$\omega=\frac{v_t}{r}$ using this equation only prove that constant angular velocity is possible

Question

Is it possible to have constant angular velocity since according to $\omega=\frac{v_t} {r}$ angular velocity is directly propotional to tangential velocity and since tangential velocity is a vector and is always changing directions in uniform rotational motion, therefore we will get different values of angular velocity depending on the direction of tangential velocity? So we can't get constant angular velocity?

Answer 1

First of all, in circular motion

$$\vert\vec\omega\vert=\frac{\vert\vec{v_t}\vert} {r}$$

Here the magnitude of angular velocity vector is directly proportional to magnitude of tangential velocity vector.

Answer 2

Angular speed is defined as $\omega=\frac{d \theta}{dt}$

Since $d\theta=\frac{ds}{r}$ (see picture) we have $\omega=\frac{ds}{dt}\frac1r=\frac{v}{r}$

In uniform circular motion, the r does not change, nor v. Thus $\omega$ must be constant.

Now this is a scalar equation. We can define vector equation as: $$\vec \omega \times \vec r=\vec v$$ Or $$\vec r \times \vec v = \vec \omega$$ You have said, that in circular motion, the velocity vector changes, which is true. But so does postion vector $\vec r$. Now how do we know for sure angular velocity is constant at all times? $$\frac {d}{dt}[\vec \omega]=\frac{d}{dt}[\vec r \times \vec v]=\frac{d\vec r}{dt}\times\vec v+\vec r\times \frac{d\vec v}{dt}=0\\\biggl[\frac{d\vec v}{dt}\parallel\vec r,\frac{d\vec r}{dt}=\vec v\biggr]$$ It implies that omega is indeed constant.

Answer 3

$\omega=\frac{v_t}{r}$ is not a vector equation for $\omega$

the vector equation is $\vec v_t = \vec r$x$\vec \omega$ The direction of $\omega$ is perpendicular to $\vec{r}$ and $\vec{v_t} $. that is perpendicular to the plane containing vectors $\vec r$ and $\vec{v_t} $. so, as long as $r$ and $v_t$ continue to remain in the same plane, $\omega$ would not change direction. Thus, for a constant $r$ and $v$, $\omega$ is a constant.

A large percentage of rotational motion is planar motion. That is : the particle rotate in one plane and $\vec r$ and $\vec v$ remain in the same plane. Thus $\vec\omega$ remains in one direction (perpendicular to the plane).