Answer
It will take $2\pi$ seconds go around the whole circle.
Work Step by Step
The angular speed is $\omega = 1~rad/s$
We can find the time to go around a whole circle:
$\theta = \omega ~t$
$t = \frac{\theta}{\omega}$
$t = \frac{2\pi~rad}{1~rad/s}$
$t = 2\pi~s$
It will take $2\pi$ seconds go around the whole circle.
