Answer
The period of one revolution would be 9.0 Earth days.
Work Step by Step
Let $r$ be the average distance from the Earth to the sun. Then,
$r = 1.5 \times 10^{11} ~m$
Then,
$\frac{v^2}{r} = g$
$v= \sqrt{gr}$
We can use the velocity to find the period of one rotation.
$T = \frac{2\pi r}{\sqrt{gr}} = 2\pi \sqrt{\frac{r}{g}}$
$T = 2 \pi \sqrt{\frac{1.5 \times 10^{11} ~m}{9.80 ~m/s^2}}$
$T = 7.77 \times 10^5 ~s$
We can convert this time to Earth days.
$T = (7.77 \times 10^5 ~s)(\frac{1 ~h}{3600 ~s})(\frac{1 ~day}{24 ~h}) = 9.0 ~days$
The period of one revolution would be 9.0 Earth days.
