Final answer:
The magnitude of the angular acceleration of the wheel is 5.45 rad/s² for a 4.50-kg wheel that is 34.5 cm in diameter rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s.
Step-by-step explanation:
To calculate the angular acceleration of the wheel, we can use the formula:
angular acceleration = (final angular velocity - initial angular velocity) / time
Given:
- Initial angular velocity (ωi) = 22.0 rad/s
- Final angular velocity (ωf) = 13.5 rad/s
- Time (t) = Unknown
Substituting the values into the formula, we have:
angular acceleration = (13.5 rad/s - 22.0 rad/s) / t
Simplifying the equation, we get:
angular acceleration = -8.5 rad/s / t
However, we are given that the wheel rotates through an angle of 13.8 rad during this time. We can use the formula:
angular displacement = angular velocity * time + 0.5 * angular acceleration * time^2
Substituting the known values, we have:
13.8 rad = (22.0 rad/s + 13.5 rad/s) * t + 0.5 * angular acceleration * t^2
Simplifying the equation, we get:
angular acceleration = 5.45 rad/s²