Angular velocity

In Physical, and more specifically in Mechanical, the angular velocity ω, also called angular frequency or pulsation , is a measurement of the Speed of Rotation. It is expressed in the international Système in Radian S by second (rad.s^-1), or more simply in s^-1 since the angles are sizes without dimension; it remains in a current way given out of turns per minute (tr/min). A complete revolution is equal to 2π radians, therefore:

\ Omega = \ frac {D \, \ theta} {D \, T} = \ frac {2 \ pi} {T} = 2 \ pi f

where the expression $\ frac {D \, \ theta} {D \, T}$ is the derivative of the angle compared to time (in rad.s^-1), T is the Period of rotation (in S) and F is the Fréquence (in s^-1).

The use the angular velocity instead of the ordinary frequency is practical in many applications because it makes it possible to avoid the excessive appearance of π. It is used, amongst other things, in many fields of physics like the quantum Mécanique and the electromagnetism like in mathematics for the Transformée of Fourier.

For a circular trajectory:

$\ Omega = \ frac {2 \ pi} {T} = \ frac {v} {R}$

T is the period (in S), R is the ray of rotation and v is the speed of the point (in m.s^-1)

One uses sometimes a angular Flight Path Vector $\ vec {\ Omega}$ . It is about the vector:

normal in the plan of rotation,
directed so that the movement is done in the positive direction,
and whose standard is worth ω.

Theorems and properties relating to the angular frequency

Composition angular velocities

Whatever the solids has, B and C, rotational frequencies are bound by:

\ vec {\ Omega} _ {A/C} = \ vec {\ Omega} _ {A/B} + \ vec {\ Omega} _ {B/C}

. Note: really it is not a question of vector since the symmetrical one in a mirror is reversed.

; Example

Is a Référentiel galiléen R.
Considérons a solid

S_1

in rotation at the angular frequency

\ omega_ {S_1/R}

, compared to the reference frame R .
Considérons a solid

S_2

in rotation compared to $S_1$ at the angular frequency

\ omega_ {S_2/S_1}

.
La number of revolutions of

S_2

compared to R,

\ omega_ {S_2/R}

will be equal to

\ omega_ {S_2/R} = \ omega_ {S_1/R} + \ omega_ {S_2/S_1}

.
Dans this case, if

\ omega_ {S_2/S_1} = \ omega_ {S_1/R}

, the solid

S_2

will be in circular Translation in reference frame R.

Relation Speed - Frequency angular

That is to say a solid S. If has and B are two points of this solid, then:

\ vec {V_A} = \ vec {V_B} + \ vec {AB} \ wedge \ vec {\ Omega} _ {A/R}

this formula shows well that “ω” (Omega) is not a speed

\ vec {AB} \ wedge \ vec {\ Omega} _ {A/R}

is a speed.

; Example

Is a disc of 1m of ray, in rotation around its axis of Symétrie at the speed

\ omega_ {D/R}

(R a reference frame galiléen). If ω is expressed in radians per seconds, then each point located on the edge of the disc will have a speed Orthogonal E with the axis of rotation (by property of the vector product) of

\ omega_ {D/R} \ times1m

. Unit: meters a second

Instantaneous center of rotation

By analogy: when a movement is not rectilinear, one can look in a specific way his speed and his direction at a given moment. In the same way, if it is not in rotation, one can consider in a specific way an angular velocity and a center of rotation.

The instantaneous Center of rotation of has compared to B, for the moment T is item I of has checking: $\ vec {V} _ {I/B} (T) = \ vec {0}$

See too

Moment (mechanical)
Pseudovecteur

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