An object is being swung around in a circle, attached by a string. If the string breaks, does the angular velocity instantly drop to zero? If an object attached to a string is being swung around in a circle, then when the string breaks, the object will continue in a straight line at a constant velocity, per Newton's First Law. If I understand correctly, the angular velocity drops to zero the instant the string breaks, correct?
 A: Its not that a particle has angular velocity only when it moves along a circular path. A particle moving along a straight line also has angular velocity about any point not lying on the line containing the path of the particle.
The angular velocity of a particle moving along a line, however, changes instant to instant so it is instantaneous angular velocity. Mathematically, if $\vec{v}$ is the linear velocity of the particle, its instantaneous angular velocity $\vec{\omega}$ about a point is given by
$$\vec{v} = \vec{\omega} \times \vec{r}$$
where $\vec{r}$ is the position vector of the particle with respect to the point.
So, no, angular velocity does not drop to zero when the string breaks. You will keep on getting non-zero angular velocity about the center of the circle for ever assuming that particle continues its straight line motion for ever.
A: No. The angular velocity would only be zero if the object's new trajectory was along a radius of the circle. But it is not - after the string breaks, the object moves along a tangent line. Its angle with respect to the centre of the circle is still changing, so it has a non-zero angular velocity.
Another way to see this is to realise that the object's angular momentum must be conserved.
