Final answer:
The angular speed of the pulley is approximately 1.71 rad/s, calculated using the relationship that angular speed equals the total angle of rotation divided by the time.
Step-by-step explanation:
To calculate the angular speed of the pulley, we use the formula ω = Θ / t, where ω is the angular speed, Θ is the angle of rotation in radians, and t is the time in seconds. In one rotation, the angle Θ is 2π radians. Since the pulley rotates 35 times in 128 seconds, the total angle Θ is 35 × 2π.
The angular speed ω = (35 × 2π radians) / (128 s). Hence, ω = (70π rad) / (128 s) = 1.71 rad/s (Approximately).