Answer
$a = \frac{F}{m}$
Work Step by Step
We can find the angular acceleration about the center of mass due to the force $F$:
$\tau = I \alpha$
$\frac{L}{6} F = (\frac{1}{12}mL^2) \alpha$
$\alpha = \frac{2F}{mL}$
We can find the linear acceleration at $O$ as a result of the angular acceleration:
$a = \alpha~\frac{L}{2}$
$a = (\frac{2F}{mL}) (\frac{L}{2})$
$a = \frac{F}{m}$
