Relative angular velocity between two bodies

Question

I was studying circular motion and came across a formula for the relative angular velocity between two bodies. I didn't find any kind of demonstration. Is it simply a determined concept, just like the vector product or the angular velocity vector?

Answer 1

take this equation:

$$\vec{v}=\vec{\omega}\times \vec{r}$$

or

$$\vec{v}\times \vec{r}=\left(\vec{\omega}\times \vec{r}\right)\times \vec{r}\tag 1$$

if you write all vectors in components of the S' coordinate system you get:

2D case

$$\vec{r}=r_x\,\hat{\vec{e}}_{x'}$$ $$\vec{v}=v_x\,\hat{\vec{e}}_{x'}+v_y\,\hat{\vec{e}}_{y'}$$ $$\vec{\omega}=\omega_z\,\hat{\vec{e}}_{z'}$$

in (1)

$$(v_x\,\hat{\vec{e}}_{x'}+v_y\,\hat{\vec{e}}_{y'})\times r_x\,\hat{\vec{e}}_{x'}=\left(\omega_z\,\hat{\vec{e}}_{z'}\times r_x\,\hat{\vec{e}}_{x'}\right)\times r_x\,\hat{\vec{e}}_{x'}\tag 2$$

thus:

$$-v_y\,r_x\,\hat{\vec{e}}_{z'}=-\omega_z\,r^2_x\,\hat{\vec{e}}_{z'}$$

or:

$$\omega_z=\frac{v_y}{r_x}\quad, \omega_{AB}= \frac{(V_{AB})_y}{r_{AB}}$$

this is your equation