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.