How Is the Formula v=wr Derived in Vector Terms?

[Chemist@]

How to derive the formula:
v=wr
where v is the tangential velocity, w is the rotational velocity, and r i the radius vector?

From the attached image, it can be concluded that (each quantity is a vector): w=r x v, also v=w x r, and r= v x w. All three vectors are perpendicular to each other, therefore the intensity of each vector can be calculated by vector multiplication. Then (each quantity is a vector modulus):
w=rv, v=wr, r=vw, this system of equations is true if w=v=r which mustn't be true. I need an explanation. What did I wrong to arrive at this incorrect equality?

 

[ShayanJ]

Look at the image below:

By definition of a radian(unit of angle), we can write s= r \theta(where \theta is in radians). Now, assuming a fixed radius, differentiation of the equation w.r.t. time will give you the desired result.

 

[SteamKing]

How to derive the formula:
v=wr
where v is the tangential velocity, w is the rotational velocity, and r i the radius vector?

From the attached image, it can be concluded that (each quantity is a vector): w=r x v, also v=w x r, and r= v x w. All three vectors are perpendicular to each other, therefore the intensity of each vector can be calculated by vector multiplication. Then (each quantity is a vector modulus):
w=rv, v=wr, r=vw, this system of equations is true if w=v=r which mustn't be true. I need an explanation. What did I wrong to arrive at this incorrect equality?

The angular velocity ω is usually a scalar, rather than a vector quantity. The magnitude of the radial velocity is given as v = ω r, where r is the magnitude of the radius vector.

For a derivation of the radial velocity vector, see this article:

http://en.wikipedia.org/wiki/Circular_motion

and note the difference between ω and the vector Ω.

You should also be aware that the cross product does not commute, so that all of these statements may not be valid simultaneously:

w=r x v, v=w x r, r= v x w.

 

[Chemist@]

Okay, but I want to derive it the way I previously posted, but it brings me nowhere and I want to make myself clear what was wrong.
The cross product does not commute, but how does that explain anything?

 

[A.T.]

The magnitude of the radial velocity is given as v = ω r,

That looks more like tangential velocity to me. Radial velocity is along the radius:
http://en.wikipedia.org/wiki/Radial_velocity

 

[A.T.]

From the attached image, it can be concluded that (each quantity is a vector): w=r x v, also v=w x r, and r= v x w.

How did you conclude that from the image? It's only true if they are all unit vectors

 

[Chemist@]

I think that you have the answer, but please explain it. How is it true only then?

 

[A.T.]

I think that you have the answer, but please explain it.

You have to explain how you concluded all that from your picture.

 

[Chemist@]

I wrote it. w=r x v, also v=w x r, and r= v x w from the picture. For example r x v gives the vector w. I got all by calculating the vector product:
<r x v>=<w>=rv*sin(pi/2)=rv. The same way I got that v=wr, r=vw.

 

[DrGreg]

I wrote it. w=r x v, also v=w x r, and r= v x w from the picture. For example r x v gives the vector w. I got all by calculating the vector product:
<r x v>=<w>=rv*sin(pi/2)=rv. The same way I got that v=wr, r=vw.

Your picture only shows three vectors perpendicular to each other. It doesn't tell you what the lengths of the vectors are. The correct equations should be<br /> \textbf{v} = \boldsymbol{\omega} \times \textbf{r}; \;<br /> \boldsymbol{\omega} = \frac{\textbf{r} \times \textbf{v}}{r^2}<br />

 

[Chemist@]

How did you get that w=rxv/r^2?

 

[A.T.]

w=r x v, also v=w x r, and r= v x w from the picture.

None of this follows from the picture. Just because 3 vectors are perpendicular, doesn't mean they necessarily represent the operands and result of a vector product.

 

[DrGreg]

How did you get that w=rxv/r^2?

See, for example, angular velocity, or any textbook on the subject.