Answer
The mass of the sphere is 23 kg.
Work Step by Step
We first find the angular acceleration $\alpha$:
$\theta = \frac{1}{2}\alpha t^2$
$\alpha = \frac{2\theta}{t^2} = \frac{(2)(2\pi~rad/rev)(160~rev))}{(15.0~s)^2}$
$\alpha = 8.936~rad/s^2$
Then, we find the moment of inertia $I$:
$\tau = I~\alpha$
$I = \frac{\tau}{\alpha} = \frac{10.8~m\cdot N}{8.936~rad/s^2} = 1.21~kg\cdot m^2$
Then, we find the mass of the sphere:
$I = \frac{2}{5}MR^2$
$M = \frac{5I}{2R^2} = \frac{(5)(1.21~kg\cdot m^2)}{(2)(0.36~m)^2}$
$M = 23~kg$
The mass of the sphere is 23 kg.