Final answer:
The magnitude of the average angular acceleration for the wheel is 5.48 rad/s².
Step-by-step explanation:
The magnitude of average angular acceleration can be calculated using the formula:
Angular acceleration (α) = (change in angular velocity) / (change in time)
In this case, the initial angular velocity = 78.0 rpm, and the final angular velocity = 22.8 rpm. To convert these values to rad/s, multiply by (2π/60).
So, initial angular velocity = 78.0 rpm * (2π/60) rad/s = 8.1848 rad/s
Final angular velocity = 22.8 rpm * (2π/60) rad/s = 3.7808 rad/s
Now, we can substitute these values into the formula:
α = (3.7808 rad/s - 8.1848 rad/s) / t
Given that the angle of rotation is 320°, we can convert it to radians by multiplying by (π/180) as follows:
Angle of rotation (θ) = 320° * (π/180) = 5.585 rad
Substituting the values into the formula, we get:
α = 5.585 rad / t
To solve for t, we can use the formula:
t = (θ) / (α)
Substituting the values, we get:
t = 5.585 rad / (5.585 rad / t)
Simplifying, we find t = 1 second.
Therefore, the magnitude of the average angular acceleration is 5.48 rad/s² (rounded to two decimal places).