# Relation between r ,ω and θ for rotation around fixed axis.

1. Nov 10, 2013

### AakashPandita

relation between r ,ω and θ for rotation around fixed axis.

$$\frac{d\textbf {r}}{dt} = \textbf {ω}$$

$$\frac{dθ}{dt} = ω$$

$$\lvert\frac{d\textbf {r}}{dt}\rvert = \frac{dθ}{dt}$$

bold means vector. Is this right?

Last edited: Nov 10, 2013
2. Nov 11, 2013

### tiny-tim

Hi AakashPandita!

(type \left and and \right before two ordinary ||s, and they resize to fit! )

$$\frac{dθ}{dt} = ω = |\textbf{ω}|$$
ω (the angular velocity vector) is along the axis of rotation (ie, out of the page)
$$\left|\frac{d\hat{\textbf {r}}}{dt}\right| = r\frac{dθ}{dt} = rω$$
and
$$\frac{d\hat{\textbf {r}}}{dt} = rω\hat{\textbf{θ}}$$
where $\hat{\textbf {r}}$ and $\hat{\textbf {θ}}$ are the unit vectors in the radial and tangential directions

3. Nov 11, 2013

### vanhees71

Let the rotation be around the $z$ axis of a Cartesian coordinate system. Then
$$\vec{r}(t)=r(t) \begin{pmatrix} \cos[\alpha(t)] \\ \sin[\alpha(t)] \\ 0 \end{pmatrix}.$$
This gives
$$\vec{v}(t)=\dot{r}(t) = \dot{r}(t) \begin{pmatrix} \cos[\alpha(t)] \\ \sin[\alpha(t)] \\ 0 \end{pmatrix} + r(t) \dot{\alpha}(t) \begin{pmatrix} -\sin[\alpha(t)] \\ \cos[\alpha(t)] \\ 0 \end{pmatrix} = \dot{r} \hat{r} + r \omega \hat{\theta}.$$
If the mass is fixed to a circle, then you have $\dot{r}=0$ and you can write
$$\vec{v}=\vec{\omega} \times \vec{r} \quad \text{with} \quad \vec{\omega}=\omega \vec{e}_z=\dot{\theta} \vec{e}_z.$$

4. Nov 11, 2013

### ZapperZ

Staff Emeritus
You should already know that something is not quite right here, because the dimensions are all wrong! Always check that first, because that is your first "line-of-defense"!

Zz.

5. Nov 11, 2013

### AakashPandita

how did you get this?

6. Nov 11, 2013

### AakashPandita

i dont understand matrices

7. Nov 11, 2013

### D H

Staff Emeritus
vanhees71 wasn't using matrices, AakashPandita. He was using vectors.

8. Nov 11, 2013

### AakashPandita

i dont understand those brackets

9. Nov 11, 2013

### vanhees71

How are you supposed to solve this question without vectors? I guess you just use a different notation than I use. In your original posting you used abstract vectors, but the equations are unfortunately not correct. So I thought, it's best to use components wrt. a Cartesian basis to explicitly calculate the derivatives.