Where the Angular Velocity Vector Coming From?

[Arman777]

I know that If we have a rigid body rotating clokwise direction,the angular velocity vector should be in the into the screen.But also I know that

$w=dθ/dt$ so..Whats the equation that tells us that $\vec w$ is into the screen ?

Is it coming from some vector product ? Or we know that its $\vec R x\vec v=\vec w$

 

[stevendaryl]

I know that If we have a rigid body rotating clokwise direction,the angular velocity vector should be in the into the screen.But also I know that

$w=dθ/dt$ so..Whats the equation that tells us that $\vec w$ is into the screen ?
I thought $\vec R x \vec {Δθ}=\vec w$ ($\vec R$ is in the upward direction).But sounds wrong...

The direction of $\vec{\omega}$ is the axis of rotation. If $\vec{v}$ is the velocity of a point on the body, and $\vec{r}$ is the vector to the center of rotation, then you can compute $\omega$ via:

$\vec{\omega} = \frac{1}{r^2} (\vec{v} \times \vec{r})$

So it's a vector that is perpendicular to the plane containing both $\vec{v}$ and $\vec{r}$.

 

[Arman777]

The direction of $\vec{\omega}$ is the axis of rotation. If $\vec{v}$ is the velocity of a point on the body, and $\vec{r}$ is the vector to the center of rotation, then you can compute $\omega$ via:

$\vec{\omega} = \frac{1}{r^2} (\vec{v} \times \vec{r})$

So it's a vector that is perpendicular to the plane containing both $\vec{v}$ and $\vec{r}$.

why there's $\frac {1} {r^2}$ ?

I see...Is there any equation that contains $\vec w$, $dθ$ and $dt$ ?

 

[stevendaryl]

why there's $\frac {1} {r^2}$ ?

I see...Is there any equation that contains $vec w$, $dθ$ and $dt$ ?

Well, the $\frac{1}{r^2}$ is intuitively correct from the following reasoning: If you have circular motion, then $v = \frac{2 \pi r}{T}$ where $T$ is the time for a complete circle. So the magnitude of $\vec{v} \times \vec{r}$ is $\frac{2 \pi r^2}{T}$. Dividing by $r^2$ gives $\frac{2\pi}{T}$ which is the same as $\frac{d\theta}{dt}$.

So the magnitude of $\vec{\omega}$ is just $\frac{d\theta}{dt}$

You can't get the direction of $\vec{\omega}$ from $d\theta$ and $dt$. You need to know what plane the rotation is in.

 

[Arman777]

I understand now.We know that $\vec w x \vec R= \vec v$
so $(\vec w x \vec R) x \vec R= \vec v x \vec R$

Which its $\vec w (\vec R.\vec R)-\vec R (\vec w.\vec R)=\vec v x \vec R$
$\vec w R^2-0=\vec v x \vec R$
$\vec w=\frac {1} {r^2} (\vec v x \vec R)$

I just should know that $\vec w$ is founding by right hand rule and $\vec w x \vec R= \vec v$

You can't get the direction of ⃗ωω→\vec{\omega} from dθdθd\theta and dtdtdt. You need to know what plane the rotation is in.

I see ok

 

[Cutter Ketch]

$w=dθ/dt$ so..Whats the equation that tells us that $\vec w$ is into the screen ?

This 1D equation is a simplification of the 3D description. In making this simplification θ is defined about an axis with a direction, so the direction is already implicitly in there.