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

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The rotation vector $\omega$ of a sphere is equal to the angular velocity vector. In this case, the sphere rotates around the z-axis with an angular velocity of 4, indicating a counterclockwise direction when viewed from the positive z-axis. The relationship is defined mathematically by the equation $\vec{\omega}=\frac{\vec{r}\times\vec{v}}{\vec{r}\cdot\vec{r}}$, where $\vec{v}$ represents the tangential velocity at a point on the sphere. Thus, the rotation vector directly corresponds to the angular velocity in this scenario. Understanding this relationship is crucial for analyzing rotational motion.
evinda
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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)
 
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evinda said:
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.
 
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