# Deriving the formula: v=wr

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1. Oct 15, 2014

### 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?

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2. Oct 15, 2014

### 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.

3. Oct 15, 2014

### SteamKing

Staff Emeritus
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.

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.

4. Oct 15, 2014

### 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?

5. Oct 15, 2014

### A.T.

That looks more like tangential velocity to me. Radial velocity is along the radius:

6. Oct 15, 2014

### A.T.

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

7. Oct 15, 2014

### Chemist@

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

8. Oct 15, 2014

### A.T.

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

9. Oct 15, 2014

### 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.

10. Oct 15, 2014

### DrGreg

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$$\textbf{v} = \boldsymbol{\omega} \times \textbf{r}; \; \boldsymbol{\omega} = \frac{\textbf{r} \times \textbf{v}}{r^2}$$

11. Oct 16, 2014

### Chemist@

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

12. Oct 16, 2014

### A.T.

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.

13. Oct 16, 2014

### DrGreg

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