Final answer:
To calculate the final angular rotation (velocity) of the gear, we use the formula for angular velocity, ending up with a final angular velocity of 116 rad/s after 20 seconds of acceleration.
Step-by-step explanation:
The student is asking for the final angular rotation of a gear after it has been accelerating for a certain time. To find the final angular velocity of the gear (rotation), we use the formula ω = ω₀ + αt, where ω is the final angular velocity, ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time. Given that the initial angular velocity (ω₀) is 12 rad/s, the angular acceleration (α) is 5.2 rad/s², and the time (t) is 20 s, we can calculate the final angular velocity as follows: ω = 12 rad/s + (5.2 rad/s² × 20 s) = 12 rad/s + 104 rad/s = 116 rad/s.