#### Answer

$I =5.2\times10^{-5} \ kgm^2$

#### Work Step by Step

We know that there are two forces on the mass. Thus, we find:
$T=mg-ma$
We also can solve for acceleration:
$\Delta y = \frac{1}{2}at^2$
Since the change in height is one, this becomes:
$a=\frac{2}{t^2}$
Finally, we find:
$\tau = I\alpha=bT$
$ I\alpha=bT$
$\frac{Ia}{r}=b(mg-ma)$
$\frac{Ia}{b}=b(mg-ma)$
$\frac{2Ia}{b^2t^2}=(mg-ma)$
$\frac{2I}{b^2t^2}=(mg-m\frac{2}{t^2})$
Thus, to get a line of slope I, we can graph $\frac{2}{b^2t^2}$ versus $(mg-m\frac{2}{t^2})$, or we could graph $\frac{2}{t^2}$ versus $b^2(mg-m\frac{2}{t^2})$. Doing this, we verify the moments of inertia. (When doing this, do not forget to subtract $I_0$ from the value of the slope.) Finally, we find their moment of inertia together to be $I =5.2\times10^{-5} \ kgm^2$ by graphing the best fit trendline and by taking the slope to be the moment of inertia.