B

To find the maximum angular velocity of the stone, we will analyze the forces acting on the stone when it is at the top of the circular path.
Step 1: Calculate the weight of the stone.
The weight (W) of the stone is given by:
W = m \cdot g
W = 0.5 \, kg \cdot 10 \, m/s^{2} = 5 \, N.
Step 2: At the top of the circular path, the tension (T) in the thread and the weight of the stone provide the centripetal force required for circular motion. The maximum tension the thread can support is 30 N, so:
T + W = \frac{m \cdot v^{2}}{r}
Maximum Tension, T = 30 N.
Therefore, we have:
30 \, N + 5 \, N = \frac{0.5 \, kg \cdot v^{2}}{2 \, m}.
Step 3: Simplifying the equation:
35 = \frac{0.5 \, v^{2}}{2}
35 = \frac{0.5}{2} v^{2}
35 = 0.25 v^{2}
Step 4: Solving for v²:
v^{2} = \frac{35}{0.25} = 140.
v = \sqrt{140} = \sqrt{4 \cdot 35} = 2 \sqrt{35}.
Step 5: Now we convert the linear velocity (v) to angular velocity (ω) using the relation: \( v = r \cdot \omega \).
Thus, \( \omega = \frac{v}{r} = \frac{2 \sqrt{35}}{2} = \sqrt{35}.
Step 6: Finally, calculate the maximum angular velocity: \( \sqrt{35} \approx 5.92 \, rad/s \).
Therefore, the maximum angular velocity of the stone is approximately 5.92 rad/s.