Final answer:
The change in the bicycle tire's angular velocity (\(\Delta \omega\)) is 4.20 rad/s, and the average angular acceleration (\(\alpha_{av}\)) is 1.75 rad/s^2, calculated over a period of 2.40 seconds as the tire changes its spin direction.
Step-by-step explanation:
The student is asking about the change in the tire's angular velocity (\(\Delta \omega\)) and the average angular acceleration (\(\alpha_{av}\)) for a bicycle tire that is first spinning clockwise at 2.10 rad/s and then stopped and spun in the opposite direction (counterclockwise) at the same magnitude of angular velocity over a period of 2.40 s.
The change in angular velocity can be calculated by taking the difference between the final and initial angular velocities. Since the tire changes direction, we must consider the final angular velocity as a positive value (as it's counterclockwise) and the initial angular velocity as a negative value (since it's clockwise, opposite to our positive direction convention).
\(\Delta \omega = \omega_{final} - \omega_{initial} = 2.10 \, rad/s - (-2.10 \, rad/s) = 4.20 \, rad/s\)
To calculate the average angular acceleration, we use the formula \(\alpha_{av} = \frac{\Delta \omega}{\Delta t}\).
\(\alpha_{av} = \frac{\Delta \omega}{\Delta t} = \frac{4.20 \, rad/s}{2.40 \, s} = 1.75 \, rad/s^2\)