The system is a torsion pendulum. The period of oscillation is $T = 2\pi\sqrt{I/\kappa}$, where $I$ is the moment of inertia and $\kappa$ is the torsional constant of the wire.

The torsional constant $\kappa$ is found from the static equilibrium condition where the applied torque $\tau$ equals the restoring torque:

$$\kappa = \frac{\tau}{\theta}$$

The moment of inertia for a solid sphere about its center is:

$$I = \frac{2}{5}mR^2$$

Substituting these into the period formula:

$$T = 2\pi\sqrt{\frac{I}{\kappa}} = 2\pi\sqrt{\frac{(2/5)mR^2}{\tau/\theta}} = 2\pi\sqrt{\frac{2mR^2\theta}{5\tau}}$$

Substitute the given values: $m = 95$ kg, $R = 0.15$ m, $\tau = 0.20$ N·m, and $\theta = 0.85$ rad.

$$T = 2\pi\sqrt{\frac{2(95 \text{ kg})(0.15 \text{ m})^2(0.85 \text{ rad})}{5(0.20 \text{ N} \cdot \text{m})}}$$