After fixing a flat tire on a bicycle you give the wheel a spin.

(a) If its initial angular speed was 5.17 rad/s and it rotated 10.6 revolutions before coming to rest, what was its average angular acceleration?
rad/s2
(b) For what length of time did the wheel rotate?
s

Sally is wierd

To solve this problem, we can use the equations of rotational kinematics. The key equations we will use are:

1. Angular displacement (θ) = (final angular position) - (initial angular position)
2. Angular velocity (ω) = (final angular speed) - (initial angular speed)
3. Angular acceleration (α) = (angular velocity change) / (time duration)

(a) To find the average angular acceleration, we need to calculate the change in angular velocity and the time duration it took for the wheel to rotate.

1. Convert the 10.6 revolutions to radians:
- 1 revolution = 2π radians
- Therefore, 10.6 revolutions = 10.6 * 2π radians

2. Find the change in angular velocity:
- The initial angular speed is given as 5.17 rad/s, and the final angular speed is 0 rad/s since the wheel came to rest.
- Therefore, the change in angular velocity = 0 rad/s - 5.17 rad/s

3. Find the average angular acceleration:
- The average angular acceleration is given by the equation: α = Δω / Δt
- Since the wheel came to rest, the final angular speed is 0 rad/s.
- Therefore, α = (-5.17 rad/s) / Δt
- Rearrange the equation to solve for Δt: Δt = (-5.17 rad/s) / α

(b) To find the length of time the wheel rotated, we need to calculate the time duration using the average angular acceleration:

1. Substitute the calculated average angular acceleration from part (a) into the equation obtained: Δt = (-5.17 rad/s) / α

2. Calculate the time duration (Δt) using the appropriate units.

Following these steps will give us the answers to both parts (a) and (b) of the question.