Calculating the velocity of a particle on a spinning disc

Question

The particle is moving towards the center as the disc is spinning, the position of the particle is described by the following expression:

$r = r(t) cos(ωt)i + r(t) sin(ωt)j$

How do I calculate the particles velocity?(not angular velocity)

Answer 1

It is easier to work in polar coordinates, where the position is $(r(t),\omega t)$ because $\theta=\omega t$. Then use the equation for velocity in polar coordinates $$\vec v=\dot r \hat r+r\dot \theta \hat \theta$$ Now if you want the result in Cartesian coordinates you can transform back.

Answer 2

you have the displacement vector $$\underline{r}=\left(\begin{matrix}r\cos\omega t\\r\sin\omega t\end{matrix}\right)$$

so the velocity vector is $$\frac{d\underline{r}}{dt}=\left(\begin{matrix}\dot{r}\cos\omega t-r\omega\sin\omega t\\ \dot{r}\sin\omega t+r\omega\cos\omega t\end{matrix}\right)$$