Recalculating angular velocity from velocity

Question

Suppose a system has angular velocity $\vec{\Omega} = \Omega_0\hat{z}$. Then according to Wikipedia, the velocity is given by $$ \vec{v} = \vec{\Omega}\times\vec{r} = \Omega_0r\sin\theta\hat{\phi} $$ where we used spherical coordinates. According to the same wikipedia page, we have that $$ \vec{\Omega} = \frac{\vec{r}\times\vec{v}}{r^2} = \frac{v}{r}\hat{r}\times\hat{v} = \Omega_0\sin\theta\hat{r}\times\hat{\phi} = -\Omega_0\sin\theta\hat{\theta} = \Omega_0\hat{z} - \Omega_0\cos\theta\hat{r} $$ Where am I wrong?

Answer 1

Here is proof of the relationship $\vec{\omega} = \frac{\vec{r} \times \vec{v} }{ \| \vec{r} \|^2 }$.

The velocity of any point, located at $\vec{r}$, if the origin is at the rotation center is $\vec{v} = \vec{\omega} \times \vec{r}$ regardless of which coordinate represntation is used. Then

$$ \require{cancel} \vec{\omega} = \frac{\vec{r} \times \vec{v} }{ \| \vec{r} \|^2 } = \frac{\vec{r} \times (\vec{\omega} \times \vec{r}) }{ \| \vec{r} \|^2 } = \frac{\vec{\omega}(\vec{r} \cdot \vec{r}) - \vec{r}\cancel{(\vec{r}\cdot \vec{\omega})}}{\| \vec{r} \|^2} = \vec{\omega} $$

_{Use the triple vector product identity $a \times (b \times c) = b(a \cdot c) - c (a\cdot b)$ where $\cdot$ is the dot prodct, and $\times$ the cross product.}

The fact that $(\vec{r} \cdot \vec{\omega}) = 0$ comes from the fact that only the perpendicular components of $\vec{\omega}$ to $\vec{r}$ contribute to linear velocity $v$.

The above is true regardless of the coordinate system used to describe it.

• Example - $\vec{r} = \pmatrix{x \\y\\0}$, $\vec{\omega} = \pmatrix{0\\0\\ \Omega}$ and so $\vec{v} = \vec{\omega}\times\vec{r}= \pmatrix{-y \Omega \\ x \Omega \\ 0}$

$$ \vec{\omega} = \frac{ \pmatrix{x \\y\\0} \times \pmatrix{-y \Omega \\ x \Omega \\ 0}}{\| \pmatrix{x \\y\\0} \|^2} = \frac{ \pmatrix{0\\0\\ \Omega (x^2+y^2)}}{x^2+y^2} = \pmatrix{0\\0\\ \Omega} \; \checkmark$$

PS - Wikipedia isn't always correct. Always check derivations yourself.

Answer 2

You should write it in the cylindrical coordinate(not spherical, because of rotation always around an axis).