit vectors is the same as that for any other choice; that is, there is onesingleinstantaneous axis of rotation to the frame, around which all points rotate at the same angular speed and in the same angular direction (clockwise or counterclockwise). The spin angular velocity of a frame or rigid body is defined to be the pseudovector whose magnitude is this common angular speed, and whose direction is along the common axis of rotation in accordance with the right-hand rule (that is, for counterclockise rotation, it points "upward" along the axis, while for clockwise rotation, it points "downward").

In larger than 3 spatial dimensions, the interpretation of spin angular velocity as a pseudovector is not valid; however, it may be characterized by a more general type of object known as an antisymmetric rank-2tensor.

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 agimbal. 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:ω1+ω2=ω2+ω1{\displaystyle \omega _{1}+\omega _{2}=\omega _{2}+\omega _{1}}.

If we choose a reference pointr0{\displaystyle {{\boldsymbol {r}}_{0}}} fixed in a rotating frame, the velocityr˙{\displaystyle {\dot {\boldsymbol {r}}}} of any point in the frame is given by

r˙=r0˙+ω×(r−r0){\displaystyle {\dot {\boldsymbol {r}}}={\dot {{\boldsymbol {r}}_{0}}}+{\boldsymbol {\omega }}\times ({\boldsymbol {r}}-{{\boldsymbol {r}}_{0}})}

Components from the basis vectors of a body-fixed frame

[edit]

Consider a rigid body rotating about a fixed point O. Construct a reference frame in the body consisting of an orthonormal set of vectorse1,e2,e3{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} 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

ω=(e˙1⋅e2)e3+(e˙2⋅e3)e1+(e˙3⋅e1)e2,{\displaystyle {\boldsymbol {\omega }}=\left({\dot {\mathbf {e} }}_{1}\cdot \mathbf {e} _{2}\right)\mathbf {e} _{3}+\left({\dot {\mathbf {e} }}_{2}\cdot \mathbf {e} _{3}\right)\mathbf {e} _{1}+\left({\dot {\mathbf {e} }}_{3}\cdot \mathbf {e} _{1}\right)\mathbf {e} _{2},}

wheree˙i=deidt{\displaystyle {\dot {\mathbf {e} }}_{i}={\frac {d\mathbf {e} _{i}}{dt}}} is the time rate of change of the frame vectorei,i=1,2,3,{\displaystyle \mathbf {e} _{i},i=1,2,3,} due to the rotation.

This formula is incompatible with the expression fororbital angular velocity

ω=r×vr2,{\displaystyle {\boldsymbol {\omega }}={\frac {{\boldsymbol {r}}\times {\boldsymbol {v}}}{r^{2}}},}

as that formula defines angular velocity for asingle point about O, while the formula in this section applies to a frame or rigid body. In the case of a rigid body asingleω{\displaystyle {\boldsymbol {\omega }}} has to account for the motion ofall particles in the body.

Components from Euler angles

[edit]
Diagram showing Euler frame in green

The components of the spin angular velocity pseudovector were first calculated byLeonhard Euler using hisEuler 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 instantaneousEuler rotations). Therefore:[7]

ω=α˙u1+β˙u2+γ˙u3{\displaystyle {\boldsymbol {\omega }}={\dot {\alpha }}\mathbf {u} _{1}+{\dot {\beta }}\mathbf {u} _{2}+{\dot {\gamma }}\mathbf {u} _{3}}

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:

ω=(α˙sin⁡βsin⁡γ+β˙cos⁡γ)i^+(α˙sin⁡βcos⁡γ−β˙sin⁡γ)j^+(α˙cos⁡β+γ˙)k^{\displaystyle {\boldsymbol {\omega }}=({\dot {\alpha }}\sin \beta \sin \gamma +{\dot {\beta }}\cos \gamma ){\hat {\mathbf {i} }}+({\dot {\alpha }}\sin \beta \cos \gamma -{\dot {\beta }}\sin \gamma ){\hat {\mathbf {j} }}+({\dot {\alpha }}\cos \beta +{\dot {\gamma }}){\hat {\mathbf {k} }}}

wherei^,j^,k^{\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}} 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

