Angular velocity and angular acceleration

Let a particle of the spinning body turn by Δθ in a time Δt. The limit of Δθ/Δt as Δt shrinks to zero is the angular velocity, ω = dθ/dt.

Every particle of the rigid body has the same ω at a given instant, so ω is the angular velocity of the whole body. Pure rotation means every part has the same angular velocity, just as pure translation means every part has the same velocity.

The speed of a particle at perpendicular distance r from the axis is v = ωr; particles on the axis (r = 0) are at rest.

Angular velocity is a vector along the axis, in the direction a right-handed screw advances when turned with the body. Reversing the sense of rotation reverses ω.

In vector form v = ω × r, with r measured from an origin on the axis; v is tangent to the particle's circle and has magnitude ωr⊥. The same relation holds for rotation about a fixed point, such as a top.

For rotation about a fixed axis the direction of ω does not change; only its magnitude can. In more general rotation both can change.

Angular acceleration is α = dω/dt. For a fixed axis it is along the axis, and the vector equation becomes the scalar α = dω/dt.