Answer
$2.1 \ m/s^2$
Work Step by Step
We know:
$\omega^2 = \frac{g}{L}$
Thus, we find:
$90^2 = \frac{g}{L}$
$1=\frac{g}{8100L}$
And
$91^2 = \frac{\sqrt{g^2+a^2}}{L}$
$ 1 = \frac{g-a}{8281L}$
Setting these equal gives:
$\frac{g}{8100L}= \frac{\sqrt{g^2+a^2}}{8281L}$
$a\approx2.1 \ m/s^2$