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This section is an excerpt fromAngular velocity tensor.[edit]

Theangular velocity tensor is askew-symmetric matrix defined by:

Ω=(0−ωzωyωz0−ωx−ωyωx0){\displaystyle \Omega ={\begin{pmatrix}0&-\omega _{z}&\omega _{y}\\\omega _{z}&0&-\omega _{x}\\-\omega _{y}&\omega _{x}&0\\\end{pmatrix}}}

The scalar elements above correspond to theangular velocity vector componentsω=(ωx,ωy,ωz){\displaystyle {\boldsymbol {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}.

This is aninfinitesimal rotation matrix.The linear mapping Ω acts as across product(ω×){\displaystyle ({\boldsymbol {\omega }}\times )}:

ω×r=Ωr{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {r}}=\Omega {\boldsymbol {r}}}

wherer{\displaystyle {\boldsymbol {r}}} is aposition vector.

When multiplied by a time difference, it results in theangular displacement tensor.

See also

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References

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  1. ^Cummings, Karen; Halliday, David (2007).Understanding physics. New Delhi: John Wiley & Sons Inc., authorized reprint to Wiley – India. pp. 449, 484, 485, 487.ISBN 978-81-265-0882-2.(UP1)
  2. ^"Angular velocity | Rotational Motion, Angular Momentum, Torque | Britannica".www.britannica.com. Retrieved5 October 2024.
  3. ^Taylor, Barry N. (2009).International System of Units (SI) (revised 2008 ed.). DIANE Publishing. p. 27.ISBN 978-1-4379-1558-7.Extract of page 27
  4. ^"Units with special names and symbols; units that incorporate special names and symbols".
  5. ^Hibbeler, Russell C. (2009).Engineering Mechanics.Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 314, 153.ISBN 978-0-13-607791-6.(EM1)
  6. ^Singh, Sunil K.Angular Velocity. Rice University. Retrieved21 May 2021 – via OpenStax.
  7. ^K.S.HEDRIH: Leonhard Euler (1707–1783) and rigid body dynamics

External links

[edit]
Look upangular velocity in Wiktionary, the free dictionary.
Wikimedia Commons has media related toAngular velocity.
Linear/translational quantitiesAngular/rotational quantities
Dimensions1LL2Dimensions1θθ2
Ttime:t
s		absement:A
m s		Ttime:t
s
1distance:d,position:r,s,x,displacement
m		area:A
m2		1angle:θ,angular displacement:θ
rad		solid angle:Ω
rad2, sr
T−1frequency:f
s−1,Hz		speed:v,velocity:v
m s−1		kinematic viscosity:ν,
specific angular momentum: h
m2 s−1		T−1frequency:f,rotational speed:n,rotational velocity:n
s−1,Hz		angular speed:ω,angular velocity:ω
rad s−1
T−2acceleration:a
m s−2		T−2rotational acceleration
s−2		angular acceleration:α
rad s−2
T−3jerk:j
m s−3		T−3angular jerk:ζ
rad s−3
Mmass:m
kg		weighted position:M ⟨x⟩ = ∑mxmoment of inertia: I
kg m2		ML
MT−1Mass flow rate:m˙{\displaystyle {\dot {m}}}
kg s−1		momentum:p,impulse:J
kg m s−1,N s		action:𝒮,actergy:ℵ
kg m2 s−1,J s		MLT−1angular momentum:L,angular impulse:ΔL
kg m rad s−1
MT−2force:F,weight:Fg
kg m s−2,N		energy:E,work:W,Lagrangian:L
kg m2 s−2,J		MLT−2torque:τ,moment:M
kg m rad s−2,N m
MT−3yank:Y
kg m s−3, N s−1		power:P
kg m2 s−3, W		MLT−3rotatum:P
kg m rad s−3, N m s−1
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