# Question about angular velocity vector

If a tangential force is applied to a spinning disk, then the tangential velocity of a particle will also change. This means that angular speed should also increase. However, force is perpendicular to the angular velocity vector, so shouldn't angular speed value remain the same?

• Only a radial force will apply no torque and thus not change angular speed. Oct 25, 2020 at 16:47

The following might be useful. You are correct that the force is perpendicular to the angular velocity vector, but in a rotating disk, the change in angular velocity is due to the torque ($$\vec{r} \times \vec {F}$$) which that force produces. In this case, the torque would be parallel to the direction of the angular velocity vector so would change the value of the angular velocity.

I hope this helps.

It is not force $$\vec F$$ that changes the angular velocity vector, it is torque $$\vec \tau$$. But force can create torque:

$$\vec \tau=\vec r \times \vec F$$

A cross-product like this creates a new vector which is not parallel to the input vectors. So the created torque will point in another direction than the force. The force being perpendicular to the angular velocity vector is thus no problem, since that force creates a torque which is not necessarily perpendicular to it.

When that torque is created, it causes angular acceleration $$\vec \alpha$$, which causes changes to the angular velocity vector, via the rotational version of Newton's 2nd law ($$I$$ is the moment-of-inerta):

$$\sum \vec \tau=I\vec \alpha$$