After fixing a flat tire on a bicycle you give the wheel a spin. Its initial angular speed was 7.35 rad/s and it rotated 15.0 revolutions before coming to rest. What was the average angular acceleration? For what length of time did the wheel rotate?
a) 0 = (7.35)^2 + 2 a ( 94.2)
a = - 0.287
b) 0 = 7.35 + -0.287t
t = 25.61 s
To find the average angular acceleration, we can use the formula:
Average angular acceleration (α) = change in angular velocity (Δω) / change in time (Δt)
The initial angular velocity (ω_i) is given as 7.35 rad/s, and the final angular velocity (ω_f) is 0 rad/s since the wheel comes to rest. Also, the number of revolutions (n) is given as 15.0.
To find the change in angular velocity (Δω), we need to calculate the difference between the initial and final angular velocities:
Δω = ω_f - ω_i
= 0 rad/s - 7.35 rad/s
= -7.35 rad/s
Now, we need to find the change in time (Δt) for the wheel to rotate 15.0 revolutions. To do this, we need to convert the number of revolutions to radians:
1 revolution = 2π radians
So, 15.0 revolutions = 15.0 * 2π radians.
Now, we can calculate the time it took for the wheel to rotate by using the formula:
Δt = (change in angle) / (average angular velocity)
where the average angular velocity is the average angular velocity for one revolution. It can be calculated as the average of the initial and final angular velocities:
average angular velocity = (ω_i + ω_f) / 2
Substituting the given values:
average angular velocity = (7.35 rad/s + 0 rad/s) / 2
= 3.675 rad/s
Now, substituting this value and the change in angle into the formula:
Δt = (15.0 * 2π radians) / (3.675 rad/s)
Calculating this expression will give us the change in time.
Once we have the change in angular velocity (Δω) and the change in time (Δt), we can calculate the average angular acceleration (α) using the formula mentioned at the beginning:
Average angular acceleration (α) = Δω / Δt
Let's solve the equations step by step to find the average angular acceleration and the length of time the wheel rotated.