How to derive linear velocity from position and angular vel.ocity

[Zak]

Hello!

I'm trying to derive the linear velocity vector from the position vector and the angular momentum vector. I've seen on the internet that V = W x R (V,W and R are all vectors and x is the cross product) but I cannot for the life of me derive it! I've tried doing it by writing out the cross product component wise and rearranging etc but I keep getting the wrong thing.

any help?

 

[Orodruin]

I'm trying to derive the linear velocity vector from the position vector and the angular momentum vector.

In general, you cannot. Knowing the angular momentum and the position is not sufficient to determine the velocity.

I've seen on the internet that V = W x R (V,W and R are all vectors and x is the cross product)

This tells us nothing unless you give actual reference to where you have seen this or at least what the vectors are supposed to represent.

 

[TomHart]

Did you mean v = rω, where ω is angular velocity?

 

[jbriggs444]

Did you mean v = rω, where ω is angular velocity?

The difficulty with this is that it prescribes the tangential velocity only. Any radial component is not accounted for.

 

[TomHart]

The difficulty with this is that it prescribes the tangential velocity only. Any radial component is not accounted for.

Isn't this question in regard to some object with a circular perimeter rotating on an axis at the center of that circle - e.g. a disk? I'm sorry, but I am missing your point. Could you please explain jbriggs444.

Or is he trying to find a velocity vector from a fixed point - say, on the road - to a point on the wheel at a certain distance from the center of the wheel - i.e. at a given radius? I guess I'm just wondering out loud at this point.

 

[jbriggs444]

Isn't this question in regard to some object with a circular perimeter rotating on an axis at the center of that circle - e.g. a disk? I'm sorry, but I am missing your point. Could you please explain jbriggs444.

I see no mention of a disc or of a circular trajectory in the original post. If the trajectory is arbitrary, the radial velocity can be non-zero.

 

[Orodruin]

If you have a rigid body with a point $A$ moving with velocity $\vec v_A$, you can always express the velocity of another point $B$ in the rigid body as $\vec v_B = \vec{v}_A + \vec \omega \times \vec r_{BA}$, where $\vec r_{BA}$ is the separation vector between $A$ and $B$. How to arrive at this result is described in Elementary Construction of the Angular Velocity, but it requires a rigid body, which is not among the prerequisites in the OP that asks for a general formula. Obviously, if $A$ is a fixed point, then $\vec v_A = 0$ and $\vec v_B = \vec \omega \times \vec r_{BA}$.