MHB Finding the Rotation Vector $\omega$ of a Sphere

[evinda]

Hello! (Wave)

A sphere with radius $10 cm$ and center $(0,0,0)$ turns around the $z$-axis with angular velocity $4$ and with such a direction that the rotation has counterclockwise direction, being seen my the positive semi-axis $z$.

I want to find the rotation-vector $\omega$.

Is this equal to the vector of the angular velocity? If so, why? (Thinking)

 

[Ackbach]

Hello! (Wave)

A sphere with radius $10 cm$ and center $(0,0,0)$ turns around the $z$-axis with angular velocity $4$ and with such a direction that the rotation has counterclockwise direction, being seen my the positive semi-axis $z$.

I want to find the rotation-vector $\omega$.

Is this equal to the vector of the angular velocity? If so, why? (Thinking)

It is - by definition. The rotation vector $\vec{\omega}$ IS the angular velocity! It's defined by
$$\vec{\omega}=\frac{\vec{r}\times\vec{v}}{\vec{r}\cdot\vec{r}},$$
where $\vec{v}$ is the regular velocity vector in the tangential direction of a point on the circle or sphere.