Final answer:
The initial angular speed of the dentist drill is 0 rad/s and the final angular speed is 800π rad/s. The angular acceleration during spin-up is 800π/2.1 rad/s². The drill rotates a total of 8000 revolutions in 20 seconds.
Step-by-step explanation:
(a) To find the initial and final angular speeds of the dentist drill, we need to convert the rpm values to radians per second (rad/s).
The initial angular speed of the drill can be calculated by multiplying the initial rpm value by 2π/60, since 1 revolution = 2π radians and 1 minute = 60 seconds. So, the initial angular speed is:
Initial angular speed = (0 rpm) * (2π/60 rad/s per rpm) = 0 rad/s.
The final angular speed can be calculated in the same way, using the value of 12,000 rpm:
Final angular speed = (12,000 rpm) * (2π/60 rad/s per rpm) = 800π rad/s.
(b) The angular acceleration during spin-up can be found using the formulas:
Angular acceleration = (change in angular speed) / (change in time).
During spin-up, the change in angular speed is the final angular speed minus the initial angular speed, and the change in time is the spin-up time of 2.1 seconds. So, the angular acceleration is:
Angular acceleration = (800π rad/s - 0 rad/s) / 2.1 s = (800π / 2.1) rad/s².
(c) The total number of revolutions the drill rotates in the total time can be calculated by multiplying the final angular speed by the total time:
Total number of revolutions = (800π rad/s) * (20 s) / (2π rad per revolution) = 8000 revolutions.